Publications
Monographs

L. Starke, K. Tabelow, Th. Niendorf, A. Pohlmann, eds., Denoising for improved parametric MRI of the kidney: Protocol for nonlocal means filtering, 2216 of Methods in Molecular Biology, Humana, New York, NY., 2021, pp. 565576, (Chapter Published), DOI 10.1007/9781071609781_34 .
Articles in Refereed Journals

S.A. Alves, J. Polzehl, N.M. Brisson, A. Bender, A.N. Agres, P. Damm, G.N. Duda, Ground reaction forces and external hip joint moments predict in vivo hip contact forces during gait, Journal of Biomechanics, 135 (2022), pp. 111037/1111037/6, DOI 10.1016/j.jbiomech.2022.111037 .

M. Coghi, W. Dreyer, P. Gajewski, C. Guhlke, P. Friz, M. Maurelli, A McKeanVlasov SDE and particle system with interaction from reflecting boundaries, SIAM Journal on Mathematical Analysis, 54 (2022), pp. 22512294, DOI 10.1137/21M1409421 .

N. Puchkin, V. Spokoiny, Structureadaptive manifold estimation, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., 23 (2022), pp. 162.
Abstract
We consider a problem of manifold estimation from noisy observations. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned weights become so corrupted that the averaged estimate shows very poor performance. We suggest a novel computationally efficient structureadaptive procedure, which simultaneously reconstructs a smooth manifold and estimates projections of the point cloud onto this manifold. The proposed approach iteratively refines the weights on each step, using the structural information obtained at previous steps. After several iterations, we obtain nearly öracle" weights, so that the final estimates are nearly efficient even in the presence of relatively large noise. In our theoretical study we establish tight lower and upper bounds proving asymptotic optimality of the method for manifold estimation under the Hausdorff loss. Our finite sample study confirms a very reasonable performance of the procedure in comparison with the other methods of manifold estimation. 
P. Dvurechensky, K. Safin, S. Shtern, M. Staudigl, Generalized selfconcordant analysis of FrankWolfe algorithms, Mathematical Programming. A Publication of the Mathematical Programming Society, (2022), published online on 29.01.2022, DOI 10.1007/s10107022017711 .
Abstract
Projectionfree optimization via different variants of the FrankWolfe method has become one of the cornerstones of large scale optimization for machine learning and computational statistics. Numerous applications within these fields involve the minimization of functions with selfconcordance like properties. Such generalized selfconcordant functions do not necessarily feature a Lipschitz continuous gradient, nor are they strongly convex, making them a challenging class of functions for firstorder methods. Indeed, in a number of applications, such as inverse covariance estimation or distanceweighted discrimination problems in binary classification, the loss is given by a generalized selfconcordant function having potentially unbounded curvature. For such problems projectionfree minimization methods have no theoretical convergence guarantee. This paper closes this apparent gap in the literature by developing provably convergent Frank?Wolfe algorithms with standard O(1/k) convergence rate guarantees. Based on these new insights, we show how these sublinearly convergent methods can be accelerated to yield linearly convergent projectionfree methods, by either relying on the availability of a local liner minimization oracle, or a suitable modification of the awaystep FrankWolfe method. 
P. Friz, T. Klose, Precise Laplace asymptotics for singular stochastic PDEs: The case of 2D gPAM, Journal of Functional Analysis, 283 (2022), pp. 109446/1109446/86, DOI 10.1016/j.jfa.2022.109446 .

O. Butkovsky, K. Dareiotis, M. Geréncser, Approximation of SDEs: A stochastic sewing approach, Probability Theory and Related Fields, 181 (2021), pp. 9751034, DOI 10.1007/s00440021010802 .

E. Vorontsova, A. Gasnikov, P. Dvurechensky, A. Ivanova, D. Pasechnyuk, Numerical methods for the resource allocation problem in a computer network problems, Computational Mathematics and Mathematical Physics, 61 (2021), pp. 297328, DOI 10.1134/S0965542521020135 .

C. Bellingeri, A. Djurdjevac, P. Friz, N. Tapia, Transport and continuity equations with (very) rough noise, SN Partial Differential Equations and Applications, 2 (2021), pp. 226, DOI 10.1007/s4298502100101y .
Abstract
Existence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established. 
C. Bellingeri, P. Friz, M. Gerencsér, Singular paths spaces and applications, Stochastic Analysis and Applications, (2021), published online on 29.10.2021, DOI 10.1080/07362994.2021.1988641 .

K. EbrahimiFard, F. Patras, N. Tapia, L. Zambotti, Wick polynomials in noncommutative probability: A grouptheoretical approach, Canadian Journal of Mathematics. Journal Canadien de Mathematiques, (2021), published online on 25.08.2021, DOI 10.4153/S0008414X21000407 .
Abstract
Wick polynomials and Wick products are studied in the context of noncommutative probability theory. It is shown that free, boolean and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf algebraic approach to cumulants and Wick products in classical probability theory. 
A. Gasnikov, D. Dvinskikh, P. Dvurechensky, D. Kamzolov, V. Matyukhin, D. Pasechnyuk, N. Tupitsa, A. Chernov, Accelerated metaalgorithm for convex optimization, Computational Mathematics and Mathematical Physics, 61 (2021), pp. 1728, DOI 10.1134/S096554252101005X .

A. Kroshnin, V. Spokoiny, A. Suvorikova, Statistical inference for BuresWasserstein barycenters, The Annals of Applied Probability, 31 (2021), pp. 12641298, DOI 10.1214/20AAP1618 .

L.Ch. Lin, Y. Chen, G. Pan, V. Spokoiny, Efficient and positive semidefinite preaveraging realized covariance estimator, Statistica Sinica, 31 (2021), pp. 14411462, DOI 10.5705/ss.202017.0489 .

M. Redmann, Ch. Bayer, P. Goyal, Lowdimensional approximations of highdimensional asset price models, SIAM Journal on Financial Mathematics, ISSN 1945497X, 12 (2021), pp. 128, DOI 10.1137/20M1325666 .
Abstract
We consider highdimensional asset price models that are reduced in their dimension in order to reduce the complexity of the problem or the effect of the curse of dimensionality in the context of option pricing. We apply model order reduction (MOR) to obtain a reduced system. MOR has been previously studied for asymptotically stable controlled stochastic systems with zero initial conditions. However, stochastic differential equations modeling price processes are uncontrolled, have nonzero initial states and are often unstable. Therefore, we extend MOR schemes and combine ideas of techniques known for deterministic systems. This leads to a method providing a good pathwise approximation. After explaining the reduction procedure, the error of the approximation is analyzed and the performance of the algorithm is shown conducting several numerical experiments. Within the numerics section, the benefit of the algorithm in the context of option pricing is pointed out. 
I. Shibaev, P. Dvurechensky, A. Gasnikov, Zerothorder methods for noisy Höldergradient functions, Optimization Letters, published online in April 2021, DOI 10.1007/s1159002101742z .

F. Stonyakin, A. Tyurin, A. Gasnikov, P. Dvurechensky, A. Agafonov, D. Dvinskikh, M. Alkousa, D. Pasechnyuk, S. Artamonov, V. Piskunova, Inexact model: A framework for optimization and variational inequalities, Optimization Methods & Software, published online in July 2021, DOI 10.1080/10556788.2021.1924714 .
Abstract
In this paper we propose a general algorithmic framework for firstorder methods in optimization in a broad sense, including minimization problems, saddlepoint problems and variational inequalities. This framework allows to obtain many known methods as a special case, the list including accelerated gradient method, composite optimization methods, levelset methods, proximal methods. The idea of the framework is based on constructing an inexact model of the main problem component, i.e. objective function in optimization or operator in variational inequalities. Besides reproducing known results, our framework allows to construct new methods, which we illustrate by constructing a universal method for variational inequalities with composite structure. This method works for smooth and nonsmooth problems with optimal complexity without a priori knowledge of the problem smoothness. We also generalize our framework for strongly convex objectives and strongly monotone variational inequalities. 
N. Tupitsa, P. Dvurechensky, A. Gasnikov, S. Guminov, Alternating minimization methods for strongly convex optimization, Journal of Inverse and IllPosed Problems, 29 (2021), pp. 721739, DOI 10.1515/jiip20200074 .
Abstract
We consider alternating minimization procedures for convex optimization problems with variable divided in many block, each block being amenable for minimization with respect to its variable with freezed other variables blocks. In the case of two blocks, we prove a linear convergence rate for alternating minimization procedure under PolyakŁojasiewicz condition, which can be seen as a relaxation of the strong convexity assumption. Under strong convexity assumption in manyblocks setting we provide an accelerated alternating minimization procedure with linear rate depending on the square root of the condition number as opposed to condition number for the nonaccelerated method. 
D. Dvinskikh, A. Gasnikov, Decentralized and parallel primal and dual accelerated methods for stochastic convex programming problems, Journal of Inverse and IllPosed Problems, 29 (2021), pp. 385405, DOI 10.1515/jiip20200068 .
Abstract
We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles, the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node takes place only up to a logarithmic factor and the notion of smoothness. By using minibatching technique, we show that the proposed methods with stochastic oracle can be additionally parallelized at each node. The considered algorithms can be applied to many data science problems and inverse problems. 
D. Dvinskikh, Stochastic approximation versus sample average approximation for Wasserstein barycenters, Optimization Methods & Software, (2021), published online on 09.09.2021, DOI 10.1080/10556788.2021.1965600 .
Abstract
In the machine learning and optimization community, there are two main approaches for the convex risk minimization problem, namely the Stochastic Approximation (SA) and the Sample Average Approximation (SAA). In terms of the oracle complexity (required number of stochastic gradient evaluations), both approaches are considered equivalent on average (up to a logarithmic factor). The total complexity depends on a specific problem, however, starting from the work [A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro, Robust stochastic approximation approach to stochastic programming, SIAM. J. Opt. 19 (2009), pp. 1574?1609] it was generally accepted that the SA is better than the SAA. We show that for the Wasserstein barycenter problem, this superiority can be inverted. We provide a detailed comparison by stating the complexity bounds for the SA and SAA implementations calculating barycenters defined with respect to optimal transport distances and entropyregularized optimal transport distances. As a byproduct, we also construct confidence intervals for the barycenter defined with respect to entropyregularized optimal transport distances in the ?2norm. The preliminary results are derived for a general convex optimization problem given by the expectation to have other applications besides the Wasserstein barycenter problem. 
CH. Bayer, D. Belomestny, P. Hager, P. Pigato, J.G.M. Schoenmakers, Randomized optimal stopping algorithms and their convergence analysis, SIAM Journal on Financial Mathematics, ISSN 1945497X, 12 (2021), pp. 12011225, DOI 10.1137/20M1373876 .
Abstract
In this paper we study randomized optimal stopping problems and consider corresponding forward and backward Monte Carlo based optimization algorithms. In particular we prove the convergence of the proposed algorithms and derive the corresponding convergence rates. 
CH. Bayer, F. Harang, P. Pigato, Logmodulated rough stochastic volatility models, SIAM Journal on Financial Mathematics, ISSN 1945497X, 12 (2021), pp. 12571284, DOI 10.1137/20M135902X .
Abstract
We propose a new class of rough stochastic volatility models obtained by modulating the powerlaw kernel defining the fractional Brownian motion (fBm) by a logarithmic term, such that the kernel retains square integrability even in the limit case of vanishing Hurst index H. The soobtained logmodulated fractional Brownian motion (logfBm) is a continuous Gaussian process even for H = 0. As a consequence, the resulting superrough stochastic volatility models can be analysed over the whole range of Hurst indices between 0 and 1/2, including H = 0, without the need of further normalization. We obtain the usual power law explosion of the skew as maturity T goes to 0, modulated by a logarithmic term, so no flattening of the skew occurs as H goes to 0. 
P. Dvurechensky, M. Staudigl, S. Shtern, Firstorder methods for convex optimization, EURO Journal on Computational Optimization, 9 (2021), pp. 100015/1100015/27, DOI 10.1016/j.ejco.2021.100015 .
Abstract
Firstorder methods for solving convex optimization problems have been at the forefront of mathematical optimization in the last 20 years. The rapid development of this important class of algorithms is motivated by the success stories reported in various applications, including most importantly machine learning, signal processing, imaging and control theory. Firstorder methods have the potential to provide low accuracy solutions at low computational complexity which makes them an attractive set of tools in largescale optimization problems. In this survey we cover a number of key developments in gradientbased optimization methods. This includes nonEuclidean extensions of the classical proximal gradient method, and its accelerated versions. Additionally we survey recent developments within the class of projectionfree methods, and proximal versions of primaldual schemes. We give complete proofs for various key results, and highlight the unifying aspects of several optimization algorithms. 
P. Friz, P. Gassiat, P. Pigato, Precise asymptotics: Robust stochastic volatility models, The Annals of Applied Probability, 31 (2021), pp. 896940, DOI 10.1214/20AAP1608 .
Abstract
We present a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to shorttime and small noise formulae for option prices. Our main tool is the theory of regularity structures, which we use in the form of Bayer et al. (Math. Finance30 (2020) 782832) In essence, we implement a Laplace method on the space of models (in the sense of Hairer), which generalizes classical works of Azencott and Ben Arous on path space and then Aida, InahamaKawabi on rough path space. When applied to rough volatility models, for example, in the setting of Bayer, Friz and Gatheral (Quant. Finance16 (2016) 887904) and FordeZhang (SIAM J. Financial Math.8 (2017) 114145), one obtains precise asymptotics for European options which refine known large deviation asymptotics. 
P. Friz, P. Gassiat, P. Pigato, Shortdated smile under rough volatility: Asymptotics and numerics, Quantitative Finance, (2021), published online on 07.12.2021, DOI 10.1080/14697688.2021.1999486 .
Abstract
In Friz et al. [Precise asymptotics for robust stochastic volatility models. Ann. Appl. Probab, 2021, 31(2), 896?940], we introduce a new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regard to shorttime and smallnoise formulae for option prices, using the framework [Bayer et al., A regularity structure for rough volatility. Math. Finance, 2020, 30(3), 782?832]. We investigate here the fine structure of this expansion in large deviations and moderate deviations regimes, together with consequences for implied volatility. We discuss computational aspects relevant for the practical application of these formulas. We specialize such expansions to prototypical rough volatility examples and discuss numerical evidence. 
P. Friz, H. Tran, Y. Yuan , Regularity of SLE in (t,k) and refined GRR estimates, Probability Theory and Related Fields, 180 (2021), pp. 71112, DOI 10.1007/s00440021010580 .
Contributions to Collected Editions

A. Agafonov, P. Dvurechensky, G. Scutari, A. Gasnikov, D. Kamzolov, A. Lukashevich, A. Daneshmand, An accelerated secondorder method for distributed stochastic optimization, in: 60th IEEE Conference on Decision and Control (CDC), IEEE, 2021, pp. 24072413, DOI 10.1109/CDC45484.2021.9683400 .

A. Daneshmand, G. Scutari, P. Dvurechensky, A. Gasnikov, Newton method over networks is fast up to the statistical precision, in: Proceedings of the 38th International Conference on Machine Learning, 139 of Proceedings of Machine Learning Research, 2021, pp. 23982409.

E. Gladin, A. Sadiev, A. Gasnikov, P. Dvurechensky, A. Beznosikov, M. Alkousa, Solving smooth minmin and minmax problems by mixed oracle algorithms, in: Mathematical Optimization Theory and Operations Research: Recent Trends, A. Strekalovsky, Y. Kochetov, T. Gruzdeva, A. Orlov , eds., 1476 of Communications in Computer and Information Science book series (CCIS), Springer International Publishing, Basel, 2021, pp. 1940.
Abstract
In this paper, we consider two types of problems that have some similarity in their structure, namely, minmin problems and minmax saddlepoint problems. Our approach is based on considering the outer minimization problem as a minimization problem with an inexact oracle. This inexact oracle is calculated via an inexact solution of the inner problem, which is either minimization or maximization problem. Our main assumption is that the available oracle is mixed: it is only possible to evaluate the gradient w.r.t. the outer block of variables which corresponds to the outer minimization problem, whereas for the inner problem, only zerothorder oracle is available. To solve the inner problem, we use the accelerated gradientfree method with zerothorder oracle. To solve the outer problem, we use either an inexact variant of Vaidya's cuttingplane method or a variant of the accelerated gradient method. As a result, we propose a framework that leads to nonasymptotic complexity bounds for both minmin and minmax problems. Moreover, we estimate separately the number of first and zerothorder oracle calls, which are sufficient to reach any desired accuracy. 
S. Guminov, P. Dvurechensky, N. Tupitsa, A. Gasnikov, On a combination of alternating minimization and Nesterov's momentum, in: Proceedings of the 38th International Conference on Machine Learning, 139 of Proceedings of Machine Learning Research, 2021, pp. 38863898, DOI 10.20347/WIAS.PREPRINT.2695 .
Abstract
Alternating minimization (AM) optimization algorithms have been known for a long time and are of importance in machine learning problems, among which we are mostly motivated by approximating optimal transport distances. AM algorithms assume that the decision variable is divided into several blocks and minimization in each block can be done explicitly or cheaply with high accuracy. The ubiquitous Sinkhorn's algorithm can be seen as an alternating minimization algorithm for the dual to the entropyregularized optimal transport problem. We introduce an accelerated alternating minimization method with a $1/k^2$ convergence rate, where $k$ is the iteration counter. This improves over known bound $1/k$ for general AM methods and for the Sinkhorn's algorithm. Moreover, our algorithm converges faster than gradienttype methods in practice as it is free of the choice of the stepsize and is adaptive to the local smoothness of the problem. We show that the proposed method is primaldual, meaning that if we apply it to a dual problem, we can reconstruct the solution of the primal problem with the same convergence rate. We apply our method to the entropy regularized optimal transport problem and show experimentally, that it outperforms Sinkhorn's algorithm. 
D. Pasechnyuk, P. Dvurechensky, S. Omelchenko, A. Gasnikov, Stochastic optimization for dynamic pricing, in: Advances in Optimization and Applications, N.N. Olenev, Y.G. Evtushenko, M. Jaćimović, M. Khachay, eds., 1514 of Communications in Computer and Information Science, Springer Nature Switzerland AG, Cham, 2021, pp. 8294, DOI 10.1007/9783030927110 .

A. Rogozin, M. Bochko, P. Dvurechensky, A. Gasnikov, V. Lukoshkin, An accelerated method for decentralized distributed stochastic optimization over timevarying graphs, in: 2021 IEEE 60th Annual Conference on Decision and Control (CDC), IEEE, 2021, pp. 33673373, DOI 10.1109/CDC45484.2021.9683400 .

A. Sadiev , A. Beznosikov, P. Dvurechensky, A. Gasnikov, Zerothorder algorithms for smooth saddlepoint problems, in: Mathematical Optimization Theory and Operations Research: Recent Trends, A. Strekalovsky, Y. Kochetov, T. Gruzdeva, A. Orlov , eds., 1476 of Communications in Computer and Information Science book series (CCIS), Springer International Publishing, Basel, 2021, pp. 7185.
Abstract
Saddlepoint problems have recently gained an increased attention from the machine learning community, mainly due to applications in training Generative Adversarial Networks using stochastic gradients. At the same time, in some applications only a zerothorder oracle is available. In this paper, we propose several algorithms to solve stochastic smooth (strongly) convexconcave saddlepoint problems using zerothorder oracles, and estimate their convergence rate and its dependence on the dimension n of the variable. In particular, our analysis shows that in the case when the feasible set is a direct product of two simplices, our convergence rate for the stochastic term is only by a factor worse than for the firstorder methods. Finally, we demonstrate the practical performance of our zerothorder methods on practical problems. 
K. Safin, P. Dvurechensky, A. Gasnikov, Adaptive gradientfree method for stochastic optimization, in: Advances in Optimization and Applications, N.N. Olenev, Y.G. Evtushenko, M. Jaćimović, M. Khachay, eds., 1514 of Communications in Computer and Information Science, Springer Nature Switzerland AG, Cham, 2021, pp. 95108, DOI 10.1007/9783030927110_7 .

D. Dvinskikh, D. Tiapkin, Improved complexity bounds in Wasserstein barycenter problem, in: 24th International Conference on Artificial Intelligence and Statistics (AISTATS), A. Banerjee, K. Fukumizu, eds., 130 of Proceedings of Machine Learning Research, Microtome Publishing, Brookline, MA, USA, 2021, pp. 17381746.
Preprints, Reports, Technical Reports

F. Galarce, K. Tabelow, J. Polzehl, Ch.P. Papanikas, V. Vavourakis, L. Lilaj, I. Sack, A. Caiazzo, Displacement and pressure reconstruction from magnetic resonance elastography images: Application to an in silico brain model, Preprint no. 2933, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2933 .
Abstract, PDF (9978 kByte)
This paper investigates a data assimilation approach for noninvasive quantification of intracranial pressure from partial displacement data, acquired through magnetic resonance elastography. Data assimilation is based on a parametrizedbackground data weak methodology, in which the state of the physical system tissue displacements and pressure fields is reconstructed from partially available data assuming an underlying poroelastic biomechanics model. For this purpose, a physicsinformed manifold is built by sampling the space of parameters describing the tissue model close to their physiological ranges, to simulate the corresponding poroelastic problem, and compute a reduced basis. Displacements and pressure reconstruction is sought in a reduced space after solving a minimization problem that encompasses both the structure of the reducedorder model and the available measurements. The proposed pipeline is validated using synthetic data obtained after simulating the poroelastic mechanics on a physiological brain. The numerical experiments demonstrate that the framework can exhibit accurate joint reconstructions of both displacement and pressure fields. The methodology can be formulated for an arbitrary resolution of available displacement data from pertinent images. It can also inherently handle uncertainty on the physical parameters of the mechanical model by enlarging the physicsinformed manifold accordingly. Moreover, the framework can be used to characterize, in silico, biomarkers for pathological conditions, by appropriately training the reducedorder model. A first application for the estimation of ventricular pressure as an indicator of abnormal intracranial pressure is shown in this contribution. 
M.G. Varzaneh, S. Riedel, A. Schmeding, N. Tapia, The geometry of controlled rough paths, Preprint no. 2926, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2926 .
Abstract, PDF (472 kByte)
We prove that the spaces of controlled (branched) rough paths of arbitrary order form a continuous field of Banach spaces. This structure has many similarities to an (infinitedimensional) vector bundle and allows to define a topology on the total space, the collection of all controlled path spaces, which turns out to be Polish in the geometric case. The construction is intrinsic and based on a new approximation result for controlled rough paths. This framework turns wellknown maps such as the rough integration map and the ItôLyons map into continuous (structure preserving) mappings. Moreover, it is compatible with previous constructions of interest in the stability theory for rough integration. 
CH. Bayer, D. Belomestny, O. Butkovsky, J.G.M. Schoenmakers, RKHS regularization of singular local stochastic volatility McKeanVlasov models, Preprint no. 2921, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2921 .
Abstract, PDF (504 kByte)
Motivated by the challenges related to the calibration of financial models, we consider the problem of solving numerically a singular McKeanVlasov equation, which represents a singular local stochastic volatility model. Whilst such models are quite popular among practitioners, unfortunately, its wellposedness has not been fully understood yet and, in general, is possibly not guaranteed at all. We develop a novel regularization approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularized model is wellposed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models. Our results are also applicable to more general McKeanVlasov equations. 
CH. Bayer, Ch. Ben Hammouda, R.F. Tempone, Numerical smoothing with hierarchical adaptive sparse grids and quasiMonte Carlo methods for efficient option pricing, Preprint no. 2917, WIAS, Berlin, 2022.
Abstract, PDF (654 kByte)
When approximating the expectation of a functional of a stochastic process, the efficiency and performance of deterministic quadrature methods, such as sparse grid quadrature and quasiMonte Carlo (QMC) methods, may critically depend on the regularity of the integrand. To overcome this issue and reveal the available regularity, we consider cases in which analytic smoothing cannot be performed, and introduce a novel numerical smoothing approach by combining a root finding algorithm with onedimensional integration with respect to a single wellselected variable. We prove that under appropriate conditions, the resulting function of the remaining variables is a highly smooth function, potentially affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e., Brownian bridge and Richardson extrapolation on the weak error). This approach facilitates the effective treatment of high dimensionality. Our study is motivated by option pricing problems, and our focus is on dynamics where the discretization of the asset price is necessary. Based on our analysis and numerical experiments, we show the advantages of combining numerical smoothing with the ASGQ and QMC methods over ASGQ and QMC methods without smoothing and the Monte Carlo approach. 
CH. Bayer, E. Hall, R.F. Tempone, Weak error rates for option pricing under linear rough volatility, Preprint no. 2916, WIAS, Berlin, 2022, DOI 10.20347/WIAS.PREPRINT.2916 .
Abstract, PDF (1685 kByte)
In quantitative finance, modeling the volatility structure of underlying assets is vital to pricing options. Rough stochastic volatility models, such as the rough Bergomi model [Bayer, Friz, Gatheral, Quantitative Finance 16(6), 887904, 2016], seek to fit observed market data based on the observation that the logrealized variance behaves like a fractional Brownian motion with small Hurst parameter, H < 1/2, over reasonable timescales. Both time series of asset prices and optionderived price data indicate that H often takes values close to 0.1 or less, i.e., rougher than Brownian motion. This change improves the fit to both option prices and time series of underlying asset prices while maintaining parsimoniousness. However, the nonMarkovian nature of the driving fractional Brownian motion in rough volatility models poses severe challenges for theoretical and numerical analyses and for computational practice. While the explicit Euler method is known to converge to the solution of the rough Bergomi and similar models, its strong rate of convergence is only H. We prove rate H + 1/2 for the weak convergence of the Euler method for the rough SteinStein model, which treats the volatility as a linear function of the driving fractional Brownian motion, and, surprisingly, we prove rate one for the case of quadratic payoff functions. Indeed, the problem of weak convergence for rough volatility models is very subtle; we provide examples demonstrating the rate of convergence for payoff functions that are well approximated by secondorder polynomials, as weighted by the law of the fractional Brownian motion, may be hard to distinguish from rate one empirically. Our proof uses TalayTubaro expansions and an affine Markovian representation of the underlying and is further supported by numerical experiments. These convergence results provide a first step toward deriving weak rates for the rough Bergomi model, which treats the volatility as a nonlinear function of the driving fractional Brownian motion. 
D. Belomestny, Ch. Bender, J.G.M. Schoenmakers, Solving optimal stopping problems via randomization and empirical dual optimization, Preprint no. 2884, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2884 .
Abstract, PDF (706 kByte)
In this paper we consider optimal stopping problems in their dual form. In this way we reformulate the optimal stopping problem as a problem of stochastic average approximation (SAA) which can be solved via linear programming. By randomizing the initial value of the underlying process, we enforce solutions with zero variance while preserving the linear programming structure of the problem. A careful analysis of the randomized SAA algorithm shows that it enjoys favorable properties such as faster convergence rates and reduced complexity as compared to the non randomized procedure. We illustrate the performance of our algorithm on several benchmark examples. 
CH. Bayer, S. Breneis, Markovian approximations of stochastic Volterra equations with the fractional kernel, Preprint no. 2868, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2868 .
Abstract, PDF (525 kByte)
We consider rough stochastic volatility models where the variance process satisfies a stochastic Volterra equation with the fractional kernel, as in the rough Bergomi and the rough Heston model. In particular, the variance process is therefore not a Markov process or semimartingale, and has quite low Hölderregularity. In practice, simulating such rough processes thus often results in high computational cost. To remedy this, we study approximations of stochastic Volterra equations using an Ndimensional diffusion process defined as solution to a system of ordinary stochastic differential equation. If the coefficients of the stochastic Volterra equation are Lipschitz continuous, we show that these approximations converge strongly with superpolynomial rate in N. Finally, we apply this approximation to compute the implied volatility smile of a European call option under the rough Bergomi and the rough Heston model. 
A. Sadiev, A. Beznosikov, P. Dvurechensky, A. Gasnikov, Zerothorder algorithms for smooth saddlepoint problems, Preprint no. 2827, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2827 .
Abstract, PDF (564 kByte)
Saddlepoint problems have recently gained an increased attention from the machine learning community, mainly due to applications in training Generative Adversarial Networks using stochastic gradients. At the same time, in some applications only a zerothorder oracle is available. In this paper, we propose several algorithms to solve stochastic smooth (strongly) convexconcave saddle point problems using zerothorder oracles, and estimate their convergence rate and its dependence on the dimension n of the variable. In particular, our analysis shows that in the case when the feasible set is a direct product of two simplices, our convergence rate for the stochastic term is only by a log n factor worse than for the firstorder methods. We also consider a mixed setup and develop 1/2thorder methods which use zerothorder oracle for the minimization part and firstorder oracle for the maximization part. Finally, we demonstrate the practical performance of our zerothorder and 1/2thorder methods on practical problems. 
CH. Bayer, M. Eigel, L. Sallandt, P. Trunschke, Pricing highdimensional Bermudan options with hierarchical tensor formats, Preprint no. 2821, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2821 .
Abstract, PDF (321 kByte)
An efficient compression technique based on hierarchical tensors for popular option pricing methods is presented. It is shown that the “curse of dimensionality" can be alleviated for the computation of Bermudan option prices with the Monte Carlo leastsquares approach as well as the dual martingale method, both using highdimensional tensorized polynomial expansions. This discretization allows for a simple and computationally cheap evaluation of conditional expectations. Complexity estimates are provided as well as a description of the optimization procedures in the tensor train format. Numerical experiments illustrate the favourable accuracy of the proposed methods. The dynamical programming method yields results comparable to recent Neural Network based methods. 
P. Ostroukhov, R. Kamalov, P. Dvurechensky, A. Gasnikov, Tensor methods for strongly convex strongly concave saddle point problems and strongly monotone variational inequalities, Preprint no. 2820, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2820 .
Abstract, PDF (302 kByte)
In this paper we propose three tensor methods for stronglyconvexstronglyconcave saddle point problems (SPP). The first method is based on the assumption of higherorder smoothness (the derivative of the order higher than 2 is Lipschitzcontinuous) and achieves linear convergence rate. Under additional assumptions of first and second order smoothness of the objective we connect the first method with a locally superlinear converging algorithm in the literature and develop a second method with global convergence and local superlinear convergence. The third method is a modified version of the second method, but with the focus on making the gradient of the objective small. Since we treat SPP as a particular case of variational inequalities, we also propose two methods for strongly monotone variational inequalities with the same complexity as the described above. 
A. Agafonov, D. Kamzolov, P. Dvurechensky, A. Gasnikov, Inexact tensor methods and their application to stochastic convex optimization, Preprint no. 2818, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2818 .
Abstract, PDF (333 kByte)
We propose a general nonaccelerated tensor method under inexact information on higher order derivatives, analyze its convergence rate, and provide sufficient conditions for this method to have similar complexity as the exact tensor method. As a corollary, we propose the first stochastic tensor method for convex optimization and obtain sufficient minibatch sizes for each derivative. 
V. Matyukhin, S. Kabanikhin, M. Shishlenin, N. Novikov, A. Vasin, A. Gasnikov, Convex optimization with inexact gradients in Hilbert space and applications to elliptic inverse problems, Preprint no. 2815, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2815 .
Abstract, PDF (1346 kByte)
In this paper we propose the gradient descent type methods to solve convex optimization problems in Hilbert space. We apply it to solve illposed Cauchy problem for Poisson equation and make a comparative analysis with Landweber iteration and steepest descent method. The theoretical novelty of the paper consists in the developing of new stopping rule for accelerated gradient methods with inexact gradient (additive noise). Note that up to the moment of stopping the method “doesn't feel the noise”. But after this moment the noise start to accumulate and the quality of the solution becomes worse for further iterations. 
P. Friz, P. Hager, N. Tapia, Unified signature cumulants and generalized Magnus expansions, Preprint no. 2814, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2814 .
Abstract, PDF (388 kByte)
The signature of a path can be described as its full noncommutative exponential. Following T. Lyons we regard its expectation, the expected signature, as path space analogue of the classical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant. We establish a universal functional relation in a general semimartingale context. Our work exhibits the importance of Magnus expansions in the algorithmic problem of computing expected signature cumulants, and further offers a farreaching generalization of recent results on characteristic exponents dubbed diamond and cumulant expansions; with motivation ranging from financial mathematics to statistical physics. From an affine process perspective, the functional relation may be interpreted as infinitedimensional, noncommutative (“Hausdorff") variation of Riccati's equation. Many examples are given. 
N. Yudin, A. Gasnikov, Flexible modification of GaussNewton method and its stochastic extension, Preprint no. 2813, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2813 .
Abstract, PDF (8261 kByte)
This work presents a novel version of recently developed GaussNewton method for solving systems of nonlinear equations, based on upper bound of solution residual and quadratic regularization ideas. We obtained for such method global convergence bounds and under natural nondegeneracy assumptions we present local quadratic convergence results. We developed stochastic optimization algorithms for presented GaussNewton method and justified sublinear and linear convergence rates for these algorithms using weak growth condition (WGC) and PolyakLojasiewicz (PL) inequality. We show that GaussNewton method in stochastic setting can effectively find solution under WGC and PL condition matching convergence rate of the deterministic optimization method. The suggested method unifies most practically used GaussNewton method modifications and can easily interpolate between them providing flexible and convenient method easily implementable using standard techniques of convex optimization. 
A. Vasin, A. Gasnikov, V. Spokoiny, Stopping rules for accelerated gradient methods with additive noise in gradient, Preprint no. 2812, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2812 .
Abstract, PDF (1129 kByte)
In this article, we investigate an accelerated firstorder method, namely, the method of similar triangles, which is optimal in the class of convex (strongly convex) problems with a Lipschitz gradient. The paper considers a model of additive noise in a gradient and a Euclidean prox structure for not necessarily bounded sets. Convergence estimates are obtained in the case of strong convexity and its absence, and a stopping criterion is proposed for not strongly convex problems. 
D. Belomestny, J.G.M. Schoenmakers, From optimal martingales to randomized dual optimal stopping, Preprint no. 2810, WIAS, Berlin, 2021.
Abstract, PDF (571 kByte)
In this article we study and classify optimal martingales in the dual formulation of optimal stopping problems. In this respect we distinguish between weakly optimal and surely optimal martingales. It is shown that the family of weakly optimal and surely optimal martingales may be quite large. On the other hand it is shown that the Doobmartingale, that is, the martingale part of the Snell envelope, is in a certain sense the most robust surely optimal martingale under random perturbations. This new insight leads to a novel randomized dual martingale minimization algorithm that does`nt require nested simulation. As a main feature, in a possibly large family of optimal martingales the algorithm efficiently selects a martingale that is as close as possible to the Doob martingale. As a result, one obtains the dual upper bound for the optimal stopping problem with low variance. 
A. Neumann, N. Peitek, A. Brechmann, K. Tabelow, Th. Dickhaus, Utilizing anatomical information for signal detection in functional magnetic resonance imaging, Preprint no. 2806, WIAS, Berlin, 2021, DOI 10.20347/WIAS.PREPRINT.2806 .
Abstract, PDF (2995 kByte)
We are considering the statistical analysis of functional magnetic resonance imaging (fMRI) data. As demonstrated in previous work, grouping voxels into regions (of interest) and carrying out a multiple test for signal detection on the basis of these regions typically leads to a higher sensitivity when compared with voxelwise multiple testing approaches. In the case of a multisubject study, we propose to define the regions for each subject separately based on their individual brain anatomy, represented, e.g., by socalled Aparc labels. The aggregation of the subjectspecific evidence for the presence of signals in the different regions is then performed by means of a combination function for pvalues. We apply the proposed methodology to real fMRI data and demonstrate that our approach can perform comparably to a twostage approach for which two independent experiments are needed, one for defining the regions and one for actual signal detection.
Talks, Poster

S. Breneis, Markovian approximations for rough volatility models, Seminar der Stochastic Numerics Research Group, King Abdullah University of Science and Technology, Stochastic Numerics Research Group, Thuval, Saudi Arabia, January 26, 2022.

O. Butkovsky, Regularisation by noise for SDEs: State of the art & open problems, Regularization by noise: Theoretical foundations, numerical methods and applications driven by Levy noise, February 13  20, 2022, Mathematisches Forschungsinstitut Oberwolfach, February 16, 2022.

N. Tapia, The moving frame method for iteratedintegrals signatures: Orthogonal invariants (online talk), Arbeitsgruppenseminar Analysis (Online Event), Universität Potsdam, Institut für Mathematik, January 28, 2022.

N. Tapia, Transport and continuity equations with (very) rough noise, Regularization by noise: Theoretical foundations, numerical methods and applications driven by Levy noise, February 13  19, 2022, Mathematisches Forschungsinstitut Oberwolfach, February 18, 2022.

CH. Bayer, Efficient Markovian approximation to rough volatility models, Rough Volatility Meeting, Imperial College London, UK, March 16, 2022.

CH. Bayer, On the existence and longtime behavior of solutions to a degenerate parabolic system (online talk), Regularization by noise: Theoretical foundations, numerical methods and applications driven by Levy noise, February 13  20, 2022, Mathematisches Forschungsinstitut Oberwolfach, February 14, 2022.

CH. Bayer, Optimal stopping, machine learning, and signatures, Seminar der Stochastic Numerics Research Group, King Abdullah University of Science and Technology, Stochastic Numerics Research Group, Thuval, Saudi Arabia, January 31, 2022.

CH. Bayer, Simulating rough volatility models (online talk), MathFinance 2022 Conference (Online Event), March 21  22, 2022, March 22, 2022.

P. Friz, A theory of rough differential equations, Webinar on Stochastic Analysis (Online Event), Beijing Institute of Technology, School of Mathematics and Statistics, China, March 31, 2022.

P. Friz, Local vol under rough vol, Rough Volatility Workshop, March 15  16, 2022, Imperial College London, UK, March 15, 2022.

TH. Koprucki, K. Tabelow, HackMD (online talk), ECoffeeLecture (Online Event), WIAS Berlin, March 25, 2022.

K. Tabelow, Neural MRI, Tandem tutorial ``Mathematics of Imaging' ', Berlin Mathematics Research Center Math+, February 18, 2022.

S. Breneis, Markovian approximations of stochastic Volterra equations with the fractional kernel, 2021 Summer School of BerlinOxford IRTG Stochastic Analysis in Interaction (Hybrid Event), September 20  24, 2021, Technische Universität Berlin, Institut für Mathematik, September 24, 2021.

S. Breneis, Markovian approximations of stochastic volatility models (online talk), 16. DoktorandInnentreffen der Stochastik (Online Event), LudwigMaximiliansUniversität München, Fakultät für Mathematik, Informatik und Statistik, July 30, 2021.

S. Breneis, On variation functions and their moduli of continuity (online talk), Methods of nonlinear analysis in differential and integral equations (Online Event), May 15  16, 2021, Rzeszów University of Technology, Department of Nonlinear Analysis, Rzeszów, Poland, May 16, 2021.

O. Butkovsky, New coupling techniques for exponential ergodicity of SPDEs in the hypoelliptic and effectively elliptic settings (online talk), Applied and Computational Mathematics Research Seminar, Tulane University, School of Science and Engineering, New Orleans, USA, April 30, 2021.

O. Butkovsky, Regularization by noise for PDEs: A stochastic sewing approach (online talk), Theory of Probability and Its Applications: P. L. Chebyshev  200 (The 6th International Conference on Stochastic Methods) (Online Event), May 17  22, 2021, Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russian Federation, May 22, 2021.

O. Butkovsky, Regularization by noise for SPDEs and SDEs: a stochastic sewing approach (online talk), BernoulliIMS 10th World Congress in Probability and Statistics (Online Event), July 19  23, 2021, Seoul National University, Korea (Republic of), July 22, 2021.

O. Butkovsky, Regularization by noise via stochastic sewing with random controls, German Probability and Statistics Days (GPSD) 2021, September 27  October 1, 2021, DMVFachgruppe Stochastik e.V., Mannheim, September 27, 2021.

O. Butkovsky, Skew fractional Brownian motion (online talk), LSA Autumn Meeting 2021 (Online Event), September 20  24, 2021, HSE University, Laboratory of Stochastic Analysis and its Applications, Moscow, Russian Federation, September 22, 2021.

O. Butkovsky, Skew fractional Brownian motion: Going beyond the CatellierGubinelli setting (online talk), 14th OxfordBerlin Young Researchers Meeting on Applied Stochastic Analysis (Online Event), February 10  12, 2021, University of Oxford, Mathematical Institute, UK, February 11, 2021.

N. Tapia, Approximation of controlled rough paths (online talk), 14th OxfordBerlin Young Researchers Meeting on Applied Stochastic Analysis (Online Event), February 10  12, 2021, University of Oxford, Mathematical Institute, UK, February 10, 2021.

N. Tapia, Iterated sums for time series classification, Leibniz MMS Summer School 2021, August 23  26, 2021, Schloss Dagstuhl, LeibnizZentrum für Informatik, Wadern, August 23, 2021.

N. Tapia, Numerical schemes for rough partial differential equations (online talk), DNA Seminar (Online Event), Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway, March 12, 2021.

N. Tapia, Robustness of ResNets, Forschungsseminar, October 25  November 29, 2021, Universität des Saarlandes, Fakultät für Mathematik und Informatik, Saarbrücken, October 28, 2021.

N. Tapia, Series iteradas para clasificación de series de tiempo (online talk), Seminario Chileno de Probabilidades (Online Event), Universidad de Chile, Centro de Modelamiento Matemático, Santiago de Chile, Chile, March 31, 2021.

N. Tapia, Transport and continuity equations with (very) rough noise (online talk), BernoulliIMS 10th World Congress in Probability and Statistics (Online Event), July 19  23, 2021, Seoul National University, Department of Statistics, Korea (Republic of), July 20, 2021.

N. Tapia, Transport and continuity equations with (very) rough noise (online talk), Seminario de Probabilidad Hispanohablante (Online Event), Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales, Argentina, March 1, 2021.

N. Tapia, Unified signature cumulants and generalized magnus expansions, Analyse stochastique trajectorielle et applications Pathwise Stochastic Analysis and Applications (Online Event), March 8  12, 2021, C.I.R.M., France, March 11, 2021.

D. Dvinskikh, D. Tiapkin, Improved complexity bounds in Wasserstein barycenter problem (online presentation), The 24th International Conference on Artificial Intelligence and Statistics (Online Event), April 13  15, 2021.

D. Dvinskikh, A clever mean: Wasserstein barycenters, Education program: Modern methods of information theory, optimization and management, July 19  August 8, 2021, Sirius University of Science and Technology, Socchi, Russian Federation, July 19, 2021.

D. Dvinskikh, Decentralized algorithms for Wasserstein barycenters (online talk), Moscow Conference on Combinatorics and Applications (Online Event), May 31  June 4, 2021, Moscow Institute of Physics and Technology, School of Applied Mathematics and Computer Science, Moscow, Russian Federation, June 2, 2021.

A. Suvorikova, Optimal transport in machine learning (onlne talk), HDI Lab Seminar, HSE University, Faculty of Computer Science (Online Event), Moskau, Russian Federation, April 29, 2021.

A. Suvorikova, Statistics for nonstatisticians (online talk), ITaS Interdisciplinary Conference 2021 (Online Event), November 15  17, 2021, Russian Academy of Sciences, Institute for Information Transmission Problems, Moscow, Russian Federation, November 16, 2021.

A. Suvorikova, Survey of methods of kmeans clustering with optimal transport (online talk), Third HSEYandex autumn school on generative models (Hybrid Event), November 23  26, 2021, Yandex SDA Campus, Moscow, Russian Federation, November 26, 2021.

CH. Bayer, A pricing BSPDE for rough volatility (online talk), MATH4UQ Seminar (Online Event), RWTH Aachen University, Mathematics for Uncertainty Quantification, April 6, 2021.

CH. Bayer, Introduction to statistical learning theory, Leibniz MMS Summer School, August 22  27, 2021, Schloss Dagstuhl, LeibnizZentrum für Informatik, August 23, 2021.

CH. Bayer, Logmodulated rough stochastic volatility models (online talk), 2021 Happening Virtually: Financial Mathematics and Engineering (Online Event), June 1  4, 2021, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, USA, June 4, 2021.

CH. Bayer, Pricing of american options using highdimensional nonlinear networks, Next Generation Models of Financial Data, September 20  22, 2021, Technischen Universität München, Fakultät für Mathematik, Burghausen, September 21, 2021.

A. Caiazzo, F. Galarce Marín, J. Polzehl, I. Sack, K. Tabelow, Physics based assimilation of displacements data from magnetic resonance elastography, Kickoff Workshop: TES Mathematics of Imaging in RealWorld Challenges (Hybrid Event), Berlin, October 6  8, 2021.

P. Dvurechensky, A short introduction to optimization (online talk), ITaS Interdisciplinary Conference 2021 (Online Event), November 15  17, 2021, Russian Academy of Sciences, Institute for Information Transmission Problems, Moscow, Russian Federation, November 15, 2021.

P. Dvurechensky, Accelerated gradient methods and their applications to Wasserstein barycenter problem (online talk), TheXIII international scientific conference and young scientist school``Theory and Numerics of Inverse and Illposed Problems'' (Online Event), April 12  22, 2021, Mathematical Center in Akademgorodok, Novosibirsk, Russian Federation, April 14, 2021.

P. Dvurechensky, Decentralize and randomize: Faster algorithm for Wasserstein barycenters, 18th International Workshop on Continuous Optimization (Online Event), July 7  9, 2021, EUROPT continuous optimization working group of EURO (The Association of European Operational Research Societies), Toulouse, France, July 7, 2021.

P. Dvurechensky, Decentralize and randomize: Faster algorithm for Wasserstein barycenters (online talk), EUROPT 2021, 18th international workshop on continuous optimization (Online Event), July 7  9, 2021, École Nationale de l'Aviation Civile, Toulouse, France, July 7, 2021.

P. Dvurechensky, Distributed optimization algorithms for Wasserstein barycenters (online talk), INFORMS Anaheim 2021, October 24  27, 2021, Institute for Operations Research and the Management Sciences (Hybrid Event), USA, October 24, 2021.

P. Dvurechensky, Newton method over networks is fast up to the statistical precision (online talk), Thirtyeighth International Conference on Machine Learning (Online Event), July 18  24, 2021, Carnegie Mellon University, Pittsburgh, USA, July 20, 2021.

P. Dvurechensky, On a combination of alternating minimization and Nesterov's momentum (online talk), Thirtyeighth International Conference on Machine Learning (Online Event), July 18  24, 2021, Carnegie Mellon University, Pittsburgh, USA, July 20, 2021.

P. Dvurechensky, Primaldual accelerated gradient methods with alternating minimization (online talk), Conference Optimization without Borders, July 12  18, 2021, Sirius University of Science and Technology, Sochi, Russian Federation, July 15, 2021.

P. Dvurechensky, Wasserstein barycenters from the computational perspective (online talk), Moscow Conference on Combinatorics and Applications (Online Event), May 31  June 4, 2021, Moscow Institute of Physics and Technology, School of Applied Mathematics and Computer Science, Moscow, Russian Federation, June 2, 2021.

P. Friz, Liouville Brownian rough paths (online talk), Probability Seminar, Universität Wien, Fakultät für Mathematik, Austria, November 14, 2021.

P. Friz, Local volatility under rough volatility, QuantMinds International 2021, December 6  9, 2021, Barcelona, Spain, December 7, 2021.

P. Friz, Mathematik: Wie wahrscheinlich ist es? Von Risiko, zufälliger Mathematik und absolutem Chaos, Aktionstag ``Einstein macht Schule'', June 18, 2021, TU Berlin, June 18, 2021.

P. Friz, New perspectives on rough paths, signatures and signature cumulants (online talk), DataSig Seminar Series (Online Event), University of Oxford, Mathematical Institute, UK, May 6, 2021.

P. Friz, On rough SDEs (online talk), International Seminar on SDEs and Related Topics (Online Event), October 29, 2021.

P. Friz, Rough stochastic differential equations, Probability Seminar, Maxwell Institute for Mathematical Science, Edinburgh, UK, October 7, 2021.

P. Friz, Rough stochastic differential equations (online talk), Analyse stochastique trajectorielle et applications Pathwise Stochastic Analysis and Applications (Online Event), March 8  12, 2021, C.I.R.M., France, March 8, 2021.

P. Friz, Unified cumulants and Magnus expansions, Noncommutative algebra, probability and analysis in action (Hybrid Event), September 20  25, 2021, Universität Greifswald, Alfried Krupp Wissenschaftskolleg, September 21, 2021.

P. Friz, What can mathematics do for artificial intelligence? (online talk), Berlin Research 50 Workshop on Artificial Intelligence in Research (Online Event), December 13, 2021, Berlin Research 50, December 13, 2021.

V. Spokoiny, Adaptive graph clustering, Statistics, Artificial Intelligence, Machine Learning, Probability, Learning Theory Event  SAMPLE, October 26  30, 2021, National Research University Higher School of Economics, International Laboratory of Stochastic Algorithms and HighDimensional Inference, Gelendzhik, Russian Federation, October 27, 2021.

V. Spokoiny, Adaptive manifold recovery, Conference Optimization without Borders, July 12  18, 2021, Sirius University of Science and Technology, Sochi, Russian Federation, July 15, 2021.

V. Spokoiny, Bayesian inference in Bernoulli model with application to ranking from pairwise comparison (online talk), Data Seminar, Université Grenoble Alpes, Laboratoire Jean Kuntzmann, France, January 21, 2021.

V. Spokoiny, Inference for nonlinear inverse problems (online talk), BernoulliIMS 10th World Congress in Probability and Statistics (Online Event), July 19  23, 2021, Seoul National University, The Korean Statistical Society, Korea (Republic of), July 20, 2021.

V. Spokoiny, Random gradient free optimization: Bayesian view, Conference Optimization without Borders, July 12  18, 2021, Sirius University of Science and Technology, Sochi, Russian Federation, July 12, 2021.

K. Tabelow, MaRDI  The mathematical research data Initiative within the NFDI (online talk), SFB 1294 Colloquium (Online Event), Universität Potsdam, Institut für Mathematik, April 16, 2021.
External Preprints

M. Alkousa, A. Gasnikov, P. Dvurechensky, A. Sadiev, L. Razouk, An approach for nonconvex uniformly concave structured saddle point problem, Preprint no. arXiv:2202.06376, Cornell University, 2022, DOI 10.48550/arXiv.2202.06376 .
Abstract
Recently, saddle point problems have received much attention due to their powerful modeling capability for a lot of problems from diverse domains. Applications of these problems occur in many applied areas, such as robust optimization, distributed optimization, game theory, and many applications in machine learning such as empirical risk minimization and generative adversarial networks training. Therefore, many researchers have actively worked on developing numerical methods for solving saddle point problems in many different settings. This paper is devoted to developing a numerical method for solving saddle point problems in the nonconvex uniformlyconcave setting. We study a general class of saddle point problems with composite structure and Höldercontinuous higherorder derivatives. To solve the problem under consideration, we propose an approach in which we reduce the problem to a combination of two auxiliary optimization problems separately for each group of variables, outer minimization problem w.r.t. primal variables, and inner maximization problem w.r.t the dual variables. For solving the outer minimization problem, we use the Adaptive Gradient Method, which is applicable for nonconvex problems and also works with an inexact oracle that is generated by approximately solving the inner problem. For solving the inner maximization problem, we use the Restarted Unified Acceleration Framework, which is a framework that unifies the highorder acceleration methods for minimizing a convex function that has Höldercontinuous higherorder derivatives. Separate complexity bounds are provided for the number of calls to the firstorder oracles for the outer minimization problem and higherorder oracles for the inner maximization problem. Moreover, the complexity of the whole proposed approach is then estimated. 
C. Bellingeri, P.K. Friz, S. Paycha, R. Preiss, Smooth rough paths, their geometry and algebraic renormalization, Preprint no. arXiv:2111.15539, Cornell University, 2022, DOI 10.48550/arXiv.2111.15539 .

F. Delarue, W. Salkeld, Probabilistic rough paths II LionsTaylor expansions and random controlled rough paths, Preprint no. arXiv:2203.01185, Cornell University, 2021, DOI 10.48550/arXiv.2203.01185 .

A. Gasnikov, A. Novitskii, V. Novitskii, F. Abdukhakimov, D. Kamzolov, A. Beznosikov, M. Takáč, P. Dvurechensky, B. Gu, The power of firstorder smooth optimization for blackbox nonsmooth problems, Preprint no. arXiv:2201.12289, Cornell University, 2022, DOI 10.48550/arXiv.2201.12289 .
Abstract
Gradientfree/zerothorder methods for blackbox convex optimization have been extensively studied in the last decade with the main focus on oracle calls complexity. In this paper, besides the oracle complexity, we focus also on iteration complexity, and propose a generic approach that, based on optimal firstorder methods, allows to obtain in a blackbox fashion new zerothorder algorithms for nonsmooth convex optimization problems. Our approach not only leads to optimal oracle complexity, but also allows to obtain iteration complexity similar to firstorder methods, which, in turn, allows to exploit parallel computations to accelerate the convergence of our algorithms. We also elaborate on extensions for stochastic optimization problems, saddlepoint problems, and distributed optimization. 
S. Mohammadi, T. Streubel, L. Klock, A. Lutti, K. Pine, S. Weber, L. Edwards, P. Scheibe, G. Ziegler, J. Gallinat, S. Kuhn, M. Callaghan, N. Weiskopf, K. Tabelow, Error quantification in multiparameter mapping facilitates robust estimation and enhanced group level sensitivity, Preprint no. bioRxiv: 2022.01.11.475846, Cold Spring Harbor Laboratory, 2022, DOI 10.1101/2022.01.11.475846 .
Abstract
MultiParameter Mapping (MPM) is a comprehensive quantitative neuroimaging protocol that enables estimation of four physical parameters (longitudinal and effective transverse relaxation rates R1 and R2*, proton density PD, and magnetization transfer saturation MTsat) that are sensitive to microstructural tissue properties such as iron and myelin content. Their capability to reveal microstructural brain differences, however, is tightly bound to controlling random noise and artefacts (e.g. caused by head motion) in the signal. Here, we introduced a method to estimate the local error of PD, R1, and MTsat maps that captures both noise and artefacts on a routine basis without requiring additional data. To investigate the method's sensitivity to random noise, we calculated the modelbased signaltonoise ratio (mSNR) and showed in measurements and simulations that it correlated linearly with an experimental rawimagebased SNR map. We found that the mSNR varied with MPM protocols, magnetic field strength (3T vs. 7T) and MPM parameters: it halved from PD to R1 and decreased from PD to MT_sat by a factor of 34. Exploring the artefactsensitivity of the error maps, we generated robust MPM parameters using two successive acquisitions of each contrast and the acquisitionspecific errors to downweight erroneous regions. The resulting robust MPM parameters showed reduced variability at the group level as compared to their singlerepeat or averaged counterparts. The error and mSNR maps may better inform powercalculations by accounting for local data quality variations across measurements. Code to compute the mSNR maps and robustly combined MPM maps is available in the opensource hMRI toolbox. 
J.M. Oeschger, K. Tabelow, S. Mohammadi, Axisymmetric diffusion kurtosis imaging with Rician bias correction: A simulation study, Preprint no. bioRxiv2022.03.15.484442, Cold Spring Harbor Laboratory, bioRxiv, 2022, DOI 10.1101/2022.03.15.484442 .

CH. Bayer, P.K. Friz, N. Tapia, Stability of deep neural networks via discrete rough paths, Preprint no. arXiv:2201.07566, Cornell University, 2022.

P. Friz, Rough stochastic differential equations, Preprint no. arXiv:2106.10340, Cornell University, 2022, DOI 10.48550/arXiv.2106.10340 .

V. Spokoiny, Finite samples inference and critical dimension for stochastically linear models, Preprint no. arXiv:2201.06327, Cornell University, 2022.
Abstract
The aim of this note is to state a couple of general results about the properties of the penalized maximum likelihood estimators (pMLE) and of the posterior distribution for parametric models in a nonasymptotic setup and for possibly large or even infinite parameter dimension. We consider a special class of stochastically linear smooth (SLS) models satisfying two major conditions: the stochastic component of the loglikelihood is linear in the model parameter, while the expected loglikelihood is a smooth function. The main results simplify a lot if the expected loglikelihood is concave. For the pMLE, we establish a number of finite sample bounds about its concentration and large deviations as well as the Fisher and Wilks expansion. The later results extend the classical asymptotic Fisher and Wilks Theorems about the MLE to the nonasymptotic setup with large parameter dimension which can depend on the sample size. For the posterior distribution, our main result states a Gaussian approximation of the posterior which can be viewed as a finite sample analog of the prominent Bernsteinvon Mises Theorem. In all bounds, the remainder is given explicitly and can be evaluated in terms of the effective sample size and effective parameter dimension. The results are dimension and coordinate free. In spite of generality, all the presented bounds are nearly sharp and the classical asymptotic results can be obtained as simple corollaries. An interesting case of logit regression with smooth or truncation priors is used to specify the results and to explain the main notions. 
O. Butkovsky, K. Dareiotis, M. Gerencsér, Optimal rate of convergence for approximations of SPDEs with nonregular drift, Preprint no. arXiv:2110.06148, Cornell University Library, arXiv.org, 2021.

O. Butkovsky, V. Margarint, Y. Yuan, Law of the SLE tip, Preprint no. arXiv:2110.11247, Cornell University Library, arXiv.org, 2021.
Abstract
We analyse the law of the SLE tip at a fixed time in capacity parametrization. We describe it as the stationary law of a suitable diffusion process, and show that it has a density which is a unique solution of a certain PDE. Moreover, we identify the phases in which the even negative moments of the imaginary value are finite. For the negative second and negative fourth moments we provide closedform expressions. 
M. Coghi, W. Dreyer, P. Gajewski, C. Guhlke, P. Friz, M. Maurelli, A McKeanVlasov SDE and particle system with interactionfrom reflecting boundaries, Preprint no. 2102.12315v1/12102.12315v1/37, Cornell University Library, arXiv.org, 2021.

A. Agafonov, P. Dvurechensky, G. Scutari, A. Gasnikov, D. Kamzolov, A. Lukashevich, A. Daneshmand, An accelerated secondorder method for distributed stochastic optimization, Preprint no. arXiv:2103.14392, Cornell University Library, arXiv.org, 2021.
Abstract
We consider distributed stochastic optimization problems that are solved with master/workers computation architecture. Statistical arguments allow to exploit statistical similarity and approximate this problem by a finitesum problem, for which we propose an inexact accelerated cubicregularized Newton's method that achieves lower communication complexity bound for this setting and improves upon existing upper bound. We further exploit this algorithm to obtain convergence rate bounds for the original stochastic optimization problem and compare our bounds with the existing bounds in several regimes when the goal is to minimize the number of communication rounds and increase the parallelization by increasing the number of workers. 
A. Beznosikov, P. Dvurechensky, A. Koloskova, V. Samokhin, S.U. Stich, A. Gasnikov, Decentralized local stochastic extragradient for variational inequalities, Preprint no. arXiv:2106.08315, Cornell University, 2021.
Abstract
We consider decentralized stochastic variational inequalities where the problem data is distributed across many participating devices (heterogeneous, or nonIID data setting). We propose a novel method  based on stochastic extragradient  where participating devices can communicate over arbitrary, possibly timevarying network topologies. This covers both the fully decentralized optimization setting and the centralized topologies commonly used in Federated Learning. Our method further supports multiple local updates on the workers for reducing the communication frequency between workers. We theoretically analyze the proposed scheme in the strongly monotone, monotone and nonmonotone setting. As a special case, our method and analysis apply in particular to decentralized stochastic minmax problems which are being studied with increased interest in Deep Learning. For example, the training objective of Generative Adversarial Networks (GANs) are typically saddle point problems and the decentralized training of GANs has been reported to be extremely challenging. While SOTA techniques rely on either repeated gossip rounds or proximal updates, we alleviate both of these requirements. Experimental results for decentralized GAN demonstrate the effectiveness of our proposed algorithm. 
A. Daneshmand, G. Scutari, P. Dvurechensky, A. Gasnikov, Newton method over networks is fast up to the statistical precision, Preprint no. arXiv:2102.06780, Cornell University, 2021.

E. Gladin, A. Sadiev, A. Gasnikov, P. Dvurechensky, A. Beznosikov, M. Alkousa, Solving smooth minmin and minmax problems by mixed oracle algorithms, Preprint no. arXiv:2103.00434, Cornell University, 2021.
Abstract
In this paper, we consider two types of problems that have some similarity in their structure, namely, minmin problems and minmax saddlepoint problems. Our approach is based on considering the outer minimization problem as a minimization problem with inexact oracle. This inexact oracle is calculated via inexact solution of the inner problem, which is either minimization or a maximization problem. Our main assumptions are that the problem is smooth and the available oracle is mixed: it is only possible to evaluate the gradient w.r.t. the outer block of variables which corresponds to the outer minimization problem, whereas for the inner problem only zerothorder oracle is available. To solve the inner problem we use accelerated gradientfree method with zerothorder oracle. To solve the outer problem we use either inexact variant of Vaydya's cuttingplane method or a variant of accelerated gradient method. As a result, we propose a framework that leads to nonasymptotic complexity bounds for both minmin and minmax problems. Moreover, we estimate separately the number of first and zerothorder oracle calls which are sufficient to reach any desired accuracy. 
E. Gorbunov, M. Danilova, I. Shibaev, P. Dvurechensky, A. Gasnikov, Nearoptimal high probability complexity bounds for nonsmooth stochastic optimization with heavytailed noise, Preprint no. arXiv:2106.05958, Cornell University, 2021.
Abstract
Thanks to their practical efficiency and random nature of the data, stochastic firstorder methods are standard for training largescale machine learning models. Random behavior may cause a particular run of an algorithm to result in a highly suboptimal objective value, whereas theoretical guarantees are usually proved for the expectation of the objective value. Thus, it is essential to theoretically guarantee that algorithms provide small objective residual with high probability. Existing methods for nonsmooth stochastic convex optimization have complexity bounds with the dependence on the confidence level that is either negativepower or logarithmic but under an additional assumption of subGaussian (lighttailed) noise distribution that may not hold in practice, e.g., in several NLP tasks. In our paper, we resolve this issue and derive the first highprobability convergence results with logarithmic dependence on the confidence level for nonsmooth convex stochastic optimization problems with nonsubGaussian (heavytailed) noise. To derive our results, we propose novel stepsize rules for two stochastic methods with gradient clipping. Moreover, our analysis works for generalized smooth objectives with Höldercontinuous gradients, and for both methods, we provide an extension for strongly convex problems. Finally, our results imply that the first (accelerated) method we consider also has optimal iteration and oracle complexity in all the regimes, and the second one is optimal in the nonsmooth setting. 
A. Kroshnin, V. Spokoiny, A. Suvorikova, Multiplier bootstrap for BuresWasserstein barycenters, Preprint no. arXiv:2111.12612, Cornell University, 2021.
Abstract
BuresWasserstein barycenter is a popular and promising tool in analysis of complex data like graphs, images etc. In many applications the input data are random with an unknown distribution, and uncertainty quantification becomes a crucial issue. This paper offers an approach based on multiplier bootstrap to quantify the error of approximating the true BuresWasserstein barycenter Q? by its empirical counterpart Qn. The main results state the bootstrap validity under general assumptions on the data generating distribution P and specifies the approximation rates for the case of subexponential P. The performance of the method is illustrated on synthetic data generated from the weighted stochastic block model. 
V. Novitskii, A. Gasnikov, Improved exploiting higher order smoothness in derivativefree optimization and continuous bandit, Preprint no. arXiv:2101.03821, Cornell University, 2021.

D. Pasechnyuk, P. Dvurechensky, S. Omelchenko, A. Gasnikov, Stochastic optimization for dynamic pricing, Preprint no. arXiv:2106.14090, Cornell University, 2021.
Abstract
We consider the problem of supply and demand balancing that is stated as a minimization problem for the total expected revenue function describing the behavior of both consumers and suppliers. In the considered market model we assume that consumers follow the discrete choice demand model, while suppliers are equipped with some quantity adjustment costs. The resulting optimization problem is smooth and convex making it amenable for application of efficient optimization algorithms with the aim of automatically setting prices for online marketplaces. We propose to use stochastic gradient methods to solve the above problem. We interpret the stochastic oracle as a response to the behavior of a random market participant, consumer or supplier. This allows us to interpret the considered algorithms and describe a suitable behavior of consumers and suppliers that leads to fast convergence to the equilibrium in a close to the real marketplace environment. 
A. Rogozin, A. Beznosikov, D. Dvinskikh, D. Kovalev, P. Dvurechensky, A. Gasnikov, Decentralized distributed optimization for saddle point problems, Preprint no. arXiv:2102.07758, Cornell University, 2021.

A. Rogozin, M. Bochko, P. Dvurechensky, A. Gasnikov, V. Lukoshkin, An accelerated method for decentralized distributed stochastic optimization over timevarying graphs, Preprint no. arXiv:2103.15598, Cornell University Library, arXiv.org, 2021.
Abstract
We consider a distributed stochastic optimization problem that is solved by a decentralized network of agents with only local communication between neighboring agents. The goal of the whole system is to minimize a global objective function given as a sum of local objectives held by each agent. Each local objective is defined as an expectation of a convex smooth random function and the agent is allowed to sample stochastic gradients for this function. For this setting we propose the first accelerated (in the sense of Nesterov's acceleration) method that simultaneously attains optimal up to a logarithmic factor communication and oracle complexity bounds for smooth strongly convex distributed stochastic optimization. We also consider the case when the communication graph is allowed to vary with time and obtain complexity bounds for our algorithm, which are the first upper complexity bounds for this setting in the literature. 
V. Tominin , Y. Tominin , E. Borodich , D. Kovalev, A. Gasnikov, P. Dvurechensky, On accelerated methods for saddlepoint problems with composite structure, Preprint no. arXiv:2103.09344, Cornell University, 2021.
Abstract
We consider stronglyconvexstronglyconcave saddlepoint problems with general nonbilinear objective and different condition numbers with respect to the primal and the dual variables. First, we consider such problems with smooth composite terms, one of which having finitesum structure. For this setting we propose a variance reduction algorithm with complexity estimates superior to the existing bounds in the literature. Second, we consider finitesum saddlepoint problems with composite terms and propose several algorithms depending on the properties of the composite terms. When the composite terms are smooth we obtain better complexity bounds than the ones in the literature, including the bounds of a recently proposed nearlyoptimal algorithms which do not consider the composite structure of the problem. If the composite terms are proxfriendly, we propose a variance reduction algorithm that, on the one hand, is accelerated compared to existing variance reduction algorithms and, on the other hand, provides in the composite setting similar complexity bounds to the nearlyoptimal algorithm which is designed for noncomposite setting. Besides that, our algorithms allow to separate the complexity bounds, i.e. estimate, for each part of the objective separately, the number of oracle calls that is sufficient to achieve a given accuracy. This is important since different parts can have different arithmetic complexity of the oracle, and it is desired to call expensive oracles less often than cheap oracles. The key thing to all these results is our general framework for saddlepoint problems, which may be of independent interest. This framework, in turn is based on our proposed Accelerated MetaAlgorithm for composite optimization with probabilistic inexact oracles and probabilistic inexactness in the proximal mapping, which may be of independent interest as well. 
P. Dvurechensky, D. Kamzolov, A. Lukashevich, S. Lee, E. Ordentlich, C.A. Uribe, A. Gasnikov, Hyperfast secondorder local solvers for efficient statistically preconditioned distributed optimization, Preprint no. arXiv:2102.08246, Cornell University, 2021.

P. Dvurechensky, M. Staudigl, S. Shtern, Firstorder methods for convex optimization, Preprint no. arXiv:2101.00935, Cornell University, 2021.
Abstract
Firstorder methods for solving convex optimization problems have been at the forefront of mathematical optimization in the last 20 years. The rapid development of this important class of algorithms is motivated by the success stories reported in various applications, including most importantly machine learning, signal processing, imaging and control theory. Firstorder methods have the potential to provide low accuracy solutions at low computational complexity which makes them an attractive set of tools in largescale optimization problems. In this survey we cover a number of key developments in gradientbased optimization methods. This includes nonEuclidean extensions of the classical proximal gradient method, and its accelerated versions. Additionally we survey recent developments within the class of projectionfree methods, and proximal versions of primaldual schemes. We give complete proofs for various key results, and highlight the unifying aspects of several optimization algorithms. 
P. Dvurechensky, M. Staudigl, Hessian barrier algorithms for nonconvex conic optimization, Preprint no. arXiv:2111.00100, Cornell University Library, arXiv.org, 2021.
Abstract
We consider the minimization of a continuous function over the intersection of a regular cone with an affine set via a new class of adaptive first and secondorder optimization methods, building on the Hessianbarrier techniques introduced in [Bomze, Mertikopoulos, Schachinger, and Staudigl, Hessian barrier algorithms for linearly constrained optimization problems, SIAM Journal on Optimization, 2019]. Our approach is based on a potentialreduction mechanism and attains a suitably defined class of approximate first or secondorder KKT points with the optimal worstcase iteration complexity O(??2) (firstorder) and O(??3/2) (secondorder), respectively. A key feature of our methodology is the use of selfconcordant barrier functions to construct strictly feasible iterates via a disciplined decomposition approach and without sacrificing on the iteration complexity of the method. To the best of our knowledge, this work is the first which achieves these worstcase complexity bounds under such weak conditions for general conic constrained optimization problems.
Research Groups
 Partial Differential Equations
 Laser Dynamics
 Numerical Mathematics and Scientific Computing
 Nonlinear Optimization and Inverse Problems
 Interacting Random Systems
 Stochastic Algorithms and Nonparametric Statistics
 Thermodynamic Modeling and Analysis of Phase Transitions
 Nonsmooth Variational Problems and Operator Equations