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Publikationen

Artikel in Referierten Journalen

  • E. Gladin, A. Gasnikov, P. Dvurechensky, Accuracy certificates for convex minimization with inexact Oracle, Journal of Optimization Theory and Applications, 204, pp. 1/1-1/23, DOI 10.1007/s10957-024-02599-9 .
    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 first-order 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 by-product, 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.

  • A. Akhavan, D. Gogolashvili, A.B. Tsybakov, Estimating the minimizer and the minimum value of a regression function under passive design, Journal of Machine Learning Research (JMLR). MIT Press, Cambridge, MA. English, English abstracts., 25 (2024), pp. 1--37.

  • D. Belomestny, J.G.M. Schoenmakers, Primal-dual regression approach for Markov decision processes with general state and action space, SIAM Journal on Control and Optimization, 62, pp. 650--679, DOI 10.1137/22M1526010 .
    Abstract
    We develop a regression based primal-dual 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 duality-based methods for optimal control problems in the literature, the Monte Carlo procedures here involved do not require nested simulations.

  • I.F.T. Ceja, Th. Gladytz, L. Starke, K. Tabelow, Th. Niendorf, H.M. Reimann, Precision fMRI and cluster-failure in the individual brain, Human Brain Mapping, 45 (2024), pp. e26813/1-- e26813/20, DOI 10.1002/hbm.26813 .

  • G. Dong, M. Flaschel, M. Hintermüller, K. Papafitsoros, C. Sirotenko, K. Tabelow, Data--driven methods for quantitative imaging, GAMM-Mitteilungen, pp. e202470014/1-- e202470014/35, DOI 10.1002/gamm.202470014 .
    Abstract
    In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative Magnetic Resonance Imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps.

  • E. Gorbunov, M. Danilova, I. Shibaev, P. Dvurechensky, A. Gasnikov, High-probability complexity bounds for non-smooth stochastic convex optimization with heavy-tailed noise, Journal of Optimization Theory and Applications, 203 (2024), pp. 2679--2738, DOI 10.1007/s10957-024-02533-z .
    Abstract
    Thanks to their practical efficiency and random nature of the data, stochastic first-order methods are standard for training large-scale machine learning models. Random behavior may cause a particular run of an algorithm to result in a highly suboptimal objective value, whereas theoretical guarantees are usually proved for the expectation of the objective value. Thus, it is essential to theoretically guarantee that algorithms provide small objective residual with high probability. Existing methods for non-smooth stochastic convex optimization have complexity bounds with the dependence on the confidence level that is either negative-power or logarithmic but under an additional assumption of sub-Gaussian (light-tailed) noise distribution that may not hold in practice, e.g., in several NLP tasks. In our paper, we resolve this issue and derive the first high-probability convergence results with logarithmic dependence on the confidence level for non-smooth convex stochastic optimization problems with non-sub-Gaussian (heavy-tailed) noise. To derive our results, we propose novel stepsize rules for two stochastic methods with gradient clipping. Moreover, our analysis works for generalized smooth objectives with Hölder-continuous gradients, and for both methods, we provide an extension for strongly convex problems. Finally, our results imply that the first (accelerated) method we consider also has optimal iteration and oracle complexity in all the regimes, and the second one is optimal in the non-smooth setting.

  • C. Kiss, L. Németh, B. Vető , Modelling the age distribution of longevity leaders, Scientific Reports, 14 (2024), pp. 20592/1--20592/13, DOI 10.1038/s41598-024-71444-w .
    Abstract
    Human longevity leaders with remarkably long lifespan play a crucial role in the advancement of longevity research. In this paper, we propose a stochastic model to describe the evolution of the age of the oldest person in the world by a Markov process, in which we assume that the births of the individuals follow a Poisson process with increasing intensity, lifespans of individuals are independent and can be characterized by a gamma?Gompertz distribution with time-dependent parameters. We utilize a dataset of the world?s oldest person title holders since 1955, and we compute the maximum likelihood estimate for the parameters iteratively by numerical integration. Based on our preliminary estimates, the model provides a good fit to the data and shows that the age of the oldest person alive increases over time in the future. The estimated parameters enable us to describe the distribution of the age of the record holder process at a future time point.

  • R.J.A. Laeven, J.G.M. Schoenmakers, N.F.F. Schweizer, M. Stadje, Robust multiple stopping -- A duality approach, Mathematics of Operations Research, published online on 15.05.2024, DOI 10.1287/moor.2021.0237 .
    Abstract
    In this paper we develop a solution method for general optimal stopping problems. Our general setting allows for multiple exercise rights, i.e., optimal multiple stopping, for a robust evaluation that accounts for model uncertainty, and for general reward processes driven by multi-dimensional jump-diffusions. Our approach relies on first establishing robust martingale dual representation results for the multiple stopping problem which satisfy appealing path-wise optimality (almost sure) properties. Next, we exploit these theoretical results to develop upper and lower bounds which, as we formally show, not only converge to the true solution asymptotically, but also constitute genuine upper and lower bounds. We illustrate the applicability of our general approach in a few examples and analyze the impact of model uncertainty on optimal multiple stopping strategies.

  • 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, 92, pp. 69--81, DOI 10.1002/mrm.30034 .

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

  • CH. Bayer, S. Breneis, Efficient option pricing in the rough Heston model using weak simulation schemes, Quantitative Finance, 24, pp. 1247--1261, DOI 10.1080/14697688.2024.2391523 .
    Abstract
    We provide an efficient and accurate simulation scheme for the rough Heston model in the standard ($H>0$) as well as the hyper-rough regime ($H > -1/2$). The scheme is based on low-dimensional 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 path-dependent European options and Bermudan options show the method's accuracy and efficiency.

  • CH. Bayer, D. Belomestny, O. Butkovsky, J.G.M. Schoenmakers, A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean--Vlasov models, Finance and Stochastics, 28, pp. 1147--1178, DOI 10.1007/s00780-024-00541-5 .
    Abstract
    Motivated by the challenges related to the calibration of financial models, we consider the problem of solving numerically a singular McKean-Vlasov equation, which represents a singular local stochastic volatility model. Whilst such models are quite popular among practitioners, unfortunately, its well-posedness has not been fully understood yet and, in general, is possibly not guaranteed at all. We develop a novel regularization approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularized model is well-posed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models. Our results are also applicable to more general McKean--Vlasov equations.

  • CH. Bayer, Ch. Ben Hammouda, R. Tempone, Multilevel Monte Carlo with numerical smoothing for robust and efficient computation of probabilities and densities, SIAM Journal on Scientific Computing, 46, pp. A1514--A1548, DOI 10.1137/22M1495718 .

  • P. Dvurechensky, P. Ostroukhov, A. Gasnikov, C.A. Uribe, A. Ivanova, Near-optimal tensor methods for minimizing the gradient norm of convex functions and accelerated primal-dual tensor methods, Optimization Methods & Software, 39, pp. 1068--1103, DOI 10.1080/10556788.2023.2296443 .

  • P. Dvurechensky, M. Staudigl, Hessian barrier algorithms for non-convex conic optimization, Mathematical Programming. A Publication of the Mathematical Programming Society, published online on 04.03.2024, DOI 10.1007/s10107-024-02062-7 .
    Abstract
    We consider the minimization of a continuous function over the intersection of a regular cone with an affine set via a new class of adaptive first- and second-order optimization methods, building on the Hessian-barrier techniques introduced in [Bomze, Mertikopoulos, Schachinger, and Staudigl, Hessian barrier algorithms for linearly constrained optimization problems, SIAM Journal on Optimization, 2019]. Our approach is based on a potential-reduction mechanism and attains a suitably defined class of approximate first- or second-order KKT points with the optimal worst-case iteration complexity O(??2) (first-order) and O(??3/2) (second-order), respectively. A key feature of our methodology is the use of self-concordant barrier functions to construct strictly feasible iterates via a disciplined decomposition approach and without sacrificing on the iteration complexity of the method. To the best of our knowledge, this work is the first which achieves these worst-case complexity bounds under such weak conditions for general conic constrained optimization problems.

  • P.K. Friz, T. Lyons, A. Seigal, Rectifiable paths with polynomial log-signature are straight lines, Bulletin of the London Mathematical Society, 56, pp. 2922--2934, DOI 10.1112/blms.13110 .
    Abstract
    The signature of a rectifiable path is a tensor series in the tensor algebra whose coefficients are definite iterated integrals of the path. The signature characterizes the path up to a generalized form of reparameterization. It is a classical result of Chen that the log-signature (the logarithm of the signature) is a Lie series. A Lie series is polynomial if it has finite degree. We show that the log-signature is polynomial if and only if the path is a straight line up to reparameterization. Consequently, the log-signature of a rectifiable path either has degree one or infinite support. Though our result pertains to rectifiable paths, the proof uses rough path theory, in particular that the signature characterizes a rough path up to reparameterization.

Beiträge zu Sammelwerken

  • E. Gorbunov, A. Sadiev, M. Danilova, S. Horváth, G. Gidel, P. Dvurechensky, A. Gasnikov, P. Richtárik, High-probability convergence for composite and distributed stochastic minimization and variational inequalities with heavy-tailed noise, in: Proceedings of The 41st International Conference on Machine Learning, 235 of Proceedings of Machine Learning Research, 2024, pp. 15951--16070.

  • L. Kastner, K. Tabelow, Ach ja --- Forschungsdatenmanagement, 32 of Mitteilungen der Deutschen Mathematiker-Vereinigung, Walter de Gruyter GmbH, Berlin/Boston, 2024, pp. 102--104, DOI 10.1515/dmvm-2024-0031 .

  • B. Schembera, F. Wübbeling, H. Kleikamp, Ch. Biedinger, 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, in: Metadata and Semantics Research, E. Garoufallou, F. Sartori, eds., Communications in Computer and Information Science, Springer, Cham, 2024, pp. 161--168, DOI 10.1007/978-3-031-65990-4_14 .
    Abstract
    In applied mathematics and related disciplines, the modeling-simulationoptimization 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.

  • P. Dvurechensky, M. Staudigl, Barrier algorithms for constrained non-convex optimization, in: Proceedings of The 41st International Conference on Machine Learning, 235 of Proceedings of Machine Learning Research, 2024, pp. 12190--12214.

  • P. Dvurechensky, J.-J. Zhu, Analysis of kernel mirror prox for measure optimization, in: Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, S. Dasgupta, S. Mandt, Y. Li, eds., 238 of Proceedings of Machine Learning Research, 2024, pp. 2350--2358.
    Abstract
    Kernel mirror prox and RKHS gradient flow for mixed functional Nash equilibrium Pavel Dvurechensky , Jia-Jie 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 infinite-dimensional continuous- limit optimization dynamics, we propose the Interacting Wasserstein-Kernel Gradient Flow, which includes the RKHS flow that is much less common than the Wasserstein gradient flow but enjoys a much simpler convexity structure. Time-discretizing this gradient flow, we propose a primal-dual 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 primal-dual convergence analysis for DRO with probability-metric constraints.

Preprints, Reports, Technical Reports

  • CH. Bayer, B. Djehiche, E. Rezvanova, R. Tempone, Continuous time stochastic optimal control under discrete time partial observations, Preprint no. 3168, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3168 .
    Abstract, PDF (1524 kByte)
    This work addresses stochastic optimal control problems where the unknown state evolves in continu- ous time while partial, noisy, and possibly controllable measurements are only available in discrete time. We develop a framework for controlling such systems, focusing on the measure-valued process of the system's state and the control actions that depend on noisy and incomplete data. Our approach uses a stochastic optimal control framework with a probability measure-valued state, which accommodates noisy measure- ments and integrates them into control decisions through a Bayesian update mechanism. We characterize the control optimality in terms of a sequence of interlaced Hamilton Jacobi Bellman (HJB) equations coupled with controlled impulse steps at the measurement times. For the case of Gaussian-controlled processes, we derive an equivalent HJB equation whose state variable is finite-dimensional, namely the state's mean and covariance. We demonstrate the effectiveness of our methods through numerical examples. These include control under perfect observations, control under no observations, and control under noisy observa- tions. Our numerical results highlight significant differences in the control strategies and their performance, emphasizing the challenges and computational demands of dealing with uncertainty in state observation.

  • CH. Bayer, M. Redmann, Dimension reduction for path signatures, Preprint no. 3163, WIAS, Berlin, 2025, DOI 10.20347/WIAS.PREPRINT.3163 .
    Abstract, PDF (740 kByte)
    This paper focuses on the mathematical framework for reducing the complexity of models using path signatures. The structure of these signatures, which can be interpreted as collections of iterated integrals along paths, is discussed and their applications in areas such as stochastic differential equations (SDEs) and financial modeling are pointed out. In particular, exploiting the rough paths view, solutions of SDEs continuously depend on the lift of the driver. Such continuous mappings can be approximated using (truncated) signatures, which are solutions of high-dimensional linear systems. In order to lower the complexity of these models, this paper presents methods for reducing the order of high-dimensional truncated signature models while retaining essential characteristics. The derivation of reduced models and the universal approxi- mation property of (truncated) signatures are treated in detail. Numerical examples, including applications to the (rough) Bergomi model in financial markets, illustrate the proposed reduction techniques and highlight their effectiveness.

  • E. Abi Jaber, Ch. Bayer, S. Breneis, State spaces of multifactor approximations of nonnegative Volterra processes, Preprint no. 3162, WIAS, Berlin, 2025.
    Abstract, PDF (594 kByte)
    We show that the state spaces of multifactor Markovian processes, coming from approximations of nonnegative Volterra processes, are given by explicit linear transformation of the nonnegative orthant. We demonstrate the usefulness of this result for applications, including simulation schemes and PDE methods for nonnegative Volterra processes.

  • A. Kroshnin, A. Suvorikova, Bernstein-type and Bennett-type inequalities for unbound matrix martingales, Preprint no. 3146, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3146 .
    Abstract, PDF (407 kByte)
    We derive explicit Bernstein-type and Bennett-type concentration inequalities for matrix-valued supermartingale processes with unbounded observations. Specifically, we assume that the $psi_alpha$- Orlicz (quasi-)norms of their difference process are bounded for some $alpha > 0$. As corollaries, we prove an empirical version of Bernstein's inequality and an extension of the bounded differences inequality, also known as McDiarmid's inequality.

  • A. Kroshnin, V. Spokoiny, A. Suvorikova, Generalized bootstrap in the Bures-Wasserstein space, Preprint no. 3145, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3145 .
    Abstract, PDF (2175 kByte)
    This study focuses on finite-sample inference on the non-linear Bures-Wasserstein manifold and introduces a generalized bootstrap procedure for estimating Bures-Wasserstein barycenters. We provide non-asymptotic statistical guarantees for the resulting bootstrap confidence sets. The proposed approach incorporates classical resampling methods, including the multiplier bootstrap highlighted as a specific example. Additionally, the paper compares bootstrap-based confidence sets with asymptotic confidence sets obtained in the work of Kroshnin et al. [2021], evaluating their statistical performance and computational complexities. The methodology is validated through experiments on synthetic datasets and real-world applications.

  • P. Bank, Ch. Bayer, P.P. Hager, S. Riedel, T. Nauen, Stochastic control with signatures, Preprint no. 3113, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3113 .
    Abstract, PDF (391 kByte)
    This paper proposes to parameterize open loop controls in stochastic optimal control problems via suit- able classes of functionals depending on the driver's path signature, a concept adopted from rough path integration theory. We rigorously prove that these controls are dense in the class of progressively mea- surable controls and use rough path methods to establish suitable conditions for stability of the controlled dynamics and target functional. These results pave the way for Monte Carlo methods to stochastic optimal control for generic target functionals and dynamics. We discuss the rather versatile numerical algorithms for computing approximately optimal controls and verify their accurateness in benchmark problems from Mathematical Finance.

  • G. Dong, M. Flaschel, M. Hintermüller, K. Papafitsoros, C. Sirotenko, K. Tabelow, Data--driven methods for quantitative imaging, Preprint no. 3105, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3105 .
    Abstract, PDF (7590 kByte)
    In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative Magnetic Resonance Imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill-posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically-oriented overview on how data-driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps.

  • E. Abi Jaber, Ch. Cuchiero, L. Pelizzari, S. Pulido, S. Svaluto-Ferro, Polynomial Volterra processes, Preprint no. 3098, WIAS, Berlin, 2024, DOI 10.20347/WIAS.PREPRINT.3098 .
    Abstract, PDF (397 kByte)
    We study the class of continuous polynomial Volterra processes, which we define as solutions to stochas- tic Volterra equations driven by a continuous semimartingale with affine drift and quadratic diffusion matrix in the state of the Volterra process. To demonstrate the versatility of possible state spaces within our framework, we construct polynomial Volterra processes on the unit ball. This construction is based on a stochastic invariance principle for stochastic Volterra equations with possibly singular kernels. Similarly to classical polynomial processes, polynomial Volterra processes allow for tractable expressions of the mo- ments in terms of the unique solution to a system of deterministic integral equations, which reduce to a system of ODEs in the classical case. By applying this observation to the moments of the finite-dimensional distributions we derive a uniqueness result for polynomial Volterra processes. Moreover, we prove that the moments are polynomials with respect to the initial condition, another crucial property shared by classical polynomial processes. The corresponding coefficients can be interpreted as a deterministic dual process and solve integral equations dual to those verified by the moments themselves. Additionally, we obtain a representation of the moments in terms of a pure jump process with killing, which corresponds to another non-deterministic dual process.

  • 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 mesh-type approach for tackling discrete-time, finite-horizon 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 “semi-tractable” 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 Linear-Quadratic Gaussian (LQG) control problems.

  • L. Schmitz, N. Tapia, Free generators and Hoffman's isomorphism for the two-parameter 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 two-parameter Hoffman exponential, we provide three classes of isomorphisms between the underlying two-parameter shuffle and quasi-shuffle algebras. In particular, we provide a Hopf algebraic connection to the (classical, one-parameter) shuffle algebra over the extended alphabet of connected matrix compositions.

  • C. Bellingeri, E. Ferrucci, N. Tapia, Branched Itô formula and natural Itô--Stratonovich 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 intrinsically-defined projection of the Connes-Kreimer 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- of-variable formula for solutions to rough differential equations. This formula, which is realised by means of an explicit morphism from the Grossman-Larson Hopf algebra to the Hopf algebra of differential operators, restricts to the well-known 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 Itô-Stratonovich correction formula for semimartingales. We place special emphasis on the one-dimensional case, in which certain rough path terms can be expressed as polynomials in the extended trace path, which for semimartingales restrict to the well-known Kailath-Segall 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 1/4-fractional Brownian motion, two of which are ?truly branched?, i.e. not quasi- geometric.

Vorträge, 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.

  • S.M. Toukam Tchoumegne, Stochastic maximum principle for McKean-Vlasov SDEs with rough drift coefficients, 20th Oxford-Berlin Young Researcher's Meeting on Applied Stochastic Analysis, December 9 - 11, 2024, University of Oxford, Mathematical Institute, UK, December 9, 2024.

  • O. Butkovsky, Approximation of additive functionals of stochastic processes given high-frequency observations, Numerical analysis and applications of SDEs, September 25 - 26, 2024, Banach Center, Będlewo, Poland, September 25, 2024.

  • O. Butkovsky, Optimal weak uniqueness for SDEs driven by fractional Brownian motion and for stochastic heat equation with distributional drift., Stochastic Dynamics and Stochastic Equations, March 25 - 27, 2024, École Polytechnique Fédérale de Lausanne, Switzerland, March 25, 2024.

  • O. Butkovsky, Optimal weak uniqueness for SDEs driven by fractional Brownian motion and for stochastic heat equation with distributional drift., 19th Annual Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, June 24 - 26, 2024, WIAS Berlin, June 25, 2024.

  • O. Butkovsky, Strong rate of convergence of the Euler scheme for SDEs with irregular drifts and approximation of additive time functionals, Mathematics, Data Science, and Education, March 13 - 15, 2024, FernUniversität in Hagen, Lehrgebiet Angewandte Stochastik, March 14, 2024.

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

  • O. Butkovsky, Weak uniqueness for singular stochastic equations (online talk), Klagenfurt-Berlin-Meeting, June 6 - 7, 2024, Universität Klagenfurt, Institut für Mathematik, Austria, June 6, 2024.

  • O. Butkovsky, Weak uniqueness for singular stochastic equations driven by fractional Brownian motion, Stochastic Afternoon at Bielefeld, Universität Bielefeld, Fakultät für Mathematik, October 30, 2024.

  • O. Butkovsky, Weak uniqueness for singular stochastic equations driven by fractional Brownian motion, 20th Oxford-Berlin Young Researcher's Meeting on Applied Stochastic Analysis, December 9 - 11, 2024, University of Oxford, Mathematical Institute, UK, December 9, 2024.

  • O. Butkovsky, Weak uniqueness for stochastic equations with singular drifts, University of Leeds, School of Mathmatics, UK, July 8, 2024.

  • W. Kenmoe Nzali, Volatile electricity market and battery storage, 7th Berlin Workshop on Mathematical Finance for Young Researchers, September 2 - 6, 2024.

  • W. Kenmoe Nzali, Volatile electricity market and battery storage, 19. Doktorand:innentreffen der Stochastik, August 27 - 30, 2024, University of Technology Cottbus-Senftenberg (BTU), August 29, 2024.

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

  • L. Németh, Data and monitoring challenges of sub-national longevity disparities: the case of Portugal, EAPS Health, Morbidity, and Mortality Working Group, September 25 - 27, 2024, Bilbao, Spain, September 26, 2024.

  • L. Németh, Ontology for R language packages, 2nd MaRDI Workshop on Scientific Computing, October 16 - 18, 2024, Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, October 18, 2024.

  • L. Pelizzari, Non-Markovian optimal stopping with signatures, Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2024, Thuwal, Saudi Arabia, May 19 - 30, 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 Oxford-Berlin Young Researcher's Meeting on Applied Stochastic Analysis, January 4 - 6, 2024, University of Oxford, Mathematical Institute, UK, January 5, 2024.

  • L. Pelizzari, Rough PDE for LSVM, Klagenfurt-Berlin-Meeting, June 6 - 7, 2024, Universität Klagenfurt, Institut für Mathematik, Austria, June 6, 2024.

  • L. Pelizzari, Rough PDEs for local stochastic volatility, Recent developments in rough paths, March 3 - 8, 2024, BI Norwegian Business School, Nydalen, Norway, March 4, 2024.

  • N. Tapia, A multiplicative surface signature through its Magnus expansion, Multi-Parameter Signatures, December 5 - 6, 2024, Norwegian University of Science and Technology, Department of Mathematical Sciences, Trondheim, Norway, December 5, 2024.

  • N. Tapia, Branched Itô formula, Directions in Rough Analysis, November 4 - 8, 2024, Mathematisches Forschungsinstitut Oberwolfach, November 6, 2024.

  • N. Tapia, Branched Itô formula and intrinsic RDEs, 20th Oxford-Berlin Young Researcher's Meeting on Applied Stochastic Analysis, December 9 - 11, 2024, University of Oxford, Mathematical Institute, UK, December 10, 2024.

  • N. Tapia, Branched Itô formula and natural Itô-Stratonovich isomorphism, Klagenfurt-Berlin-Meeting, June 6 - 8, 2024, Universität Klagenfurt, Institut für Mathematik, Austria, June 6, 2024.

  • N. Tapia, Stabilitiy of deep neural networks via discrete rough paths, Mathematics, Data Science, and Education, March 13 - 15, 2024, FernUniversität Hagen, March 13, 2024.

  • N. Tapia, Stability of deep neural networks via discrete rough paths, Mathematics of data streams: signatures, neural differential equations, and diffusion models, April 10 - 13, 2024, Universität Greifswald, Institut für Mathematik, April 12, 2024.

  • N. Tapia, The signature of an image through its Magnus expansion, 19th Annual Berlin-Oxford Young Researchers Meeting on Applied Stochastic Analysis, June 24 - 26, 2024, WIAS Berlin, June 26, 2024.

  • N. Tapia, Unified signature cumulants and generalized magnus expansions, 70 Years of Magnus Series, July 1 - 5, 2024, Universitat Jaume I, Departament de Matemàtiques, Castellón, Spain, July 2, 2024.

  • CH. Bayer, A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean---Vlasov models, 24th MathFinance conference, September 19 - 20, 2024, Burg Reichenstein, Trechtingshausen, September 19, 2024.

  • CH. Bayer, Signature methods in finance, Bachelier Seminar, École Polytechnique (CMAP), Palaiseau, France, February 28, 2024.

  • P. Dvurechensky, Hessian barrier algorithms for non-convex conic optimization (semi-plenary talk), Workshop on Nonsmooth Optimization and Applications (NOPTA 2024), April 8 - 12, 2024, University of Antwerp, Department of Mathematics, Belgium, April 10, 2024.

  • V. Spokoiny, Inference for nonlinear problems, University of Electronic Science and Technology of China (UEST), School of Mathematical Sciences, Chengdu, China, May 14, 2024.

  • A. Suvorikova, Bernstein type inequality for unbounded martingales, Statistics & Learning Theory in the Era of AI, June 23 - 28, 2024, Mathematisches Forschungszentrum Oberwolfach (MFO), June 28, 2024.

  • CH. Bayer, W. Kenmoe Nzali, D. Kreher, M. Landstorfer, Volatile electricity markets and battery storage: A model-based approach for optimal control, MATH+ Day, Urania Berlin, October 18, 2024.

  • CH. Bayer, P.K. Friz, Path signatures and rough path analysis (minicourse), Spring School 2024, SFB 1481 Sparsity and Singular Structures, May 13 - 17, 2024, RWTH Aachen University.

  • CH. Bayer, A Reproducing Kernel Hilbert Space approach to singular local stochastic volatility McKean-Vlasov models, Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2024, May 25 - June 1, 2024, King Abdullah University of Science and Technology, Stochastic Numerics Research Group, Thuwal, Saudi Arabia, May 27, 2024.

  • CH. Bayer, A reproducing kernel Hilbert space approach to singular local stochastic volatility McKean-Vlasov models, University of Oslo, Department of Mathematics, Norway, August 27, 2024.

  • CH. Bayer, An adaptive algorithm for rough differential equations, Directions in Rough Analysis, November 4 - 8, 2024, Mathematisches Forschungsinstitut Oberwolfach, November 8, 2024.

  • CH. Bayer, Efficient Markovian approximations of rough volatility models, Finance and Stochastics Seminar, Imperial College London, Department of Mathematics, UK, January 30, 2024.

  • CH. Bayer, Markovian approximations to rough volatility models, Research in Options: RiO 2024, December 4 - 8, 2024, FGV EMAp - School of Applied Mathematics, Rio de Janeiro, Brazil, December 5, 2024.

  • CH. Bayer, Primal and dual optimal stopping with signatures, International Conference on Scientific Computation and Differential Equations, July 15 - 19, 2024, National University of Singapore, Singapore, July 16, 2024.

  • CH. Bayer, Primal and dual optimal stopping with signatures, London Mathematical Finance Seminar, Imperial College London, Department of Mathematics, UK, February 1, 2024.

  • CH. Bayer, Primal and dual optimal stopping with signatures (Plenarvortrag), International Conference on Computational Finance (ICCF24), April 2 - 5, 2024, Utrecht University, Mathematical Institute, Netherlands, April 2, 2024.

  • CH. Bayer, RKHS regularization of singular local stochstic volatility McKean-Vlasov models, Stochastics in Mathematical Finance and Physics Conference, October 21 - 25, 2024, Tunis El Manar University, Department of Mathematics, Hammamet, Tunisia, October 21, 2024.

  • CH. Bayer, Signatures for stochastic optimal control, 2024 Conference on Modern Topics in Stochastic Analysis and Applications (in honour of Terry Lyons' 70th birthday), April 22 - 26, 2024, Imperial College London, UK, April 22, 2024.

  • P. Dvurechensky, Barrier algorithms for constrained non-convex optimization, The International Conference on Machine Learning (ICML), Wien, Austria, July 21 - 26, 2024.

  • P. Dvurechensky, Barrier algorithms for constrained non-convex optimization, Europt 2024, 21st Conference on Advances in Continuous Optimization, June 26 - 28, 2024, Lund University, Department of Automatic Control, Sweden, June 26, 2024.

  • P. Dvurechensky, Decentralized local stochastic extra-gradient for variational inequalities, French-German-Spanish Conference on Optimization 2024, June 20 - 21, 2024, University of Oviedo, Departament of Mathematics, Spain, June 20, 2024.

  • P. Dvurechensky, High-probability convergence for composite and distributed stochastic minimization and variational inequalities with heavy-tailed noise, The International Conference on Machine Learning (ICML), Wien, Austria, July 21 - 26, 2024.

  • P. Dvurechensky, On some results on minimization involving self-concordant functions and barriers, ALGOPT2024 workshop on Algorithmic Optimization: Tools for AI and Data Science, August 27 - 30, 2024, Université catholique de Louvain, School of Engineering, Louvain-la-Neuve, Belgium, August 27, 2024.

  • P. Friz, On the analysis of some SPDEs via RSDEs, Stochastic Dynamics and Stochastic Equations, March 25 - 27, 2024, École Polytechnique Fédérale de Lausanne, Switzerland, March 25, 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.

  • P.K. Friz, Brownian rough paths in irregular geometries, Stochastics and Geometry Workshop, September 8 - 13, 2024, Banff International Research Station (BIRS), Canada, September 11, 2024.

  • P.K. Friz, On signatures, log-signatures, moments and cumulants, Advances in Probability Theory and Interacting Particle Systems: A Conference in Honor of S. R. Srinivasa Varadhan, August 26 - 28, 2024, Harvard University, Department of Mathematics, Cambridge, USA, August 26, 2024.

  • P.K. Friz, Rough analysis of rough volatility models, Stochastic Numerics and Statistical Learning: Theory and Applications Workshop 2024, May 24 - 27, 2024, King Abdullah University of Science and Technology, Stochastic Numerics Research Group, Thuwal, Saudi Arabia, May 26, 2024.

  • P.K. Friz, Rough analysis of rough volatility models, 2024 Conference on Modern Topics in Stochastic Analysis and Applications (in honour of Terry Lyons' 70th birthday), April 22 - 25, 2024, Imperial College London, UK, April 23, 2024.

  • P.K. Friz, Simulation and weak error bounds for local stochastic volatility models, Quant Minds 2024, November 18 - 21, 2024, London, UK, November 21, 2024.

  • P.K. Friz, Some things related to signatures, Signatures of paths and images, June 10 - 14, 2024, The Norwegian Academy of Science and Letters, Centre for Advanced Study (CAS), Oslo, Norway, June 11, 2024.

  • P.K. Friz, Rough Paths with Jumps (Minicourse), Recent developments in rough paths, March 3 - 8, 2024, BI Norwegian Business School, Oslo, Norway, March 4, 2024.

  • A. Kroshnin, Tropical optimal transport, Maslov dequantization, and large deviation principle, University of Pisa, Department of Mathematics, Italy, November 29, 2024.

  • J.G.M. Schoenmakers, Optimal stopping with randomly arriving opportunities to stop, DYNSTOCH 2024, May 22 - 24, 2024, Christian-Albrechts-Universität zu Kiel (CAU), May 22, 2024.

  • V. Spokoiny, Gaussian variational inference in high dimension, Mohamed Bin Zayed University of Artificial Intelligence (MBZUAI), Department of Machine Learning, Abu Dhabi, United Arab Emirates, March 12, 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.

  • V. Spokoiny, Inference for nonlinear inverse problems, New challenges in high-dimensional statistics, December 16 - 20, 2024, Centre International de Rencontres Mathématiques (CIRM), Luminy, France, December 16, 2024.

  • V. Spokoiny, Non-asymptotic and non-minimax estimation of smooth functionals, Statistics & Learning Theory in the Era of AI, June 23 - 28, 2024, Mathematisches Forschungszentrum Oberwolfach (MFO), June 28, 2024.

  • V. Spokoiny, Statistical inference for nonlinear inverse problems, Statistical Aspects of Non-Linear Inverse Problems, September 17 - 19, 2024, University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, UK.

  • V. Spokoiny, Uncertainty quantification under self-concordance, ALGOPT2024 workshop on Algorithmic Optimization: Tools for AI and Data Science, August 27 - 30, 2024, Université catholique de Louvain, School of Engineering, Louvain-la-Neuve, Belgium, August 29, 2024.

Preprints im Fremdverlag

  • O. Butkovsky, L. Mytnik, Weak uniqueness for singular stochastic equations, Preprint no. arXiv:2405.13780, Cornell University, 2024, DOI 10.48550/arXiv.2405.13780 .

  • S. Athreya, O. Butkovsky, K. Lê, L. Mytnik, Analytically weak and mild solutions to stochastic heat equation with irregular drift, Preprint no. arXiv:2410.06599, Cornell University, 2024, DOI 2410.06599 .

  • I. Chevyrev, J. Diehl, K. Ebrahimi-Fard, N. Tapia, A multiplicative surface signature through its Magnus expansion, Preprint no. arXiv:2406.16856, Cornell University, 2024, DOI 10.48550/arXiv.2406.16856 .
    Abstract
    In the last decade, the concept of path signature has found great success in data science applications, where it provides features describing the path. This is partly explained by the fact that it is possible to compute the signature of a path in linear time, owing to a dynamic programming principle, based on Chen's identity. The path signature can be regarded as a specific example of product or time-/path-ordered integral. In other words, it can be seen as a 1-parameter object build on iterated integrals over a path. Increasing the number of parameters by one, which amounts to considering iterated integrals over surfaces, is more complicated. An observation that is familiar in the context of higher gauge theory where multiparameter iterated integrals play an important role. The 2-parameter case is naturally related to a non-commutative version of Stokes' theorem, which is understood to be fundamentally linked to the concept of crossed modules of groups. Indeed, crossed modules with non-trivial kernel of the feedback map permit to compute features of a surface that go beyond what can be expressed by computing line integrals along the boundary of a surface. A good candidate for the crossed analog of free Lie algebra then seems to be a certain free crossed module over it. Building on work by Kapranov, we study the analog to the classical path signature taking values in such a free crossed module of Lie algebra. In particular, we provide a Magnus-type expression for the logarithm of surface signature as well as a sewing lemma for the crossed module setting.

  • E. Gladin, P. Dvurechensky, A. Mielke , J.-J. Zhu, Interaction-force transport gradient flows, Preprint no. arXiv:2405.17075, Cornell University, 2024, DOI 10.48550/arXiv.2405.17075 .
    Abstract
    This paper presents a new type of gradient flow geometries over non-negative and probability measures motivated via a principled construction that combines the optimal transport and interaction forces modeled by reproducing kernels. Concretely, we propose the interaction-force transport (IFT) gradient flows and its spherical variant via an infimal convolution of the Wasserstein and spherical MMD Riemannian metric tensors. We then develop a particle-based optimization algorithm based on the JKO-splitting scheme of the mass-preserving spherical IFT gradient flows. Finally, we provide both theoretical global exponential convergence guarantees and empirical simulation results for applying the IFT gradient flows to the sampling task of MMD-minimization studied by Arbel et al. [2019]. Furthermore, we prove that the spherical IFT gradient flow enjoys the best of both worlds by providing the global exponential convergence guarantee for both the MMD and KL energy.

  • 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, bioRxiv, 2024, DOI 10.1101/2023.06.21.545891 .

  • B. Schembera, F. Wübbeling, H. Kleikamp, B. Schmidt, A. Shehu, M. Reidelbach, Ch. Biedinger, J. Fiedler, Th. Koprucki, D. Iglezakis, D. Göddeke, Towards a knowledge graph for models and algorithms in applied mathematics, Preprint no. arXiv:2408.10003, Cornell University, 2024, DOI 10.48550/arXiv.2408.10003 .
    Abstract
    Mathematical models and algorithms are an essential part of mathematical research data, as they are epistemically grounding numerical data. In order to represent models and algorithms as well as their relationship semantically to make this research data FAIR, two previously distinct ontologies were merged and extended, becoming a living knowledge graph. The link between the two ontologies is established by introducing computational tasks, as they occur in modeling, corresponding to algorithmic tasks. Moreover, controlled vocabularies are incorporated and a new class, distinguishing base quantities from specific use case quantities, was introduced. Also, both models and algorithms can now be enriched with metadata. Subject-specific metadata is particularly relevant here, such as the symmetry of a matrix or the linearity of a mathematical model. This is the only way to express specific workflows with concrete models and algorithms, as the feasible solution algorithm can only be determined if the mathematical properties of a model are known. We demonstrate this using two examples from different application areas of applied mathematics. In addition, we have already integrated over 250 research assets from applied mathematics into our knowledge graph.

  • P. Dvurechensky, C. Geiersbach, M. Hintermüller, A. Kannan, S. Kater, G. Zöttl, A Cournot-Nash model for a coupled hydrogen and electricity market, Preprint no. arxiv:2410.20534, Cornell University, 2024, DOI 10.48550/arXiv.2410.20534 .
    Abstract
    We present a novel model of a coupled hydrogen and electricity market on the intraday time scale, where hydrogen gas is used as a storage device for the electric grid. Electricity is produced by renewable energy sources or by extracting hydro- gen from a pipeline that is shared by non-cooperative agents. The resulting model is a generalized Nash equilibrium problem. Under certain mild assumptions, we prove that an equilibrium exists. Perspectives for future work are presented.

  • P. Dvurechensky, Y. Nesterov, Improved global performance guarantees of second-order methods in convex minimization, Preprint no. arXiv:2408.11022, Cornell University, 2024.

  • P. Dvurechensky, M. Staudigl, Barrier algorithms for constrained non-convex optimization, Preprint no. arXiv:2404.18724, Cornell University, 2024, DOI 10.48550/arXiv.2404.18724 .

Forschungsgruppen