TY - GEN A1 - Raharinirina, N. Alexia A1 - Weber, Marcus A1 - Birk, Ralph A1 - Fackeldey, Konstantin A1 - Klasse, Sarah M. A1 - Richter, Tonio Sebastian T1 - Different Tools and Results for Correspondence Analysis N2 - This is a list of codes generated from ancient egyptian texts. The codes are used for a correspondence analysis (CA). Codes and CA software are available from the linked webpage. Y1 - 2021 U6 - https://doi.org/10.12752/8257 N1 - A detailed description of the software can be found in the code repository at https://github.com/AlexiaNomena/Correspondence_Analysis_User_Friendly (repository version of CA software might include updates). ER - TY - JOUR A1 - Ribera Borrell, Enric A1 - Quer, Jannes A1 - Richter, Lorenz A1 - Schütte, Christof T1 - Improving control based importance sampling strategies for metastable diffusions via adapted metadynamics JF - SIAM Journal on Scientific Computing (SISC) N2 - Sampling rare events in metastable dynamical systems is often a computationally expensive task and one needs to resort to enhanced sampling methods such as importance sampling. Since we can formulate the problem of finding optimal importance sampling controls as a stochastic optimization problem, this then brings additional numerical challenges and the convergence of corresponding algorithms might as well suffer from metastabilty. In this article we address this issue by combining systematic control approaches with the heuristic adaptive metadynamics method. Crucially, we approximate the importance sampling control by a neural network, which makes the algorithm in principle feasible for high dimensional applications. We can numerically demonstrate in relevant metastable problems that our algorithm is more effective than previous attempts and that only the combination of the two approaches leads to a satisfying convergence and therefore to an efficient sampling in certain metastable settings. KW - importance sampling KW - stochastic optimal control KW - rare event simulation KW - metastability KW - neural networks KW - metadynamics Y1 - 2023 U6 - https://doi.org/10.1137/22M1503464 VL - 89 IS - 1 ER - TY - CHAP A1 - Berner, Julius A1 - Richter, Lorenz A1 - Ullrich, Karen T1 - An optimal control perspective on diffusion-based generative modeling T2 - Transactions on Machine Learning Research N2 - We establish a connection between stochastic optimal control and generative models based on stochastic differential equations (SDEs) such as recently developed diffusion probabilistic models. In particular, we derive a Hamilton-Jacobi-Bellman equation that governs the evolution of the log-densities of the underlying SDE marginals. This perspective allows to transfer methods from optimal control theory to generative modeling. First, we show that the evidence lower bound is a direct consequence of the well-known verification theorem from control theory. Further, we develop a novel diffusion-based method for sampling from unnormalized densities -- a problem frequently occurring in statistics and computational sciences. Y1 - 2024 UR - https://openreview.net/forum?id=oYIjw37pTP ER - TY - JOUR A1 - Nüsken, Nikolas A1 - Richter, Lorenz T1 - Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems JF - Journal of Machine Learning N2 - Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and engineering. In recent years, a great number of computational approaches have been developed, most of them relying on a combination of Monte Carlo sampling and deep learning based approximation. For elliptic and parabolic problems, existing methods can broadly be classified into those resting on reformulations in terms of backward stochastic differential equations (BSDEs) and those aiming to minimize a regression-type L2-error (physics-informed neural networks, PINNs). In this paper, we review the literature and suggest a methodology based on the novel diffusion loss that interpolates between BSDEs and PINNs. Our contribution opens the door towards a unified understanding of numerical approaches for high-dimensional PDEs, as well as for implementations that combine the strengths of BSDEs and PINNs. The diffusion loss furthermore bears close similarities to (least squares) temporal difference objectives found in reinforcement learning. We also discuss eigenvalue problems and perform extensive numerical studies, including calculations of the ground state for nonlinear Schr ¨odinger operators and committor functions relevant in molecular dynamics. Y1 - 2023 U6 - https://doi.org/0.4208/jml.220416 VL - 2 IS - 1 SP - 31 EP - 64 ER - TY - JOUR A1 - Hartmann, Carsten A1 - Richter, Lorenz T1 - Nonasymptotic bounds for suboptimal importance sampling JF - SIAM/ASA Journal on Uncertainty Quantification N2 - Importance sampling is a popular variance reduction method for Monte Carlo estimation, where an evident question is how to design good proposal distributions. While in most cases optimal (zero-variance) estimators are theoretically possible, in practice only suboptimal proposal distributions are available and it can often be observed numerically that those can reduce statistical performance significantly, leading to large relative errors and therefore counteracting the original intention. Previous analysis on importance sampling has often focused on asymptotic arguments that work well in a large deviations regime. In this article, we provide lower and upper bounds on the relative error in a nonasymptotic setting. They depend on the deviation of the actual proposal from optimality, and we thus identify potential robustness issues that importance sampling may have, especially in high dimensions. We particularly focus on path sampling problems for diffusion processes with nonvanishing noise, for which generating good proposals comes with additional technical challenges. We provide numerous numerical examples that support our findings and demonstrate the applicability of the derived bounds. Y1 - 2024 U6 - https://doi.org/10.1137/21M1427760 VL - 12 IS - 2 SP - 309 EP - 346 ER - TY - CHAP A1 - Richter, Lorenz A1 - Berner, Julius T1 - Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning T2 - Proceedings of the 39th International Conference on Machine Learning, PMLR N2 - The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations based on associated stochastic differential equations (SDEs), which allow the minimization of corresponding losses using gradient-based optimization methods. In respective numerical implementations it is therefore crucial to rely on adequate gradient estimators that exhibit low variance in order to reach convergence accurately and swiftly. In this article, we rigorously investigate corresponding numerical aspects that appear in the context of linear Kolmogorov PDEs. In particular, we systematically compare existing deep learning approaches and provide theoretical explanations for their performances. Subsequently, we suggest novel methods that can be shown to be more robust both theoretically and numerically, leading to substantial performance improvements. Y1 - 2022 VL - 162 SP - 18649 EP - 18666 ER - TY - THES A1 - Richter, Stephanie T1 - Mathematische Behandlung von Dienstreihenfolgeregeln in der Wochenschemaoptimierung Y1 - 2013 ER - TY - GEN A1 - Hartmann, Carsten A1 - Richter, Lorenz A1 - Schütte, Christof A1 - Zhang, Wei T1 - Variational characterization of free energy: Theory and algorithms N2 - The article surveys and extends variational formulations of the thermodynamic free energy and discusses their information-theoretic content from the perspective of mathematical statistics. We revisit the well-known Jarzynski equality for nonequilibrium free energy sampling within the framework of importance sampling and Girsanov change-of-measure transformations. The implications of the different variational formulations for designing efficient stochastic optimization and nonequilibrium simulation algorithms for computing free energies are discussed and illustrated. T3 - ZIB-Report - 17-52 KW - Importance sampling KW - Donsker-Varadhan principle KW - thermodynamic free energy KW - nonequilibrium molecular dynamics KW - stochastic approximation KW - cross-entropy method Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-65045 SN - 1438-0064 ER - TY - JOUR A1 - Raya-Moreno, Martí A1 - Dasch, Noah Alexy A1 - Farahani, Nasrin A1 - Gonzalez Oliva, Ignacio A1 - Gulans, Andris A1 - Hossain, Manoar A1 - Kleine, Hannah A1 - Kuban, Martin A1 - Lubeck, Sven A1 - Maurer, Benedikt A1 - Pavone, Pasquale A1 - Peschel, Fabian A1 - Popova-Gorelova, Daria A1 - Qiao, Lu A1 - Richter, Elias A1 - Rigamonti, Santiago A1 - Rodrigues Pelá, Ronaldo A1 - Sinha, Kshitij A1 - Speckhard, Daniel T. A1 - Tillack, Sebastian A1 - Tumakov, Dmitry A1 - Hong, Seokhyun A1 - Uzulis, Janis A1 - Voiculescu, Mara A1 - Vona, Cecilia A1 - Yang, Mao A1 - Draxl, Claudia T1 - An exciting approach to theoretical spectroscopy Y1 - 2026 ER -