Publications

Monographs

  • A. Bayandina, P. Dvurechensky, A. Gasnikov, Mirror descent and convex optimization problems with non-smooth inequality constraints, in: Large Scale and Distributed Optimization, P. Giselsson, A. Rantzer, eds., Lecture Notes in Mathematics 2227, Springer Nature Switzerland AG, Cham, 2018, pp. 181--215, (Chapter Published), DOI 10.1007/978-3-319-97478-1_8 .

  • D. Belomestny, J. Schoenmakers, Advanced Simulation-Based Methods for Optimal Stopping and Control: With Applications in Finance, Macmillan Publishers Ltd., London, 2018, 364 pages, (Monograph Published), DOI 10.1057/978-1-137-03351-2 .

Articles in Refereed Journals

  • V. Spokoiny, N. Willrich, Bootstrap tuning in Gaussian ordered model selection, The Annals of Applied Probability, 47 (2019), pp. 1351--1380, DOI 10.1214/18-AOS1717 .
    Abstract
    In the problem of model selection for a given family of linear estimators, ordered by their variance, we offer a new “smallest accepted” approach motivated by Lepski's device and the multiple testing idea. The procedure selects the smallest model which satisfies the acceptance rule based on comparison with all larger models. The method is completely data-driven and does not use any prior information about the variance structure of the noise: its parameters are adjusted to the underlying possibly heterogeneous noise by the so called “propagation condition” using bootstrap multiplier method. The validity of the bootstrap calibration is proved for finite samples with an explicit error bound. We provide a comprehensive theoretical study of the method and describe in details the set of possible values of the selector ( hatm ). We also establish some precise oracle error bounds for the corresponding estimator ( hattheta = tildetheta_hatm ) which equally applies to estimation of the whole parameter vectors, its subvector or linear mapping, as well as estimation of a linear functional.

  • K. Tabelow, E. Balteau, J. Ashburner, M.F. Callaghan, B. Draganski, G. Helms, F. Kherif, T. Leutritz, A. Lutti, Ch. Phillips, E. Reimer, L. Ruthotto, M. Seif, N. Weiskopf, G. Ziegler, S. Mohammadi, hMRI -- A toolbox for quantitative MRI in neuroscience and clinical research, NeuroImage, (2019), published online on 21.01.2019, DOI 10.1016/j.neuroimage.2019.01.029 .
    Abstract
    Quantitative magnetic resonance imaging (qMRI) finds increasing application in neuroscience and clinical research due to its sensitivity to micro-structural properties of brain tissue, e.g. axon, myelin, iron and water concentration. We introduce the hMRI--toolbox, an easy-to-use open-source tool for handling and processing of qMRI data presented together with an example dataset. This toolbox allows the estimation of high-quality multi-parameter qMRI maps (longitudinal and effective transverse relaxation rates R1 and R2*, proton density PD and magnetisation transfer MT) that can be used for calculation of standard and novel MRI biomarkers of tissue microstructure as well as improved delineation of subcortical brain structures. Embedded in the Statistical Parametric Mapping (SPM) framework, it can be readily combined with existing SPM tools for estimating diffusion MRI parameter maps and benefits from the extensive range of available tools for high-accuracy spatial registration and statistical inference. As such the hMRI--toolbox provides an efficient, robust and simple framework for using qMRI data in neuroscience and clinical research.

  • W. Dreyer, P. Friz, P. Gajewski, C. Guhlke, M. Maurelli, Stochastic many-particle model for LFP electrodes, Continuum Mechanics and Thermodynamics, 30 (2018), pp. 593--628, DOI 10.1007/s00161-018-0629-7 .
    Abstract
    In the framework of non-equilibrium thermodynamics we derive a new model for porous electrodes. The model is applied to LiFePO4 (LFP) electrodes consisting of many LFP particles of nanometer size. The phase transition from a lithium-poor to a lithium-rich phase within LFP electrodes is controlled by surface fluctuations leading to a system of stochastic differential equations. The model is capable to derive an explicit relation between battery voltage and current that is controlled by thermodynamic state variables. This voltage-current relation reveals that in thin LFP electrodes lithium intercalation from the particle surfaces into the LFP particles is the principal rate limiting process. There are only two constant kinetic parameters in the model describing the intercalation rate and the fluctuation strength, respectively. The model correctly predicts several features of LFP electrodes, viz. the phase transition, the observed voltage plateaus, hysteresis and the rate limiting capacity. Moreover we study the impact of both the particle size distribution and the active surface area on the voltagecharge characteristics of the electrode. Finally we carefully discuss the phase transition for varying charging/discharging rates.

  • P. Pigato, Tube estimates for diffusion processes under a weak Hörmander condition, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 54 (2018), pp. 299--342, DOI 10.1214/16-AIHP805 .
    Abstract
    We consider a diffusion process under a local weak Hörmander condition on the coefficients. We find Gaussian estimates for the density in short time and exponential lower and upper bounds for the probability that the diffusion remains in a small tube around a deterministic trajectory (skeleton path). These bounds depend explicitly on the radius of the tube and on the energy of the skeleton path. We use a norm which reflects the non-isotropic structure of the problem, meaning that the diffusion propagates in R2 with different speeds in the directions ? and [?,b]. We establish a connection between this norm and the standard control distance.

  • M. Redmann, P. Kürschner, An output error bound for time-limited balanced truncation, Systems & Control Letters, 121 (2018), pp. 1--6, DOI 10.1016/j.sysconle.2018.08.004 .
    Abstract
    When solving partial differential equations numerically, usually a high order spatial discretization is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretized systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT). However, if one aims at finding a good ROM on a certain finite time interval only, time-limited BT (TLBT) can be a more accurate alternative. So far, no error bound on TLBT has been proved. In this paper, we close this gap in the theory by providing an output error bound for TLBT with two different representations. The performance of the error bound is then shown in several numerical experiments.

  • M. Redmann, Energy estimates and model order reduction for stochastic bilinear systems, International Journal of Control, (2018), Published online: 8.11.2018, DOI 10.1080/00207179.2018.1538568 .
    Abstract
    In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique called balanced truncation (BT) to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. When BT is applied the reduced system is obtained by removing these states from the original system. The main contribution of this paper is an L2-error bound for BT for stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far.

  • M. Redmann, Type II balanced truncation for deterministic bilinear control systems, SIAM Journal on Control and Optimization, 56 (2018), pp. 2593--2612, DOI 10.1137/17M1147962 .
    Abstract
    When solving partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is balanced truncation (BT), a method which has been extensively studied for deterministic linear systems. As so-called type I BT it has already been extended to bilinear equations, an important subclass of nonlinear systems. We provide an alternative generalisation of the linear setting to bilinear systems which is called type II BT. The Gramians that we propose in this context contain information about the control. It turns out that the new approach delivers energy bounds which are not just valid in a small neighbourhood of zero. Furthermore, we provide an ℋ∞-error bound which so far is not known when applying type I BT to bilinear systems.

  • M. Redmann, Type II singular perturbation approximation for linear systems with Lévy noise, SIAM Journal on Control and Optimization, 56 (2018), pp. 2120--2158, DOI 10.1137/17M113160X .
    Abstract
    When solving linear stochastic partial differential equations numerically, usually a high order spatial discretisation is needed. Model order reduction (MOR) techniques are often used to reduce the order of spatially-discretised systems and hence reduce computational complexity. A particular MOR technique to obtain a reduced order model (ROM) is singular perturbation approximation (SPA), a method which has been extensively studied for deterministic systems. As so-called type I SPA it has already been extended to stochastic equations. We provide an alternative generalisation of the deterministic setting to linear systems with Lévy noise which is called type II SPA. It turns out that the ROM from applying type II SPA has better properties than the one of using type I SPA. In this paper, we provide new energy interpretations for stochastic reachability Gramians, show the preservation of mean square stability in the ROM by type II SPA and prove two different error bounds for type II SPA when applied to Lévy driven systems

  • M.A. Freitag, M. Redmann, Balanced model order reduction for linear random dynamical systems driven by Lévy noise, Journal of Computational Dynamics, 5 (2018), pp. 33--59, DOI 10.3934/jcd.2018002 .

  • D. Belomestny, J.G.M. Schoenmakers, Projected particle methods for solving McKean--Vlaslov equations, SIAM Journal on Numerical Analysis, 56 (2018), pp. 3169--3195, DOI 10.1137/17M1111024 .
    Abstract
    We propose a novel projection-based particle method for solving McKean--Vlasov stochastic differential equations. Our approach is based on a projection-type estimation of the marginal density of the solution in each time step. The projection-based particle method leads in many situations to a significant reduction of numerical complexity compared to the widely used kernel density estimation algorithms. We derive strong convergence rates and rates of density estimation. The convergence analysis, particularly in the case of linearly growing coefficients, turns out to be rather challenging and requires some new type of averaging technique. This case is exemplified by explicit solutions to a class of McKean--Vlasov equations with affine drift. The performance of the proposed algorithm is illustrated by several numerical examples.

  • K. Chouk, P. Friz, Support theorem for a singular SPDE: The case of gPAM, Annales de l'Institut Henri Poincare. Probabilites et Statistiques, 54 (2018), pp. 202-219.
    Abstract
    We consider the generalized parabolic Anderson equation (gPAM) in 2 dimensions with periodic boundary. This is an example of a singular semilinear stochastic partial differential equation in the subcritical regime, with (renormalized) solutions only recently understood via Hairer?s regularity structures and, in some cases equivalently, paracontrollled distributions by Gubinelli, Imkeller and Perkowski. In the present paper we utilise the paracontrolled machinery and obtain a (Stroock?Varadhan) type support description for the law of gPAM. In the spirit of rough paths, the crucial step is to identify the support of the enhanced noise in a sufficiently fine topology. The renormalization is seen to affect the support description.

  • A. Gasnikov, P. Dvurechensky, M. Zhukovskii, S. Kim, S. Plaunov, D. Smirnov, F. Noskov, About the power law of the PageRank vector distribution. Part 2. Backley--Osthus model, power law verification for this model and setup of real search engines, Numerical Analysis and Applications, 11 (2018), pp. 16--32, DOI 10.1134/S1995423918010032 .

  • B. Hofmann, P. Mathé, Tikhonov regularization with oversmoothing penalty for non-linear ill-posed problems in Hilbert scales, Inverse Problems. An International Journal on the Theory and Practice of Inverse Problems, Inverse Methods and Computerized Inversion of Data, 34 (2018), pp. 015007/1--015007/14.

  • V. Krätschmer, M. Ladkau, R.J.A. Laeven, J.G.M. Schoenmakers, M. Stadje, Optimal stopping under uncertainty in drift and jump intensity, Mathematics of Operations Research, (2018), published online on 09.08.2018, urlhttps://doi.org/101287/moor.2017.0899., DOI 10.1287/moor.2017.0899 .
    Abstract
    This paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a method to practically solve this problem in a general setting, allowing for general time-consistent ambiguity averse preferences and general payoff processes driven by jump-diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem %represented by the Snell envelope using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We analyze the asymptotic behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples.

  • A. Lejay, P. Pigato, Statistical estimation of the oscillating Brownian motion, Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 24 (2018), pp. 3568--3602, DOI 10.3150/17-BEJ969 .
    Abstract
    We study the asymptotic behavior of estimators of a two-valued, discontinuous diffusion coefficient in a Stochastic Differential Equation, called an Oscillating Brownian Motion. Using the relation of the latter process with the Skew Brownian Motion, we propose two natural consistent estimators, which are variants of the integrated volatility estimator and take the occupation times into account. We show the stable convergence of the renormalized errors? estimations toward some Gaussian mixture, possibly corrected by a term that depends on the local time. These limits stem from the lack of ergodicity as well as the behavior of the local time at zero of the process. We test both estimators on simulated processes, finding a complete agreement with the theoretical predictions.

  • I. Silin, V. Spokoiny, Bayesian inference for spectral projectors of covariance matrix, Electronic Journal of Statistics, 12 (2018), pp. 1948--1987, DOI 10.1214/18-EJS1451 .

  • CH. Bayer, P. Friz, A. Gulisashvili, B. Horvath, B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Quantitative Finance, published online on 13.11.2018, DOI 10.1080/14697688.2018.1529420 .
    Abstract
    We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter H < ½. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order t1/2 (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

  • CH. Bayer, H. Mai, J.G.M. Schoenmakers, Forward-reverse expectation-maximization algorithm for Markov chains: Convergence and numerical analysis, Advances in Applied Probability, 2 (2018), pp. 621--644, DOI 10.1017/apr.2018.27 .
    Abstract
    We develop a forward-reverse expectation-maximization (FREM) algorithm for estimating parameters of a discrete-time Markov chain evolving through a certain measurable state-space. For the construction of the FREM method, we develop forward-reverse representations for Markov chains conditioned on a certain terminal state. We prove almost sure convergence of our algorithm for a Markov chain model with curved exponential family structure. On the numerical side, we carry out a complexity analysis of the forward-reverse algorithm by deriving its expected cost. Two application examples are discussed.

  • P. Dvurechensky, A. Gasnikov, A. Lagunovskaya, Parallel algorithms and probability of large deviation for stochastic convex optimization problems, Numerical Analysis and Applications, 11 (2018), pp. 33--37, DOI 10.1134/S1995423918010044 .

  • P. Friz, H. Zhang, Differential equations driven by rough paths with jumps, Journal of Differential Equations, 264 (2018), pp. 6226--6301, DOI 10.1016/j.jde.2018.01.031 .
    Abstract
    We develop the rough path counterpart of Itô stochastic integration and differential equations driven by general semimartingales. This significantly enlarges the classes of (Itô/forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.

  • TH. Koprucki, M. Kohlhase, K. Tabelow, D. Müller, F. Rabe, Model pathway diagrams for the representation of mathematical models, Optical and Quantum Electronics, 50 (2018), pp. 70/1--70/9, DOI 10.1007/s11082-018-1321-7 .
    Abstract
    Mathematical models are the foundation of numerical simulation of optoelectronic devices. We present a concept for a machine-actionable as well as human-understandable representation of the mathematical knowledge they contain and the domain-specific knowledge they are based on. We propose to use theory graphs to formalize mathematical models and model pathway diagrams to visualize them. We illustrate our approach by application to the van Roosbroeck system describing the carrier transport in semiconductors by drift and diffusion. We introduce an approach for the block-based composition of models from simpler components.

Contributions to Collected Editions

  • Y. Bruned, I. Chevyrev, P. Friz, Examples of renormalized SDEs, in: Stochastic Partial Differential Equations and Related Fields, A. Eberle, M. Grothaus, W. Hoh, M. Kassmann, W. Stannat, G. Trutnau, eds., 229 of Springer Proceedings in Mathematics & Statistics, Springer Nature Switzerland AG, Cham, 2018, pp. 303--317, DOI 10.1007/978-3-319-74929-7 .

  • P. Dvurechensky, A. Gasnikov, A. Kroshnin, Computational optimal transport: Complexity by accelerated gradient descent is better than by Sinkhorn's algorithm, in: Proceedings of the 35th International Conference on Machine Learning, J. Dy, A. Krause, eds., 80 of Proceedings of Machine Learning Research, 2018, pp. 1367--1376.

  • TH. Koprucki, A. Maltsi, T. Niermann, T. Streckenbach, K. Tabelow, J. Polzehl, Towards model-based geometry reconstruction of quantum dots from TEM, in: Proceedings of the 18th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2018), A. Djurišić, J. Piprek, eds., IEEE Conference Publications Management Group, Piscataway, 2018, pp. 115--116.

  • O. Marquardt, P. Mathé, Th. Koprucki, M. Caro, M. Willatzen, Data-driven electronic structure calculations in semiconductor nanostructures -- beyond the eight-band k.p formalism, in: Proceedings of the 18th International Conference on Numerical Simulation of Optoelectronic Devices (NUSOD 2018), A. Djurišić, J. Piprek, eds., IEEE Conference Publications Management Group, 2018, pp. 55--56.

  • P. Mathé, S. Agapiou, Posterior contraction in Bayesian inverse problems under Gaussian priors, in: New Trends in Parameter Identification for Mathematical Models, B. Hofmann, A. Leitao, J. Passamani Zubelli, eds., Trends in Mathematics, Springer, Basel, 2018, pp. 1--29, DOI 10.1007/978-3-319-70824-9 .

  • H.-J. Mucha, H.-G. Bartel, I. Reiche, K. Müller, Multivariate statistische Auswertung von archäometrischen Messwerten von Mammut-Elfenbein, in: Archäometrie und Denkmalpflege 2018, L. Glaser, ed., Verlag Deutsches Elektronen-Synchrotron, Hamburg, 2018, pp. 162--165, DOI 10.3204/DESY-PROC-2018-01 .

  • H.-J. Mucha, H.-G. Bartel, Distance and data transformation, in: The SAS Encyclopedia of Archaeological Sciences, S.L. López Varela, ed., JohnWiley & Sons, Inc., 2018, pp. published online on 26.11.2018, DOI 10.1002/9781119188230.saseas0194 .

Preprints, Reports, Technical Reports

  • CH. Bayer, B. Stemper, Deep calibration of rough stochastic volatility models, Preprint no. 2547, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2547 .
    Abstract, PDF (3663 kByte)
    Sparked by Alòs, León und Vives (2007); Fukasawa (2011, 2017); Gatheral, Jaisson und Rosenbaum (2018), so-called rough stochastic volatility models such as the rough Bergomi model by Bayer, Friz und Gatheral (2016) constitute the latest evolution in option price modeling. Unlike standard bivariate diffusion models such as Heston (1993), these non-Markovian models with fractional volatility drivers allow to parsimoniously recover key stylized facts of market implied volatility surfaces such as the exploding power-law behaviour of the at-the-money volatility skew as time to maturity goes to zero. Standard model calibration routines rely on the repetitive evaluation of the map from model parameters to Black-Scholes implied volatility, rendering calibration of many (rough) stochastic volatility models prohibitively expensive since there the map can often only be approximated by costly Monte Carlo (MC) simulations (Bennedsen, Lunde & Pakkanen, 2017; McCrickerd & Pakkanen, 2018; Bayer et al., 2016; Horvath, Jacquier & Muguruza, 2017). As a remedy, we propose to combine a standard Levenberg-Marquardt calibration routine with neural network regression, replacing expensive MC simulations with cheap forward runs of a neural network trained to approximate the implied volatility map. Numerical experiments confirm the high accuracy and speed of our approach.

  • CH. Bayer, M. Redmann, J.G.M. Schoenmakers, Dynamic programming for optimal stopping via pseudo-regression, Preprint no. 2532, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2532 .
    Abstract, PDF (337 kByte)
    We introduce new variants of classical regression-based algorithms for optimal stopping problems based on computation of regression coefficients by Monte Carlo approximation of the corresponding L2 inner products instead of the least-squares error functional. Coupled with new proposals for simulation of the underlying samples, we call the approach "pseudo regression". We show that the approach leads to asymptotically smaller errors, as well as less computational cost. The analysis is justified by numerical examples.

  • D. Belomestny, J.G.M. Schoenmakers, V. Spokoiny, Y. Tavyrikov, Optimal stopping via deeply boosted backward regression, Preprint no. 2530, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2530 .
    Abstract, PDF (209 kByte)
    In this note we propose a new approach towards solving numerically optimal stopping problems via boosted regression based Monte Carlo algorithms. The main idea of the method is to boost standard linear regression algorithms in each backward induction step by adding new basis functions based on previously estimated continuation values. The proposed methodology is illustrated by several numerical examples from finance.

  • J. Polzehl, K. Papafitsoros, K. Tabelow, Patch-wise adaptive weights smoothing, Preprint no. 2520, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2520 .
    Abstract, PDF (28 MByte)
    Image reconstruction from noisy data has a long history of methodological development and is based on a variety of ideas. In this paper we introduce a new method called patch-wise adaptive smoothing, that extends the Propagation-Separation approach by using comparisons of local patches of image intensities to define local adaptive weighting schemes for an improved balance of reduced variability and bias in the reconstruction result. We present the implementation of the new method in an R package aws and demonstrate its properties on a number of examples in comparison with other state-of-the art image reconstruction methods.

  • CH. Bayer, D. Belomestny, M. Redmann, S. Riedel, J.G.M. Schoenmakers, Solving linear parabolic rough partial differential equations, Preprint no. 2506, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2506 .
    Abstract, PDF (4827 kByte)
    We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity α with ⅓ < α ≤ ½ . Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.

  • L. Antoine, P. Pigato, Maximum likelihood drift estimation for a threshold diffusion, Preprint no. 2497, WIAS, Berlin, 2018, DOI 10.20347/WIAS.PREPRINT.2497 .
    Abstract, PDF (425 kByte)
    We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called the drifted Oscillating Brownian motion. The asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations.

Talks, Poster

  • CH. Bayer, A regularity structure for rough volatility, Vienna Seminar in Mathematical Finance and Probability, TU Wien, Research Unit of Financial and Actuarial Mathematics, Austria, January 10, 2019.

  • V. Spokoiny, Inference for spectral projectors, RTG Kolloquium, Universität Heidelberg, Institut ür angewandte Mathematik, January 10, 2019.

  • A. Maltsi, Th. Koprucki, T. Niermann, T. Streckenbach, K. Tabelow, J. Polzehl, Computing TEM images of semiconductor nanostructures, Applied Mathematics and Simulation for Semiconductors (AMaSiS 2018), WIAS Berlin, October 8 - 10, 2018.

  • A. Suvorikova, CLT for barycenters in 2-Wasserstein space, Mass TransportiationTheory: Opening perspectives in Statistics, Probability and Computer Science, June 3 - 10, 2018, Universidad de Valladolid, Departamento de Estadística e Investigación Operativa, Spain, June 5, 2018.

  • A. Suvorikova, Central limit theorem for Wasserstein barycenters of Gaussian measures, 4th Conference of the International Society for Nonparametric Statistics, June 11 - 15, 2018, University of Salerno, Italy, June 15, 2018.

  • A. Suvorikova, Central limit theorem for barycenters in 2-Wasserstein space, Ruhr-Universität Bochum, May 30, 2018.

  • A. Suvorikova, Construction of non-asymptotic confidence sets in 2-Wasserstein space, Structural Learning Seminar, Skolkovo Institute of Science and Technology, Moscow, Russian Federation, May 10, 2018.

  • A. Suvorikova, Gaussian process forecast with multidimensional distributional input, Haindorf Seminar 2018, January 23 - 27, 2018, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, January 25, 2018.

  • A. Suvorikova, Statistical inference with optimal transport, Spring School ``Structural Inference 2018'' and Closing Workshop, March 4 - 9, 2018.

  • A. Suvorikova, Statistical inference with optimal transport, WIAS-Day, February 22 - 23, 2018.

  • N. Buzun, Sein's method, Haindorf Seminar 2018, January 24 - 27, 2018, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, January 26, 2018.

  • M. Coghi, Mean field limit of interacting filaments for 3D Euler equations, Bielefeld-Edinburgh-Swansea Stochastic Spring, March 26 - 28, 2018, Universität Bielefeld, Fakultät für Mathematik, March 27, 2018.

  • M. Coghi, Mean field limit of interacting filaments for 3D Euler equations, BSDEs, Information and McKean-Vlasov equations, September 10 - 12, 2018, University of Leeds, School of Mathematics, UK, September 10, 2018.

  • M. Coghi, Non-local stochastic scalar conservation laws, Bielefeld Stochastic Afternoon, Universität Bielefeld, Fakultät für Mathematik, October 16, 2018.

  • M. Coghi, Pathwise McKean--Vlasov theory, 10th Oxford-Berlin Young Researchers Meeting O Applied Stochastic Analysis, November 29 - December 1, 2018, University of Oxford, Mathematical Institute, UK, December 1, 2018.

  • M. Maurelli, McKean-Vlasov SDEs with irregular drift: Large deviations for particle approximation, University of Oxford, Mathematical Institute, UK, March 5, 2018.

  • M. Maurelli, Sanov theorem for Brownian rough paths and an application to interacting particles, Università di Roma La Sapienza, Dipartimento di Matematica Guido Castelnuovo, Italy, January 18, 2018.

  • M. Maurelli , A McKean-Vlasov SDE with reflecting boundaries, CASA Colloquium, Eindhoven University of Technology, Department of Mathematics and Computer Science, Netherlands, January 10, 2018.

  • P. Pigato, Asymptotic analysis of rough volatility models, Probability Seminar, L'Università di Milano-Bicocca, Dipartimento di Matematica e Applicazioni, Italy, July 13, 2018.

  • P. Pigato, Asymptotic analysis of rough volatility models, Seminar of the Research Training Group 2131, Ruhr-Universität Bochum, Research Training Group 2131, June 25, 2018.

  • P. Pigato, Density and tube estimates for diffusion processes under Hormander-type conditions, Séminaire (de calcul) stochastique, Université de Strasbourg, Institut de recherche mathématique avancée), France, November 23, 2018.

  • P. Pigato, Estimation of piecewise-constant coefficients in a stochastic differential equation, The 40th Conference on Stochastic Processes and their Applications - SPA 2018, University of Gothenburg, Göteborg, Sweden, June 13, 2018.

  • P. Pigato, Faits stilisés et modelisation de la volatilité, École Polytechnique, Palaiseau, France, April 20, 2018.

  • P. Pigato, Faits stilisés et modelisation de la volatilitë, Seminaire, Institut de Science Financière et d'Assurances - Université Lyon 1, France, May 14, 2018.

  • P. Pigato, Precise asymptotics of rough stochastic volatility models, University of Trento, Department of Mathematics, November 16, 2018.

  • P. Pigato, Short dated option pricing under rough volatility, Berlin-Paris Young Researchers Workshop Stochastic Analysis with applications in Biology and Finance, May 2 - 4, 2018, Institut des Systèmes Complexes de Paris Ile-de-France (ISC-PIF), National Center for Scientific Research, Paris, France, May 4, 2018.

  • P. Pigato, Some elements of statistics and modeling for financial markets, University of Trento, Department of Mathmatics, Italy, December 19, 2018.

  • M. Redmann, Beyond the theory of ordinary differential equations, Seminar of the Deparment of Mathematics and Computer Science, University of Southern Denmark, Department of Mathematics and Computer Science, Odense, Denmark, February 22, 2018.

  • M. Redmann, Beyond the theory of ordinary differential equations, Institute's Seminar, February 22, 2018, University of Southern Denmark, Department of Mathematics and Computer Science (IMADA), Odense, Denmark, February 22, 2018.

  • M. Redmann, Numerical approximations of parabolic rough PDEs, Harmonic Analysis for Stochastic PDEs, July 10 - 13, 2018, Delft University of Technology, Netherlands, July 10, 2018.

  • M. Redmann, Solving linear parabolic rough partial differential equations, 13th German Probability and Statistics Days 2018, February 27 - March 2, 2018, Albert-Ludwigs-Universität Freiburg, Abteilung für Mathematische Stochastik, March 1, 2018.

  • M. Redmann, Solving parabolic rough partial differential equations using regression, 13th International Conference in Monte Carlo & Quasi-Monte Carlo Methods in Scientific Computing, July 1 - 6, 2018, University of Rennes, Faculty of Economics, France, July 5, 2018.

  • M. Redmann, Solving stochastic partial differential equations, Universität Greifswald, Institut für Mathematik und Informatik, April 19, 2018.

  • B. Stemper, Calibration of the rough Bergomi model via neural networks, 9th Annual ERC Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, June 14 - 16, 2018, WIAS Berlin, June 14, 2018.

  • B. Stemper, Pricing under rough volatility, Premia Meeting, Centre de recherche INRIA Paris, France, March 12, 2018.

  • B. Stemper, Short-time near-the-money skew in rough volatility models, 10th World Congress of the Bachelier Finance Society, July 16 - 20, 2018, Bachelier Finance Society, Dublin, Ireland, July 19, 2018.

  • D. Dvinskikh, Decentralize and randomize: Faster algorithm for Wasserstein barycenters, Thirty-second Conference on Neural Information Processing Systems, December 3 - 8, 2018, Montréal, Canada, December 6, 2018.

  • CH. Bayer, Shortime near-the-money skew in rough fractional volatility models, 9-th International Workshop on Applied Probability, June 18 - 21, 2018, Eörvös Loránd University (ELU), Budapest, Hungary, June 19, 2018.

  • CH. Bayer, Smoothing the payoff for computation of basket options, Berlin-Paris Young Researchers Workshop Stochastic Analysis with applications in Biology and Finance, May 2 - 4, 2018, Institut des Systèmes Complexes de Paris Ile-de-France (ISC-PIF), National Center for Scientific Research, Paris, France, May 3, 2018.

  • CH. Bayer, Smoothing the payoff for computation of basket options, 13th International Conference in Monte Carlo & Quasi-Monte Carlo Methods in Scientific Computing, July 1 - 6, 2018, University of Rennes, Faculty of Economics, France, July 3, 2018.

  • CH. Bayer, Smoothing the payoff for computation of basket options, Stochastic Methods in Finance and Physics, July 23 - 27, 2018, National Technical University of Athens, Department of Mathematics, Heraklion, Greece.

  • P. Dvurechennky, Universal method for variational inequalities with Holder-continuous monotone operator, Mathematics and Image Analysis, January 15 - 17, 2018.

  • P. Dvurechensky, D. Dvinskikh, Decentralize and randomize: Faster algorithm for Wasserstein barycenters, Thirty-Second Conference On Neural Information Processing Systems, Montréal, Canada, December 3 - 8, 2018.

  • P. Dvurechensky, Computational optimal transport: Accelerated gradient descent vs Sinkhorn, ISMP 2018 Bordeaux, July 1 - 6, 2018, University of Bordeaux, Institut de Mathématiques, France, July 4, 2018.

  • P. Dvurechensky, Computational optimal transport: Accelerated gradient descent vs Sinkhorn's algorithm, Statistical Optimal Transport, July 24 - 25, 2018, Skolkovo Institute of Science and Technology, Moskau, Russian Federation, July 25, 2018.

  • P. Dvurechensky, Computational optimal transport: Complexity by accelerated gradient descent is better than by Sinkhorn's algorithm, The 35th International Conference on Machine Learning (ICML 2018), Stockholm, Sweden, July 9 - 15, 2018.

  • P. Dvurechensky, Computational optimal transport: Complexity by accelerated gradient descent is better than by Sinkhorn's algorithm, The 35th International Conference on Machine Learning (ICML 2018), July 9 - 15, 2018, International Machine Learning Society (IMLS), Stockholm, Sweden, July 11, 2018.

  • P. Dvurechensky, Decentralize and randomize: Faster algorithm for Wasserstein barycenters, Thirty-second Conference on Neural Information Processing Systems, December 3 - 8, 2018, Montréal, Canada, December 6, 2018.

  • P. Dvurechensky, Faster algorithms for (regularized) optimal transport, Grenoble Optimization Days 2018, June 28 - 29, 2018, Université Grenoble Alpes, Laboratoire Jean Kuntzmann, France, June 29, 2018.

  • P. Dvurechensky, Faster algorithms for (regularized) optimal transport, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow region, Russian Federation, March 30, 2018.

  • P. Dvurechensky, Primal-dual methods for solving infinite-dimensional games, Games, Dynamics and Optimization GDO2018, March 13 - 15, 2018, Universität Wien, Fakultät für Mathematik, Austria, March 15, 2018.

  • P. Dvurechensky, Statistical optimal transport, WIAS-Day, February 23, 2018.

  • P. Friz, Analysis of rough volatility via rough paths / regularity structures, METE - Mathematics and Economics: Trends and Explorations. A conference celebrating Mete Soner's 60th birthday and his contributions to Analysis, Control, Finance and Probability, June 4 - 8, 2018, Eidgenössische Technische Hochschule Zürich, Forschungsinstitut für Mathematik, Switzerland, June 5, 2018.

  • P. Friz, From rough paths and regularity structures to short dated option pricing under rough volatility, Workshop on Mathematical Finance and related Issues, Osaka University, Nakanoshima Center, Japan, March 15, 2018.

  • P. Friz, Rough path analysis of rough volatility, 9-th International Workshop On Applied Probability, June 18 - 22, 2018, Eörvös Loránd University (ELU), Budapest, Hungary, June 18, 2018.

  • P. Friz, Rough path analysis of rough volatility, Stochastic Analysis Seminar, Imperial College London, Department of Mathematics, Stochastic Analysis Group, UK, February 13, 2018.

  • P. Friz, Rough paths, stochastics and PDE's, ECMath Colloquium, July 6, 2018, Humboldt-Universität zu Berlin, July 6, 2018.

  • P. Friz, Stepping stoch vol and related topics, 25th Global Derivatives Trading & Risk Management 2018, Volatility modelling & Trading, May 14 - 18, 2018, Lissabon, Portugal, May 16, 2018.

  • P. Friz, Varieties of signature tensors, Workshop on Stochastic Analysis, Geometry and Statistics, June 21 - 22, 2018, Imperial College London, UK, June 22, 2018.

  • A. Koziuk, Gaussian comparison on a family of Eucledian balls, Haindorf Seminar 2018, January 23 - 27, 2018, Humboldt-Universität zu Berlin, Wirtschaftswissenschaftliche Fakultät, Hejnice, Czech Republic, January 26, 2018.

  • P. Mathé, Bayesian inverse problems with non-commuting operators, Chemnitz September of Applied Mathematics 2018, Chemnitz Symposium on Inverse Problems, September 27 - 28, 2018, Technische Universität Chemnitz, Fakultät für Mathematik, September 28, 2018.

  • P. Mathé, Complexity of linear ill-posed problems in Hilbert space, Stochastisches Kolloquium, Georg-August-Universität Göttingen, Institut für Mathematische Stochastik, February 7, 2018.

  • P. Mathé, Complexity of linear ill-posed problems in Hilbert space, University of Cyprus, Department of Mathematics and Statistics, Nicosia, Cyprus, December 7, 2018.

  • P. Mathé, Relating posterior contraction for direct and inverse Bayesian problems by the modulus of continuity, International Conference on Inverse Problems, October 12 - 14, 2018, Fudan University, School of Mathematical Sciences, Shanghai, China, October 13, 2018.

  • H.-J. Mucha, Multivariate statistische Auswertung von archäometrischen Messwerten von Mammut-Elfenbein, Tagung Archäometrie und Denkmalpflege 2018, Deutsches Elektronen-Synchrotron DESY, Hamburg, March 20 - 24, 2018.

  • J. Polzehl, High resolution magnetic resonance imaging experiments - lessons in nonlinear statistical modeling, 3rd Leibniz MMS Days, February 28 - March 2, 2018, "Wissenschaftszentrum Leipzig", March 1, 2018.

  • J. Polzehl, Modeling high dimensional data, Leibniz MMS Summer School 2018, September 3 - 7, 2018, Leibniz MMS Network, Oberwolfach.

  • J. Polzehl, Towards in-vivo histology of the brain - some statistical contributions, CMRR Seminar, University of Minnesota, Center for Magnetic Resonance Research (CMRR), Minneapolis, USA, October 15, 2018.

  • V. Spokoiny, Adaptive nonparametric clustering, Multiscale Problems in Materials Science and Biology: Analysis and Computation, January 8 - 12, 2018, Tsinghua University, Yau Mathematical Sciences Center, Sanya, Hainan, China, January 10, 2018.

  • V. Spokoiny, Bootstrap confidence sets for spectral projectors of sample covariance, 12th International Vilnius Conference on Probability Theory and Mathematical Statistics and 2018 IMS Annual Meeting on Probability and Statistics, July 2 - 6, 2018, Vilnius University, Lithuanian Mathematical Society and the Institute of Mathematical Statistics, Lithuania, July 5, 2018.

  • V. Spokoiny, Gaussian process forecast with multidimensional distributional input, 4th Conference of the International Society for Nonparametric Statistics, June 11 - 15, 2018, University of Salerno, Italy, June 15, 2018.

  • V. Spokoiny, Inference for spectral projectors, Statistical Inference for Structured High-dimensional Models, March 11 - 16, 2018, Mathematisches Forschungsinstitut Oberwolfach, March 14, 2018.

  • V. Spokoiny, Large ball probability with applications in statistics, Mathematical Workshop of the School of Applied Mathematics and Computer Science, Moscow Institute of Physics and Technology, Dolgoprudny, Russian Federation, November 30, 2018.

  • V. Spokoiny, Large ball probability with applications to statistical inference, 6th Princeton Day of Statistics, November 9, 2018, Princeton University, Department of Operations Research and Financial Engineering, USA, November 9, 2018.

  • V. Spokoiny, Manifold learning: Theory and applications, Pennsylvania State University, Department of Mathematics, State College, USA, November 14, 2018.

  • V. Spokoiny, Prior impact in Bayesian inference, Internatioal Seminar & Workshop:``Stochastic dynamics on large networks: Prediction and inference", October 15 - 16, 2018, Max-Planck-Institut für Physik komplexer Systeme, Dresden, October 15, 2018.

  • V. Spokoiny, Structured nonparametric and high-dimensional statistics, 2018 Joint Meeting of the Korean Mathematical Society and the German Mathematical Society, October 3 - 6, 2018, Korean Mathematical Society, Seoul, Korea (Republic of), October 5, 2018.

  • K. Tabelow, Structural adaptation for noise reduction in magnetic resonance imaging, SIAM Conference on Imaging Science, Minisymposium MS5 ``Learning and Adaptive Approaches in Image Processing'', June 5 - 8, 2018, Bologna, Italy, June 5, 2018.

External Preprints

  • A. Kroshnin, D. Dvinskikh, P. Dvurechensky, N. Tupitsa, C. Uribe, On the complexity of approximating Wasserstein barycenter, Preprint no. . arXiv:1901.08686, Cornell University Library, arXiv.org, 2019.

  • J.Y. Park, J. Polzehl, S. Chatterjee, A. Brechmann, ET AL., Semiparametric modeling of time-varying activation and connectivity in task-based fMRI data, Discussion paper, http://works.bepress.com/mfiecas/20, 2018.

  • A. Ivanova, P. Dvurechensky, A. Gasnikov, Composite optimization for the resource allocation problem, Preprint no. arXiv:1810.00595, Cornell University Library, arXiv.org, 2018.
    Abstract
    In this paper we consider resource allocation problem stated as a convex minimization problem with linear constraints. To solve this problem, we use gradient and accelerated gradient descent applied to the dual problem and prove the convergence rate both for the primal iterates and the dual iterates. We obtain faster convergence rates than the ones known in the literature. We also provide economic interpretation for these two methods. This means that iterations of the algorithms naturally correspond to the process of price and production adjustment in order to obtain the desired production volume in the economy. Overall, we show how these actions of the economic agents lead the whole system to the equilibrium.

  • C. A. Uribe, D. Dvinskikh, P. Dvurechensky, A. Gasnikov, A. Nedić, Distributed computation of Wasserstein barycenters over networks, Preprint no. arXiv:1803.02933, Cornell University Library, arXiv.org, 2018.

  • F. Bachoc, A. Suvorikova , J.-M. Loubes, V. Spokoiny, Gaussian process forecast with multidimensional distributional entries, Preprint no. arXiv:1805.00753v1, Cornell University Library, arXiv.org, 2018.

  • D. Belomestny, J.G.M. Schoenmakers, V. Spokoiny, Y. Tavyrikov, Optimal stopping via reinforced regression, Preprint no. arXiv:1808.02341v2, Cornell University Library, arXiv.org, 2018.

  • M. Boeckel, V. Spokoiny, A. Suvorikova, Multivariate Brenier cumulative distribution functions and their application to non-parametric testing, Preprint no. arXiv:1809.04090v1, Cornell University Library, arXiv.org, 2018.

  • I. Chevyrev, P. Friz, A. Korepanov, I. Melbourne, H. Zhang, Multiscale systems, homogenization, and rough paths, Preprint no. arXiv:1712.01343, Cornell University Library, arXiv.org, 2018.

  • F. Götze , A. Naumov, V. Spokoiny, V. Ulyanov, Large ball probability, Gaussian comparison and anti-concentration, Preprint no. arXiv:1708.08663v2, Cornell University Library, arXiv.org, 2018.

  • B. Hofmann, S. Kindermann, P. Mathé, Penalty-based smoothness conditions in convex variational regularization, Preprint no. arXiv:1805.01320, Cornell University Library, arXiv.org, 2018.
    Abstract
    The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.

  • Y. Nesterov, A. Gasnikov, S. Guminov, P. Dvurechensky, Primal-dual accelerated gradient descent with line search for convex and nonconvex optimization problems, Preprint no. arXiv:1809.05895, Cornell University Library, arXiv.org, 2018.
    Abstract
    In this paper a new variant of accelerated gradient descent is proposed. The proposed method does not require any information about the objective function, uses exact line search for the practical accelerations of convergence, converges according to the well-known lower bounds for both convex and non-convex objective functions, possesses primal-dual properties and can be applied in the non-euclidian set-up. As far as we know this is the first such method possessing all of the above properties at the same time. We demonstrate how in practice one can efficiently use the combination of line-search and primal-duality by considering a convex optimization problem with a simple structure (for example, affinely constrained).

  • N. Puchkin, V. Spokoiny, Adaptive multiclass nearest neighbor classifier, Preprint no. arXiv:1804.02756, Cornell University Library, arXiv.org, 2018.

  • K. Andzhey, V. Spokoiny, Instrumental variables regression, Preprint no. arXiv:1806.06111v1, Cornell University Library, arXiv.org, 2018.

  • CH. Bayer, R. Tempone, S. Wolfers, Pricing american options by exercise rate optimization, Preprint no. arXiv:1809.07300, Cornell University Library, arXiv.org, 2018.
    Abstract
    We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal emphexercise regions, which consist of points in time and space at which the option is exercised. In contrast, our method determines emphexercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. Starting in a neutral strategy with constant exercise rate then allows us to globally optimize this function in a gradual manner. Numerical experiments on vanilla put options in the multivariate Black--Scholes model and preliminary theoretical analysis underline the efficiency of our method both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of exercise rates. Finally, the flexibility of our method is demonstrated by numerical experiments on max call options in the Black--Scholes model and vanilla put options in Heston model and the non-Markovian rough Bergomi model.

  • P. Dvurechensky, A. Gasnikov, E. Gorbunov, An accelerated directional derivative method for smooth stochastic convex optimization, Preprint no. arXiv: 1804.02394, Cornell University Library, arXiv.org, 2018.
    Abstract
    We consider smooth stochastic convex optimization problems in the context of algorithms which are based on directional derivatives of the objective function. This context can be considered as an intermediate one between derivative-free optimization and gradient-based optimization. We assume that at any given point and for any given direction, a stochastic approximation for the directional derivative of the objective function at this point and in this direction is available with some additive noise. The noise is assumed to be of an unknown nature, but bounded in the absolute value. We underline that we consider directional derivatives in any direction, as opposed to coordinate descent methods which use only derivatives in coordinate directions. For this setting, we propose a non-accelerated and an accelerated directional derivative method and provide their complexity bounds. Despite that our algorithms do not use gradient information, our non-accelerated algorithm has a complexity bound which is, up to a factor logarithmic in problem dimension, similar to the complexity bound of gradient-based algorithms. Our accelerated algorithm has a complexity bound which coincides with the complexity bound of the accelerated gradient-based algorithm up to a factor of square root of the problem dimension, whereas for existing directional derivative methods this factor is of the order of problem dimension. We also extend these results to strongly convex problems. Finally, we consider derivative-free optimization as a particular case of directional derivative optimization with noise in the directional derivative and obtain complexity bounds for non-accelerated and accelerated derivative-free methods. Complexity bounds for these algorithms inherit the gain in the dimension dependent factors from our directional derivative methods.

  • P. Dvurechensky, A. Gasnikov, E. Gorbunov, An accelerated method for derivative-free smooth stochastic convex optimization, Preprint no. arXiv: 1802.09022, Cornell University Library, arXiv.org, 2018.
    Abstract
    We consider an unconstrained problem of minimization of a smooth convex function which is only available through noisy observations of its values, the noise consisting of two parts. Similar to stochastic optimization problems, the first part is of a stochastic nature. On the opposite, the second part is an additive noise of an unknown nature, but bounded in the absolute value. In the two-point feedback setting, i.e. when pairs of function values are available, we propose an accelerated derivative-free algorithm together with its complexity analysis. The complexity bound of our derivative-free algorithm is only by a factor of n??? larger than the bound for accelerated gradient-based algorithms, where n is the dimension of the decision variable. We also propose a non-accelerated derivative-free algorithm with a complexity bound similar to the stochastic-gradient-based algorithm, that is, our bound does not have any dimension-dependent factor. Interestingly, if the solution of the problem is sparse, for both our algorithms, we obtain better complexity bound if the algorithm uses a 1-norm proximal setup, rather than the Euclidean proximal setup, which is a standard choice for unconstrained problems.

  • P. Dvurechensky, A. Gasnikov, A. Kroshnin, Computational optimal transport: Complexity by accelerated gradient descent is better than by Sinkhorn's algorithm, Preprint no. arXiv: 1802.04367, Cornell University Library, arXiv.org, 2018.

  • P. Dvurechensky, A. Gasnikov, F. Stonyakin, A. Titov, Generalized mirror prox: Solving variational inequalities with monotone operator, inexact Oracle, and unknown Hölder parameters, Preprint no. arXiv:1806.05140, Cornell University Library, arXiv.org, 2018.

  • P. Dvurechensky, D. Dvinskikh, A. Gasnikov, C.A. Uribe, A. Nedić, Decentralize and randomize: Faster algorithm for Wasserstein barycenters, Preprint no. arXiv:1806.03915, Cornell University Library, arXiv.org, 2018.
    Abstract
    We study the problem of decentralized distributed computation of a discrete approximation for regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. Particularly, we assume that there is a network of agents/machines/computers where each agent holds a private continuous probability measure, and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop and theoretically analyze a novel accelerated primal-dual stochastic gradient method for general stochastic convex optimization problems with linear equality constraints. Then, we apply this method to the decentralized distributed optimization setting to propose a new algorithm for the distributed semi-discrete regularized Wasserstein barycenter problem. The proposed algorithm can be executed over arbitrary networks that are undirected, connected and static, using the local information only. Moreover, we show explicit non-asymptotic complexity in terms of the problem parameters. Finally, we show the effectiveness of our method on the distributed computation of the regularized Wasserstein barycenter of univariate Gaussian and von Mises distributions, as well as on some applications to image aggregation.

  • P. Friz, P. Gassiat, P. Pigato, Precise asymptotics: Robust stochastic volatility models, Preprint no. arXiv:1811.00267, Cornell University Library, arXiv.org, 2018.

  • A. Koziuk, V. Spokoiny, Toolbox: Gaussian comparison on Eucledian balls, Preprint no. arXiv:1804.00601v1, Cornell University Library, arXiv.org, 2018.

  • P. Mathé, Bayesian inverse problems with non-commuting operators, Preprint no. arXiv:1801.09540, Cornell University Library, arXiv.org, 2018.
    Abstract
    The Bayesian approach to ill-posed operator equations in Hilbert space recently gained attraction. In this context, and when the prior distribution is Gaussian, then two operators play a significant role, the one which governs the operator equation, and the one which describes the prior covariance. Typically it is assumed that these operators commute. Here we extend this analysis to non-commuting operators, replacing the commutativity assumption by a link condition. We discuss its relation to the commuting case, and we indicate that this allows to use interpolation type results to obtain tight bounds for the contraction of the posterior Gaussian distribution towards the data generating element.