Publications
Monographs

N. Tupitsa, P. Dvurechensky, D. Dvinskikh, A. Gasnikov, Section: Computational Optimal Transport, P.M. Pardalos, O.A. Prokopyev, eds., Encyclopedia of Optimization, Springer International Publishing, Cham, published online on 11.07.2023 pages, (Chapter Published), DOI 10.1007/9783030546212_8611 .

J. Polzehl, K. Tabelow, Magnetic Resonance Brain Imaging: Modeling and Data Analysis using R, 2nd Revised Edition, Series: Use R!, Springer International Publishing, Cham, 2023, 258 pages, (Monograph Published), DOI 10.1007/9783031389498 .
Abstract
This book discusses the modeling and analysis of magnetic resonance imaging (MRI) data acquired from the human brain. The data processing pipelines described rely on R. The book is intended for readers from two communities: Statisticians who are interested in neuroimaging and looking for an introduction to the acquired data and typical scientific problems in the field; and neuroimaging students wanting to learn about the statistical modeling and analysis of MRI data. Offering a practical introduction to the field, the book focuses on those problems in data analysis for which implementations within R are available. It also includes fully worked examples and as such serves as a tutorial on MRI analysis with R, from which the readers can derive their own data processing scripts. The book starts with a short introduction to MRI and then examines the process of reading and writing common neuroimaging data formats to and from the R session. The main chapters cover three common MR imaging modalities and their data modeling and analysis problems: functional MRI, diffusion MRI, and MultiParameter Mapping. The book concludes with extended appendices providing details of the nonparametric statistics used and the resources for R and MRI data.The book also addresses the issues of reproducibility and topics like data organization and description, as well as open data and open science. It relies solely on a dynamic report generation with knitr and uses neuroimaging data publicly available in data repositories. The PDF was created executing the R code in the chunks and then running LaTeX, which means that almost all figures, numbers, and results were generated while producing the PDF from the sources. 
CH. Bayer, P.K. Friz, M. Fukasawa, J. Gatheral, A. Jacquier, M. Rosenbaum, eds., Rough Volatility, Society for Industrial and Applied Mathematics, Philadelphia, 2023, xxviii + 263 pages, (Monograph Published), DOI 10.1137/1.9781611977783 .
Articles in Refereed Journals

D. Belomestny, J.G.M. Schoenmakers, Primaldual regression approach for Markov decision processes with general state and action space, SIAM Journal on Control and Optimization, 62, pp. 650679, DOI 10.1137/22M1526010 .
Abstract
We develop a regression based primaldual martingale approach for solving finite time horizon MDPs with general state and action space. As a result, our method allows for the construction of tight upper and lower biased approximations of the value functions, and, provides tight approximations to the optimal policy. In particular, we prove tight error bounds for the estimated duality gap featuring polynomial dependence on the time horizon, and sublinear dependence on the cardinality/dimension of the possibly infinite state and action space. From a computational point of view the proposed method is efficient since, in contrast to usual dualitybased methods for optimal control problems in the literature, the Monte Carlo procedures here involved do not require nested simulations. 
J.M. Oeschger, K. Tabelow, S. Mohammadi, Investigating apparent differences between standard DKI and axisymmetric DKI and its consequences for biophysical parameter estimates, Magnetic Resonance in Medicine, published online on 03.02.2024, DOI 10.1002/mrm.30034 .

A. Rogozin, A. Beznosikov, D. Dvinskikh, D. Kovalev, P. Dvurechensky, A. Gasnikov, Decentralized saddle point problems via nonEuclidean mirror prox, Optimization Methods & Software, published online in Jan. 2024, DOI 10.1080/10556788.2023.2280062 .

P. Dvurechensky, A. Gasnikov, P. Ostroukhov, C.A. Uribe, A. Ivanova, Nearoptimal tensor methods for minimizing the gradient norm of convex function, Optimization Methods & Software, Published online on 05.02.2024, DOI 10.1080/10556788.2023.2296443 .

O. Butkovsky, K. Dareiotis, M. Gerencsér, Optimal rate of convergence for approximations of SPDEs with nonregular drift, SIAM Journal on Numerical Analysis, 61 (2023), pp. 11031137, DOI 10.1137/21M1454213 .

O. Butkovsky, V. Margarint, Y. Yuan, Law of the SLE tip, Electronic Journal of Probability, 28 (2023), pp. 126/1126/25, DOI 10.1214/23EJP1015 .

F. Galarce Marín, 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, SIAM Journal on Imaging Sciences, 16 (2023), pp. 9961027, DOI 10.1137/22M149363X .
Abstract
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. 
O. Yufereva, M. Persiianov, P. Dvurechensky, A. Gasnikov, D. Kovalev, Decentralized convex optimization on timevarying networks with application to Wasserstein barycenters, Computational Management Science, published online on 16.12.2023, DOI 10.1007/s10287023004939 .

A. Agafonov, D. Kamzolov, P. Dvurechensky, A. Gasnikov, Inexact tensor methods and their application to stochastic convex optimization, Optimization Methods & Software, published online in Nov. 2023, DOI 10.1080/10556788.2023.2261604 .

D. Belomestny, J.G.M. Schoenmakers, From optimal martingales to randomized dual optimal stopping, Quantitative Finance, 23 (2023), pp. 10991113, DOI 10.1080/14697688.2023.2223242 .
Abstract
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. 
F. Bourgey, S. De Marco, P.K. Friz, P. Pigato, Local volatility under rough volatility, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, 33 (2023), pp. 11191145, DOI 10.1111/mafi.12392 .

J. Diehl, K. EbrahimiFard, N. Tapia, Generalized iteratedsums signatures, Journal of Algebra, 632 (2023), pp. 801824, DOI 10.1016/j.jalgebra.2023.06.007 .
Abstract
We explore the algebraic properties of a generalized version of the iteratedsums signature, inspired by previous work of F. Király and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasishuffle product of words on the latter. We introduce three nonlinear transformations on iteratedsums signatures, close in spirit to Machine Learning applications, and show some of their properties. 
K. EbrahimiFard, F. Patras, N. Tapia, L. Zambotti, Shifted substitution in noncommutative multivariate power series with a view toward free probability, SIGMA. Symmetry, Integrability and Geometry. Methods and Applications, 19 (2023), pp. 038/1038/17, DOI 10.3842/SIGMA.2023.038 .

N. Kornilov, A. Gasnikov, P. Dvurechensky, D. Dvinskikh, Gradient free methods for nonsmooth convex stochastic optimization with heavytailed noise on convex compact, Computational Management Science, 20 (2023), pp. 37/137/43, DOI 10.1007/s10287023004702 .

A. Vasin, A. Gasnikov, P. Dvurechensky, V. Spokoiny, Accelerated gradient methods with absolute and relative noise in the gradient, Optimization Methods & Software, published online in June 2023, DOI 10.1080/10556788.2023.2212503 .

S. Athreya, O. Butkovsky, K. Lê, L. Mytnik, Wellposedness of stochastic heat equation with distributional drift and skew stochastic heat equation, Communications on Pure and Applied Mathematics, published online on 29.11.2023, DOI 10.1002/cpa.22157 .

CH. Bayer, P. Hager, S. Riedel, J.G.M. Schoenmakers, Optimal stopping with signatures, The Annals of Applied Probability, 33 (2023), pp. 238273, DOI 10.1214/22AAP1814 .
Abstract
We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process. We consider classic and randomized stopping times represented by linear functionals of the associated rough path signature, and prove that maximizing over the class of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature. The only assumption on the process is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion which fail to be either semimartingales or Markov processes. 
CH. Bayer, M. Eigel, L. Sallandt, P. Trunschke, Pricing highdimensional Bermudan options with hierarchical tensor formats, SIAM Journal on Financial Mathematics, ISSN 1945497X, 14 (2023), pp. 383406, DOI 10.1137/21M1402170 .

CH. Bayer, P. Friz, N. Tapia, Stability of deep neural networks via discrete rough paths, SIAM Journal on Mathematics of Data Science, 5 (2023), pp. 5076, DOI 10.1137/22M1472358 .
Abstract
Using rough path techniques, we provide a priori estimates for the output of Deep Residual Neural Networks. In particular we derive stability bounds in terms of the total pvariation of trained weights for any p ≥ 1. 
CH. Bayer, Ch. Ben Hammouda, R.F. Tempone, Numerical smoothing with hierarchical adaptive sparse grids and quasiMonte Carlo methods for efficient option pricing, Quantitative Finance, 23 (2023), pp. 209227, DOI 10.1080/14697688.2022.2135455 .
Abstract
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. 
P.K. Friz, Th. Wagenhofer, Reconstructing volatility: Pricing of index options under rough volatility, Mathematical Finance. An International Journal of Mathematics, Statistics and Financial Economics, 33 (2023), pp. 1940, DOI 10.1111/mafi.12374 .

P.K. Friz, P. ZorinKranich , Rough semimartingales and $p$variation estimates for martingale transforms, The Annals of Applied Probability, 51 (2023), pp. 397441, DOI 10.1214/22AOP1598 .

V. Spokoiny, Dimension free nonasymptotic bounds on the accuracy of high dimensional Laplace approximation, SIAM/ASA Journal on Uncertainty Quantification, 11 (2023), pp. 10441068, DOI 10.1137/22M1495688 .
Abstract
This note attempts to revisit the classical results on Laplace approximation in a modern nonasymptotic and dimension free form. Such an extension is motivated by applications to high dimensional statistical and optimization problems. The established results provide explicit nonasymptotic bounds on the quality of a Gaussian approximation of the posterior distribution in total variation distance in terms of the so called empheffective dimension ( dimL ). This value is defined as interplay between information contained in the data and in the prior distribution. In the contrary to prominent Bernstein  von Mises results, the impact of the prior is not negligible and it allows to keep the effective dimension small or moderate even if the true parameter dimension is huge or infinite. We also address the issue of using a Gaussian approximation with inexact parameters with the focus on replacing the Maximum a Posteriori (MAP) value by the posterior mean and design the algorithm of Bayesian optimization based on Laplace iterations. The results are specified to the case of nonlinear regression.
Contributions to Collected Editions

R. Danabalan, M. Hintermüller, Th. Koprucki, K. Tabelow, MaRDI: Building research data infrastructures for mathematics and the mathematical sciences, in: 1st Conference on Research Data Infrastructure (CoRDI)  Connecting Communities, Y. SureVetter, C. Goble, eds., 1 of Proceedings of the Conference on Research Data Infrastructure, TIB Open Publishing, Hannover, 2023, pp. 69/169/4, DOI 10.52825/cordi.v1i.397 .
Abstract
MaRDI is building a research data infrastructure for mathematics and beyond based on semantic technologies (metadata, ontologies, knowledge graphs) and data repositories. Focusing on the algorithms, models and workflows, the MaRDI infrastructure will connect with other disciplines and NFDI consortia on data processing methods, solving real world problems and support mathematicians on research datamanagement 
S. Abdurakhmon, M. Danilova, E. Gorbunov, S. Horvath, G. Gauthier, P. Dvurechensky, P. Richtarik, Highprobability bounds for stochastic optimization and variational inequalities: The case of unbounded variance, in: Proceedings of the 40th International Conference on Machine Learning, A. Krause, E. Brunskill, K. Cho, B. Engelhardt, S. Sabato, J. Scarlett, eds., 202 of Proceedings of Machine Learning Research, 2023, pp. 2956329648.

T. Boege, R. Fritze, Ch. Görgen, J. Hanselman, D. Iglezakis, L. Kastner, Th. Koprucki, T. Krause, Ch. Lehrenfeld, S. Polla, M. Reidelbach, Ch. Riedel, J. Saak, B. Schembera, K. Tabelow, M. Weber, Researchdata management planning in the German mathematical community, 130 of EMS Magazine, European Mathematical Society, 2023, pp. 4047, DOI 10.4171/MAG/152 .

P. Dvurechensky, A. Gasnikov, A. Tiurin, V. Zholobov, Unifying framework for accelerated randomized methods in convex optimization, in: Foundations of Modern Statistics. FMS 2019, D. Belomestny, C. Butucea, E. Mammen, E. Moulines , M. Reiss, V.V. Ulyanov, eds., 425 of Springer Proceedings in Mathematics & Statistics, Springer, Cham, 2023, pp. 511561, DOI 10.1007/9783031301148_15 .
Preprints, Reports, Technical Reports

D. Belomestny, J.G.M. Schoenmakers, V. Zorina, Weighted mesh algorithms for general Markov decision processes: Convergence and tractability, Preprint no. 3088, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3088 .
Abstract, PDF (401 kByte)
We introduce a meshtype approach for tackling discretetime, finitehorizon Markov Decision Processes (MDPs) characterized by state and action spaces that are general, encompassing both finite and infinite (yet suitably regular) subsets of Euclidean space. In particular, for bounded state and action spaces, our algorithm achieves a computational complexity that is tractable in the sense of Novak & Wozniakowski, and is polynomial in the time horizon. For unbounded state space the algorithm is “semitractable” in the sense that the complexity is proportional to ε ^{c} with some dimension independent c ≥ 2, for achieving an accuracy ε and polynomial in the time horizon with degree linear in the underlying dimension. As such the proposed approach has some flavor of the randomization method by Rust which deals with infinite horizon MDPs and uniform sampling in compact state space. However, the present approach is essentially different due to the finite horizon and a simulation procedure due to general transition distributions, and more general in the sense that it encompasses unbounded state space. To demonstrate the effectiveness of our algorithm, we provide illustrations based on LinearQuadratic Gaussian (LQG) control problems. 
L. Schmitz, N. Tapia, Free generators and Hoffman's isomorphism for the twoparameter shuffle algebra, Preprint no. 3087, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3087 .
Abstract, PDF (239 kByte)
Signature transforms have recently been extended to data indexed by two and more parameters. With free Lyndon generators, ideas from B_{∞}algebras and a novel twoparameter Hoffman exponential, we provide three classes of isomorphisms between the underlying twoparameter shuffle and quasishuffle algebras. In particular, we provide a Hopf algebraic connection to the (classical, oneparameter) shuffle algebra over the extended alphabet of connected matrix compositions. 
C. Bellingeri, E. Ferrucci, N. Tapia, Branched Itô formula and natural ItôStratonovich isomorphism, Preprint no. 3083, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3083 .
Abstract, PDF (510 kByte)
Branched rough paths define integration theories that may fail to satisfy the usual integration by parts identity. The intrinsicallydefined projection of the ConnesKreimer Hopf algebra onto its primitive elements defined by Broadhurst and Kreimer, and further studied by Foissy, allows us to view it as a commutative B?algebra and thus to write an explicit change ofvariable formula for solutions to rough differential equations. This formula, which is realised by means of an explicit morphism from the GrossmanLarson Hopf algebra to the Hopf algebra of differential operators, restricts to the wellknown Itô formula for semimartingales. We establish an isomorphism with the shuffle algebra over primitives, extending Hoffman?s exponential for the quasi shuffle algebra, and in particular the usual ItôStratonovich correction formula for semimartingales. We place special emphasis on the onedimensional case, in which certain rough path terms can be expressed as polynomials in the extended trace path, which for semimartingales restrict to the wellknown KailathSegall polynomials. We end by describing an algebraic framework for generating examples of branched rough paths, and, motivated by the recent literature on stochastic processes, exhibit a few such examples above scalar 1/4fractional Brownian motion, two of which are ?truly branched?, i.e. not quasi geometric. 
CH. Bayer, L. Pelizzari, J.G.M. Schoenmakers, Primal and dual optimal stopping with signatures, Preprint no. 3068, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3068 .
Abstract, PDF (458 kByte)
We propose two signaturebased methods to solve the optimal stopping problem  that is, to price American options  in nonMarkovian frameworks. Both methods rely on a global approximation result for Lpfunctionals on rough pathspaces, using linear functionals of robust, rough path signatures. In the primal formulation, we present a nonMarkovian generalization of the fa mous LongstaffSchwartz algorithm, using linear functionals of the signature as regression basis. For the dual formulation, we parametrize the space of squareintegrable martingales using linear functionals of the signature, and apply a sample average approximation. We prove convergence for both methods and present first numerical examples in nonMarkovian and nonsemimartingale regimes. 
J.A. Dekker, R.J.A. Laeven, J.G.M. Schoenmakers, M.H. Vellekoop, Optimal stopping with randomly arriving opportunities to stop, Preprint no. 3056, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3056 .
Abstract, PDF (701 kByte)
We develop methods to solve general optimal stopping problems with opportunities to stop that arrive randomly. Such problems occur naturally in applications with market frictions. Pivotal to our approach is that our methods operate on random rather than deterministic time scales. This enables us to convert the original problem into an equivalent discretetime optimal stopping problem with natural number valued stopping times and a possibly infinite horizon. To numerically solve this problem, we design a random times least squares Monte Carlo method. We also analyze an iterative policy improvement procedure in this setting. We illustrate the efficiency of our methods and the relevance of randomly arriving opportunities in a few examples. 
CH. Bayer, S. Breneis, Efficient option pricing in the rough Heston model using weak simulation schemes, Preprint no. 3045, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3045 .
Abstract, PDF (569 kByte)
We provide an efficient and accurate simulation scheme for the rough Heston model in the standard ($H>0$) as well as the hyperrough regime ($H > 1/2$). The scheme is based on lowdimensional Markovian approximations of the rough Heston process derived in [Bayer and Breneis, arXiv:2309.07023], and provides weak approximation to the rough Heston process. Numerical experiments show that the new scheme exhibits second order weak convergence, while the computational cost increases linear with respect to the number of time steps. In comparison, existing schemes based on discretization of the underlying stochastic Volterra integrals such as Gatheral's HQE scheme show a quadratic dependence of the computational cost. Extensive numerical tests for standard and pathdependent European options and Bermudan options show the method's accuracy and efficiency. 
CH. Bayer, S. Breneis, Weak Markovian approximations of rough Heston, Preprint no. 3044, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3044 .
Abstract, PDF (834 kByte)
The rough Heston model is a very popular recent model in mathematical finance; however, the lack of Markov and semimartingale properties poses significant challenges in both theory and practice. A way to resolve this problem is to use Markovian approximations of the model. Several previous works have shown that these approximations can be very accurate even when the number of additional factors is very low. Existing error analysis is largely based on the strong error, corresponding to the L2 distance between the kernels. Extending earlier results by [Abi Jaber and El Euch, SIAM Journal on Financial Mathematics 10(2):309?349, 2019], we show that the weak error of the Markovian approximations can be bounded using the L1error in the kernel approximation for general classes of payoff functions for European style options. Moreover, we give specific Markovian approximations which converge superpolynomially in the number of dimensions, and illustrate their numerical superiority in option pricing compared to previously existing approximations. The new approximations also work for the hyperrough case H > 1/2. 
P. Bank, Ch. Bayer, P. Friz, L. Pelizzari, Rough PDEs for local stochastic volatility models, Preprint no. 3034, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3034 .
Abstract, PDF (575 kByte)
In this work, we introduce a novel pricing methodology in general, possibly nonMarkovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a timeinhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to socalled rough partial differential equations (RPDEs), through a FeynmanKac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for nonMarkovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models. 
P. Dvurechensky, J.J. Zhu, Kernel mirror prox and RKHS gradient flow for mixed functional Nash equilibrium, Preprint no. 3032, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3032 .
Abstract, PDF (436 kByte)
Kernel mirror prox and RKHS gradient flow for mixed functional Nash equilibrium Pavel Dvurechensky , JiaJie Zhu Abstract The theoretical analysis of machine learning algorithms, such as deep generative modeling, motivates multiple recent works on the Mixed Nash Equilibrium (MNE) problem. Different from MNE, this paper formulates the Mixed Functional Nash Equilibrium (MFNE), which replaces one of the measure optimization problems with optimization over a class of dual functions, e.g., the reproducing kernel Hilbert space (RKHS) in the case of Mixed Kernel Nash Equilibrium (MKNE). We show that our MFNE and MKNE framework form the backbones that govern several existing machine learning algorithms, such as implicit generative models, distributionally robust optimization (DRO), and Wasserstein barycenters. To model the infinitedimensional continuous limit optimization dynamics, we propose the Interacting WassersteinKernel Gradient Flow, which includes the RKHS flow that is much less common than the Wasserstein gradient flow but enjoys a much simpler convexity structure. Timediscretizing this gradient flow, we propose a primaldual kernel mirror prox algorithm, which alternates between a dual step in the RKHS, and a primal step in the space of probability measures. We then provide the first unified convergence analysis of our algorithm for this class of MKNE problems, which establishes a convergence rate of O(1/N ) in the deterministic case and O(1/√N) in the stochastic case. As a case study, we apply our analysis to DRO, providing the first primaldual convergence analysis for DRO with probabilitymetric constraints. 
CH. Bayer, S. Breneis, T. Lyons, An adaptive algorithm for rough differential equations, Preprint no. 3013, WIAS, Berlin, 2023, DOI 10.20347/WIAS.PREPRINT.3013 .
Abstract, PDF (838 kByte)
We present an adaptive algorithm for effectively solving rough differential equations (RDEs) using the log ODE method. The algorithm is based on an error representation formula that accurately describes the contribution of local errors to the global error. By incorporating a cost model, our algorithm efficiently deter mines whether to refine the time grid or increase the order of the logODE method. In addition, we provide several examples that demonstrate the effectiveness of our adapctive algorithm in solving RDEs.
Talks, Poster

O. Butkovsky, New developments in regularization by noise for SDEs, Finnish Mathematical Days 2024, January 4  5, 2024, Aalto University, Finnish Mathematical Society, Helsinki, Finland, January 4, 2024.

O. Butkovsky, Weak and strong wellposedness and local times for SDEs driven by fractional Brownian motion with integrable drift (online talk), 18th OxfordBerlin Young Researcher's Meeting on Applied Stochastic Analysis, University of Oxford, Mathematical Institute, UK, January 6, 2024.

A. Kroshnin, Robust kmeans in metric spaces, Seminar Modern Methods in Applied Stochastics and Nonparametric Statistics, WIAS Berlin, February 6, 2024.

L. Pelizzari, Optimal control in energy markets using rough analysis & deep networks (online talk), MATH+ Spotlight talk (online event), January 24, 2024.

L. Pelizzari, Primal and dual optimal stopping with signatures, 18th OxfordBerlin Young Researcher's Meeting on Applied Stochastic Analysis, January 4  6, 2024, University of Oxford, Mathematical Institute, UK, January 5, 2024.

P.K. Friz, Analyzing classes of SPDEs via RSDEs, Stochastic Partial Differential Equations, February 12  16, 2024, Universtität Wien, Erwin Schrödinger International Institute for Mathematics and Physics (ESI), Austria, February 16, 2024.

V. Spokoiny, Inference for nonlinear inverse problems, The Mathematics of Data, January 21  26, 2024, National University of Singapore, Institute for Mathematical Sciences, Singapore, January 23, 2024.

A. Shehu, MaRDI: The mathematical research data initiative, 2023 NFDI4DS Conference and Consortium Meeting, November 10, 2023, Fraunhofer FOKUS, November 10, 2023.

S. Breneis, American options under rough Heston, 11th General AMaMeF Conference, June 26  30, 2023, Universität Bielefeld, Center for Mathematical Economics, June 30, 2023.

S. Breneis, Pathdependent options under rough Heston, 4th Workshop on Stochastic Methods in Finance and Physics, Heraklion, Kreta, Greece, July 17  21, 2023.

S. Breneis, Pricing American options under rough Heston, Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2023, Thuwal, Saudi Arabia, May 21  June 1, 2023.

S. Breneis, Pricingpath dependent options under rough Heston, CDTIRTG Summer School 2023, September 3  8, 2023, Templin, September 3, 2023.

S. Breneis, Pricingpath dependent options under rough Heston, 18. Doktorand:innentreffen der Stochastik 2023, August 21  23, 2023, Universität Heidelberg, Fachbereich Mathematik, August 23, 2023.

S. Breneis, Weak Markovian approximations of rough Heston, 17th OxfordBerlin Young Researcher's Meeting on Applied Stochastic Analysis, April 27  29, 2023, WIAS & TU Berlin, April 27, 2023.

O. Butkovsky, Regularization by noise for SDEs and SPDEs beyond the Brownian case, Probability Seminar, Université ParisSaclay, CentraleSupélec, France, May 25, 2023.

O. Butkovsky, Stochastic equations with singular drift driven by fractional Brownian motion, 17th OxfordBerlin Young Researcher's Meeting on Applied Stochastic Analysis, April 27  29, 2023, WIAS & TU Berlin, April 28, 2023.

O. Butkovsky, Stochastic equations with singular drift driven by fractional Brownian motion, 43rd Conference on Stochastic Processes and their Applications, August 23  July 28, 2023, Bernoulli Society, Portugal, July 25, 2023.

O. Butkovsky, Stochastic equations with singular drift driven by fractional Brownian motion (online talk), Nonlocal Operators, Probability and Singularities (online event), researchseminars.org, April 4, 2023.

O. Butkovsky, Stochastic sewing, JohnNirenberg inequality, and taming singularities for regularization by noise, Mean Field, Interactions with Singular Kernels and their Approximations 2023, December 18, 2023, Institut Henri Poincaré, Paris, France, December 18, 2023.

O. Butkovsky, Stochastic sewing, JohnNirenberg inequality, and taming singularities for regularization by noise: A very practical guide, SDEs with Low Regularity Coefficients: Theory and Numerics, September 20  22, 2023, University of Torino, Department of Mathematics, Italy, September 22, 2023.

O. Butkovsky, Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise, 14th Conference on Monte Carlo Methods and Applications, June 26  30, 2023, Sorbonne University, Paris, France, June 29, 2023.

L. Pelizzari, Primaldual optimal stopping with signatures, Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2023, Thuwal, Saudi Arabia, May 26  June 1, 2023.

L. Pelizzari, Rough PDEs and local stochastic volatility, Volatility is rough, Isle of Skye Workshop, May 21  25, 2023, Sabhal Mòr Ostaig, Sleat, Isle of Skye, UK, May 25, 2023.

L. Pelizzari, Rough PDEs for local stochastic volatility models, 17th OxfordBerlin Young Researcher's Meeting on Applied Stochastic Analysis, April 27  29, 2023, WIAS & TU Berlin, April 27, 2023.

N. Tapia, Branched Itô formula, SFI: Structural Aspects of Signatures and Rough Paths, August 28  September 1, 2023, The Norwegian Academy of Science and Letters, Centre for Advanced Study (CAS), Oslo, Norway, August 31, 2023.

N. Tapia, Branched Itô formula, MiniWorkshop ``Combinatorial and Algebraic Structures in Rough Analysis and Related Fields'', November 26  December 2, 2023, Mathematisches Forschungsinstitut Oberwolfach, November 30, 2023.

N. Tapia, Branched Itô formula, Imperial College London, Mathematical Institute, UK, November 7, 2023.

N. Tapia, Stability of deep neural networks via discrete rough paths, Oxford Stochastic Analysis and Mathematical Finance Seminar, University of Oxford, Mathematical Institute, UK, February 13, 2023.

A. Kroshnin, Sobolev space of measurevalued functions, Variational and Information Flows in Machine Learning and Optimal Transport, November 19  24, 2023, Mathematisches Forschungsinstitut Oberwolfach, November 20, 2023.

CH. Bayer, D. Kreher, M. Landstorfer, W. Kenmoe Nzali, Volatile electricity markets and battery storage: A modelbased approach for optimal control, MATH+ Day, HumboldtUniversität zu Berlin, October 20, 2023.

CH. Bayer, P. Friz, J.G.M. Schoenmakers, V. Spokoiny, N. Tapia, L. Pelizzari, Optimal control in energy markets using rough analysis and deep networks, MATH+ Day, HumboldtUniversität zu Berlin, October 20, 2023.

CH. Bayer, Markovian approximations to rough volatility models, Volatility is rough, Isle of Skye Workshop, May 21  25, 2023, Sabhal Mòr Ostaig, Sleat, Isle of Skye, UK, May 25, 2023.

CH. Bayer, Markovian approximations to rough volatility models, Stochastics around Finance, August 28  30, 2023, Kanazawa University, Natural Science and Technology, Kanazawa, Japan, August 28, 2023.

CH. Bayer, Markovian approximations to rough volatility models, HeriotWatt University, Mathematical Institute, Edinburgh, UK, November 15, 2023.

CH. Bayer, Optimal stopping with signatures, Probabilistic Methods, Signatures, Cubature and Geometry, January 9  11, 2023, University of York, Department of Mathematics, UK, January 9, 2023.

CH. Bayer, Optimal stopping with signatures, Quantitative Finance Conference, April 12  15, 2023, University of Cambridge, Centre for Financial Research, UK, April 13, 2023.

CH. Bayer, Optimal stopping with signatures, 10th International Congress on Industrial and Applied Mathematics (ICIAM 2023), Minisymposium 00322 ``Methodological Advancement in Rough Paths and Data Science'', August 20  25, 2023, Waseda University, Tokyo, Japan, August 24, 2023.

CH. Bayer, Optimal stopping with signatures, Workshop on Stochastic Control Theory, October 25  26, 2023, KTH Royal Institute of Technology, Department of Mathematics, Stockholm, Sweden, October 26, 2023.

CH. Bayer, Optimal stopping with signatures, University of Dundee, School of Science and Engineering, UK, November 13, 2023.

CH. Bayer, Optimal stopping with signatures (online talk), North British Probability Seminar, University of Edinburgh, UK, March 29, 2023.

CH. Bayer, Rough PDEs for local stochastic volatility models, Rough Volatility Workshop, November 21  22, 2023, Sorbonne Université, Institut Henri Poincaré, Paris, France.

CH. Bayer, Signatures and applications, 4th Workshop on Stochastic Methods in Finance and Physics, July 17  21, 2023, Institute of Applied and Computational Mathematics (IACM), Heraklion, Kreta, Greece.

CH. Bayer, NonMarkovian models in finance, Stochastic Numerics and Statistical Learning: Theory and Applications 2023 Workshop, May 26  June 1, 2023, King Abdullah University of Science and Technology, Computer, Electrical and Mathematical Sciences, Thuwal, Saudi Arabia, May 27, 2023.

C. Cárcamo Sanchez, F. Galarce Marín, A. Caiazzo, I. Sack, K. Tabelow, Quantitative tissue pressure imaging via PDEinformed assimilation of MRdata, MATH+ Day, HumboldtUniversität zu Berlin, October 20, 2023.

P. Dvurechensky, C. Geiersbach, M. Hintermüller, A. Kannan, S. Kater, Equilibria for distributed multimodal energy systems under uncertainty, MATH+ Day, HumboldtUniversität zu Berlin, October 20, 2023.

P. Dvurechensky, Decentralized local stochastic extragradient for variational inequalities, Thematic Einstein Semester Conference on Mathematical Optimization for Machine Learning, September 13  15, 2023, Mathematics Research Cluster MATH+, Berlin, September 14, 2023.

P. Dvurechensky, Decentralized local stochastic extragradient for variational inequalities, European Conference on Computational Optimization (EUCCO), Session ``Optimization under Uncertainty'', September 25  27, 2023, Universität Heidelberg, September 25, 2023.

P. Dvurechensky, Hessian barrier algorithms for nonconvex conic optimization, 20th Workshop on Advances in Continuous Optimization, August 22  25, 2023, Corvinus University, Institute of Mathematical Statistics and Modelling, Budapest, August 25, 2023.

P.K. Friz, On rough stochastic differential equations, SPDEs, Optimal Control and Mean Field Games  Analysis, Numerics and Applications, July 10  14, 2023, Universität Bielefeld, Center for Interdisciplinary Research (ZiF), July 11, 2023.

P.K. Friz, Rough paths for local (possibly rough) stochastic volatility, LieStørmer Colloquium Analytic and Probabilistic Aspects of Rough Paths, November 27  29, 2023, Norwegian Academy of Science and Letters, Oslo, Norway, November 27, 2023.

A. Kroshnin, Robust kmeans clustering in metric spaces, Workshop on Statistics in Metric Spaces, October 11  13, 2023, Center for Research in Economics and Statistics (CREST), UMR 9194, Palaiseau, France, October 12, 2023.

A. Kroshnin, Robust kmeans clustering in metric spaces, Rencontres de Statistique Mathématique, December 18  22, 2023, Centre International de Rencontres Mathématiques (CIRM), Marseille, France, December 20, 2023.

A. Kroshnin, Entropic Wasserstein barycenters, Interpolation of Measures, January 24  25, 2023, Lagrange Mathematics and Computation Research Center, Huawei, Paris, France, January 24, 2023.

J.G.M. Schoenmakers, Optimal stopping with randomly arriving opportunities, Stochastische Analysis und Stochastik der Finanzmärkte, HumboldtUniversität zu Berlin, Institut für Mathematik, November 23, 2023.

J.G.M. Schoenmakers, Primaldual regression approach for Markov decision processes with general state and action spaces, SPDEs, Optimal Control and Mean Field Games  Analysis, Numerics and Applications, July 11  14, 2023, Universität Bielefeld, Center for Interdisciplinary Research (ZiF), July 12, 2023.

V. Spokoiny, Bayesian inference for complex models, MIA 2023  Mathematics and Image Analysis, February 1  3, 2023, Berlin, February 3, 2023.

V. Spokoiny, Bayesian inference using mixed Laplace approximation with applications to errorinoperator models, New York University, Courant Institute of Mathematical Sciences and Center for Data Science, USA, October 3, 2023.

V. Spokoiny, Estimation and inference for errorinoperator model, Mathematics in Armenia: Advances and Perspectives, July 2  8, 2023, Yerevan State University and National Academy of Sciences, Institute of Mathematics, Yerevan, Armenia, July 3, 2023.

V. Spokoiny, Estimation and inference for errorinoperator model, Lecture Series Trends in Statistics, National University of Singapore, Department of Mathematics, Singapore, August 25, 2023.

V. Spokoiny, Estimation and inference for errorinoperator model, Massachusetts Institute of Technology, Department of Mathematics, Cambridge, USA, September 29, 2023.

V. Spokoiny, Inference in errorinoperator model, Tel Aviv University, Department of Statistics, Israel, March 30, 2023.

V. Spokoiny, Marginal Laplace approximation and Gaussian mixtures, Optimization and Statistical Learning, OSL2023, January 15  20, 2023, Les Houches School of Physics, France, January 17, 2023.

K. Tabelow, MaRDI: Building research data infrastructures for mathematics and the mathematical science, 1st Conference on Research Data Infrastructure (CoRDI), September 12  14, 2023, Karlsruhe Institute of Technology (KIT), September 12, 2023.

K. Tabelow, Mathematical research data management in interdisciplinary research, Workshop on Biophysicsbased Modeling and Data Assimilation in Medical Imaging (Hybrid Event), WIAS Berlin, August 31, 2023.

J.J. Zhu, From gradient flow forcebalance to robust machine learning, Basque Center for Applied Mathematics, Bilbao, Spain, October 31, 2023.
External Preprints

O. Butkovsky, S. Gallay, Weak existence for SDEs with singular drifts and fractional Brownian or Levy noise beyond the subcritical regime, Preprint no. arXiv:2311.12013, Cornell University, 2023, DOI 10.48550/arXiv.2311.12013 .

O. Butkovsky, K. Lê, L. Mytnik, Stochastic equations with singular drift driven by fractional Brownian motion, Preprint no. arXiv:2302.11937, Cornell University, 2023, DOI 10.48550/arXiv.2302.11937 .

O. Yufereva, M. Persiianov, P. Dvurechensky, A. Gasnikov, D. Kovalev, Decentralized convex optimization on timevarying networks with application to Wasserstein barycenters, Preprint no. arXiv:2205.15669, Cornell University, 2023, DOI 10.48550/arXiv.2205.15669 .

K. EbrahimiFard, F. Patras, N. Tapia, L. Zambotti, Shifted substitution in noncommutative multivariate power series with a view toward free probability, Preprint no. arXiv:2204.01445, Cornell University, 2023, DOI 10.48550/arXiv.2204.01445 .

D. Gergely, B. Fricke, J.M. Oeschger, L. Ruthotto, P. Freund, K. Tabelow, S. Mohammadi, ACID: A comprehensive toolbox for image processing and modeling of brain, spinal cord, and ex vivo diffusion MRI data, Preprint no. bioRxiv:2023.10.13.562027, Cold Spring Harbor Laboratory, 2023, DOI 10.1101/2023.10.13.562027 .

E. Gladin, A. Gasnikov, P. Dvurechensky, Accuracy certificates for convex minimization with inexact Oracle, Preprint no. arXiv:2310.00523, Cornell University, 2023, DOI 10.48550/arXiv.2310.00523 .
Abstract
Accuracy certificates for convex minimization problems allow for online verification of the accuracy of approximate solutions and provide a theoretically valid online stopping criterion. When solving the Lagrange dual problem, accuracy certificates produce a simple way to recover an approximate primal solution and estimate its accuracy. In this paper, we generalize accuracy certificates for the setting of inexact firstorder oracle, including the setting of primal and Lagrange dual pair of problems. We further propose an explicit way to construct accuracy certificates for a large class of cutting plane methods based on polytopes. As a byproduct, we show that the considered cutting plane methods can be efficiently used with a noisy oracle even thought they were originally designed to be equipped with an exact oracle. Finally, we illustrate the work of the proposed certificates in the numerical experiments highlighting that our certificates provide a tight upper bound on the objective residual. 
E. Gorbunov, A. Sadiev, D. Dolinova, S. Horvát, G. Gidel, P. Dvurechensky, A. Gasnikov, P. Richtárik, Highprobability convergence for composite and distributed stochastic minimization and variational inequalities with heavytailed noise, Preprint no. arXiv:2310.01860, Cornell University, 2023, DOI 10.48550/arXiv.2310.01860 .
Abstract
Highprobability analysis of stochastic firstorder optimization methods under mild assumptions on the noise has been gaining a lot of attention in recent years. Typically, gradient clipping is one of the key algorithmic ingredients to derive good highprobability guarantees when the noise is heavytailed. However, if implemented naïvely, clipping can spoil the convergence of the popular methods for composite and distributed optimization (ProxSGD/Parallel SGD) even in the absence of any noise. Due to this reason, many works on highprobability analysis consider only unconstrained nondistributed problems, and the existing results for composite/distributed problems do not include some important special cases (like strongly convex problems) and are not optimal. To address this issue, we propose new stochastic methods for composite and distributed optimization based on the clipping of stochastic gradient differences and prove tight highprobability convergence results (including nearly optimal ones) for the new methods. Using similar ideas, we also develop new methods for composite and distributed variational inequalities and analyze the highprobability convergence of these methods. 
N. Kornilov, A. Gasnikov, P. Dvurechensky, D. Dvinskikh, Gradient free methods for nonsmooth convex stochastic optimization with heavytailed noise on convex compact, Preprint no. arXiv:2304.02442, Cornell University, 2023, DOI 10.48550/arXiv.2304.02442 .

N. Kornilov, E. Gorbunov, M. Alkousa, F. Stonyakin, P. Dvurechensky, A. Gasnikov, Intermediate gradient methods with relative inexactness, Preprint no. arXiv:2310.00506, Cornell University, 2023, DOI 10.48550/arXiv.2310.00506 .
Abstract
This paper is devoted to firstorder algorithms for smooth convex optimization with inexact gradi ents. Unlike the majority of the literature on this topic, we consider the setting of relative rather than absolute inexactness. More precisely, we assume that an additive error in the gradient is propor tional to the gradient norm, rather than being globally bounded by some small quantity. We propose a novel analysis of the accelerated gradient method under relative inexactness and strong convex ity and improve the bound on the maximum admissible error that preserves the linear convergence of the algorithm. In other words, we analyze how robust is the accelerated gradient method to the relative inexactness of the gradient information. Moreover, based on the Performance Estimation Problem (PEP) technique, we show that the obtained result is optimal for the family of accelerated algorithms we consider. Motivated by the existing intermediate methods with absolute error, i.e., the methods with convergence rates that interpolate between slower but more robust nonaccelerated algorithms and faster, but less robust accelerated algorithms, we propose an adaptive variant of the intermediate gradient method with relative error in the gradient. 
J.M. Oeschger, K. Tabelow, S. Mohammadi, Investigating apparent differences between standard DKI and axisymmetric DKI and its consequences for biophysical parameter estimates, Preprint no. bioRxiv:2023.06.21.545891, Cold Spring Harbor Laboratory, 2023, DOI 10.1101/2023.06.21.545891 .

D.A. Pasechnyuk, M. Persiianov, P. Dvurechensky, A. Gasnikov, Algorithms for Euclideanregularised optimal transport, Preprint no. arXiv:2307.00321, Cornell University, 2023, DOI 10.48550/arXiv.2307.00321 .

A. Sadiev, E. Gorbunov, S. Horváth, G. Gidel, P. Dvurechensky, A. Gasnikov, P. Peter, Highprobability bounds for stochastic optimization and variational inequalities: The case of unbounded variance, Preprint no. arXiv:2302.00999, Cornell University, 2023, DOI 10.48550/arXiv.2302.00999 .

B. Schembera, F. Wübbeling, H. Kleikamp, Ch. Bledinger, J. Fiedler, M. Reidelbach, A. Shehu, B. Schmidt, Th. Koprucki, D. Iglezakis, D. Göddeke, Ontologies for models and algorithms in applied mathematics and related disciplines, Preprint no. arXiv:2310.20443, Cornell University, 2023, DOI 10.48550/arXiv.2310.20443 .
Abstract
In applied mathematics and related disciplines, the modelingsimulationoptimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data, the Mathematical Research Data Initiative has developed, merged and implemented ontologies and knowledge graphs. This contributes to making mathematical research data FAIR by introducing semantic technology and documenting the mathematical foundations accordingly. Using the concrete example of microfracture analysis of porous media, it is shown how the knowledge of the underlying mathematical model and the corresponding numerical algorithms for its solution can be represented by the ontologies. 
V. Spokoiny, Concentration of a high dimensional subGaussian vector, Preprint no. arXiv:2305.07885, Cornell University, 2023, DOI 10.48550/arXiv.2305.07885 .

V. Spokoiny, Mixed Laplace approximation for marginal posterior and Bayesian inference in errorinoperator model, Preprint no. arXiv:2305.08193, Cornell University, 2023, DOI 10.48550/arXiv.2305.09336 .

V. Spokoiny, Nonlinear regression: Finite sample guarantees, Preprint no. arXiv:2305.08193, Cornell University, 2023, DOI 10.48550/arXiv.2305.08193 .
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