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The graph limit of the minimizer of the Onsager-Machlup functional and its computation

Science China mathematics 2021年第2期64卷 239-280页

作者： Qiang Du Tiejun Li Xiaoguang Li Weiqing Ren Department of Applied Phgysics and Applied Mathematics Columbia UniversityNeu YorkNY 10027USA LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Beijing Computational Science Research Certer Beijing 100193China MOE-LCSM School of Mathematics and StatisticsHunan Normal UniversityChangsha 410081China Department of Mathematics National University of SingaporeSingapore 119076Singapore

The Onsager-Machlup(OM)functional is well known for characterizing the most probable transition path of a diffusion process with non-vanishing ***,it suffers from a notorious issue that the functional is unbounded below when the specified transition time T goes to *** hinders the interpretation of the results obtained by minimizing the OM *** provide a new perspective on this *** mild conditions,we show that although the infimum of the OM functional becomes unbounded when T goes to infinity,the sequence of minimizers does contain convergent subsequences on the space of *** graph limit of this minimizing subsequence is an extremal of the abbreviated action functional,which is related to the OM functional via the Maupertuis principle with an optimal *** further propose an energy-climbing geometric minimization algorithm(EGMA)which identifies the optimal energy and the graph limit of the transition path *** algorithm is successfully applied to several typical examples in rare event *** interesting comparisons with the Freidlin-Wentzell action functional are also made.

关键词： Onsager-Machlup functional Freidlin-Wentzell functional graph limit geometric minimization Maupertuis principle

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Complex Gaussianity of long-distance random wave processes

arXiv

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arXiv 2024年

作者： Bal, Guillaume Nair, Anjali Departments of Statistics and Mathematics and Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Department of Statistics and Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

Interference of randomly scattered classical waves naturally leads to familiar speckle patterns, where the wave intensity follows an exponential distribution while the wave field itself is described by a circularly symmetric complex normal distribution. In the Itô-Schrödinger paraxial model of wave beam propagation, we demonstrate how a deterministic incident beam transitions to such a fully developed speckle pattern over long distances in the so-called scintillation (weak-coupling) regime. Copyright © 2024, The Authors. All rights reserved.

关键词： Normal distribution

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Long distance propagation of light in random media with partially coherent sources

arXiv

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arXiv 2024年

Optical beam propagation in random media is characterized by familiar speckle patterns generated by intricate interference effects. Such patterns may be modified and possibly attenuated for partially coherent incident beam profiles. In the weak-coupling regime of the Itô-Schrödinger paraxial model of wave propagation, we show how the spatio-temporal statistics of the partially coherent beams interact with the statistics of the random medium to enhance or suppress scintillation effects. Copyright © 2024, The Authors. All rights reserved.

关键词： Scintillation

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A FOLIATED AND REVERSIBLE FINSLER MANIFOLD IS DETERMINED BY ITS BROKEN SCATTERING RELATION

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Pure and applied Analysis 2021年第4期3卷 789-811页

作者： DE HOOP, Maarten V. Ilmavirta, Joonas Lassas, Matti Saksala, Teemu Departments of Computational and Applied Mathematics Earth Science Rice University Houston TX United States Department of Mathematics and Statistics University of Jyväskylä Jyväskylä Finland Department of Mathematics and Statistics University of Helsinki Helsinki Finland Department of Mathematics North Carolina State University Raleigh NC United States

The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds satisfying a convex foliation condition have the same broken scattering relation, then they are isometric. This implies that some anisotropic material parameters of the earth can be in principle reconstructed from single scattering measurements at the surface. © 2021, Mathematical Sciences Publishers. All rights reserved.

关键词： anisotropic elasticity distance functions Finsler manifold scattering relation

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CP-PINNs: Data-Driven Changepoints Detection in PDEs Using Online Optimized Physics-Informed Neural Networks

CP-PINNs: Data-Driven Changepoints Detection in PDEs Using O...

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2024 IEEE International Conferences of AI, Science, Engineering, and Technology, AIxSET 2024

作者： Dong, Zhikang P., Polak Stony Brook University Department of Applied Mathematics and Statistics Stony Brook United States Institute for Advanced Computational Science Center of Excellence in Wireless and Information Technology Stony Brook University Department of Applied Mathematics and Statistics Stony Brook United States

ISBN: (纸本)9798350390995

We investigate the inverse problem for Partial Differential Equations (PDEs) in scenarios where the parameters of the given PDE dynamics may exhibit changepoints at random time. We employ Physics-Informed Neural Networks (PINNs) - universal approximators capable of estimating the solution of any physical law described by a system of PDEs, which serves as a regularization during neural network training, restricting the space of admissible solutions and enhancing function approximation accuracy. We demonstrate that when the system exhibits sudden changes in the PDE dynamics, this regularization is either insufficient to accurately estimate the true dynamics, or it may result in model miscalibration and failure. Consequently, we propose a PINNs extension using a Total-Variation penalty, which allows to accommodate multiple changepoints in the PDE dynamics and significantly improves function approximation. These changepoints can occur at random locations over time and are estimated concurrently with the solutions. Additionally, we introduce an online convex optimization method for re-weighting loss function terms dynamically. Through empirical analysis using examples of various equations with parameter changes, we showcase the advantages of our proposed model. In the absence of changepoints, the model reverts to the original PINNs model. However, when changepoints are present, our approach yields superior parameter estimation, improved model fitting, and reduced training error compared to the original PINNs model. © 2024 IEEE.

关键词： Inverse problems

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Optimal Configurations in Coverage Control with Polynomial Costs

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IFAC-PapersOnLine 2018年第12期51卷 106-111页

作者： Bopardikar, Shaunak D. Mehta, Dhagash Hauenstein, Jonathan D. United Technologies Research Center 411 Silver Lane East HartfordCT06118 United States Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States

We revisit the static coverage control problem for placement of vehicles with simple motion on the real line, under the assumption that the cost is a polynomial function of the locations of the vehicles. The main contribution of this paper is to demonstrate the use of tools from numerical algebraic geometry, in particular, a numerical polynomial homotopy continuation method that guarantees to find all solutions of polynomial equations, in order to characterize the global minima for the coverage control problem. The results are then compared against a classic distributed approach involving the use of Lloyd descent, which is known to converge only to a local minimum under certain technical conditions. © 2018

关键词： Polynomials Geometry Global optimization Numerical methods Coverage control Distributed approaches Homotopies Homotopy continuation methods Numerical algebraic geometry Polynomial equation Polynomial functions Technical conditions

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Depth Separation in Norm-Bounded Infinite-Width Neural Networks 37

Depth Separation in Norm-Bounded Infinite-Width Neural Netwo...

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37th Annual Conference on Learning Theory, COLT 2024

作者： Parkinson, Suzanna Ongie, Greg Willett, Rebecca Shamir, Ohad Srebro, Nathan Committee on Computational and Applied Mathematics University of Chicago United States Department of Mathematical and Statistical Sciences Marquette University United States Department of Statistics Department of Computer Science University of Chicago United States Department of Computer Science and Applied Mathematics Weizmann Institute of Science Israel Toyota Technological Institute Chicago United States

We study depth separation in infinite-width neural networks, where complexity is controlled by the overall squared 2-norm of the weights (sum of squares of all weights in the network). Whereas previous depth separation results focused on separation in terms of width, such results do not give insight into whether depth determines if it is possible to learn a network that generalizes well even when the network width is unbounded. Here, we study separation in terms of the sample complexity required for learnability. Specifically, we show that there are functions that are learnable with sample complexity polynomial in the input dimension by norm-controlled depth-3 ReLU networks, yet are not learnable with sub-exponential sample complexity by norm-controlled depth-2 ReLU networks (with any value for the norm). We also show that a similar statement in the reverse direction is not possible: any function learnable with polynomial sample complexity by a norm-controlled depth-2 ReLU network with infinite width is also learnable with polynomial sample complexity by a norm-controlled depth-3 ReLU network. © 2024 S. Parkinson, G. Ongie, R. Willett, O. Shamir & N. Srebro.

关键词： Source separation

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A Partial RKDG Method for Solving the 2D Ideal MHD Equations Written in Semi-Lagrangian Formulation on Moving Meshes with Exactly Divergence-Free Magnetic Field

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Advances in applied mathematics and Mechanics 2023年第4期15卷 932-963页

作者： Shijun Zou Xijun Yu Zihuan Dai Fang Qing Xiaolong Zhao School of Mathematical Sciences Peking UniversityBeijing 100871China Institute of Applied Physics and Computational Mathematics Beijing 100088China Department of Mathematics Southern University of Science and TechnologyShenzhenGuangdong 518055China School of Mathematics and Statistics Zhengzhou UniversityZhengzhouHenan 450001China

A partial Runge-Kutta Discontinuous Galerkin(RKDG)method which preserves the exactly divergence-free property of the magnetic field is proposed in this paper to solve the two-dimensional ideal compressible magnetohydrodynamics(MHD)equations written in semi-Lagrangian formulation on moving quadrilateral *** this method,the fluid part of the ideal MHD equations along with zcomponent of the magnetic induction equation is discretized by the RKDG method as our previous paper[47].The numerical magnetic field in the remaining two directions(i.e.,x and y)are constructed by using the magnetic flux-freezing principle which is the integral form of the magnetic induction equation of the ideal *** the divergence of the magnetic field in 2D is independent of its z-direction component,an exactly divergence-free numerical magnetic field can be obtained by this *** propose a new nodal solver to improve the calculation accuracy of velocities of the moving meshes.A limiter is presented for the numerical solution of the fluid part of the MHD equations and it can avoid calculating the complex eigen-system of the MHD *** numerical examples are presented to demonstrate the accuracy,non-oscillatory property and preservation of the exactly divergence-free property of our method.

关键词： Ideal compressible MHD equations semi-Lagrangian formulation exactly divergence-free magnetic field Runge-Kutta discontinuous Galerkin method

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Reconstructing the Absorption Function in a Quasi-Linear Sorption Dynamic Model via an Iterative Regularizing Algorithm

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Advances in applied mathematics and Mechanics 2024年第1期16卷 237-252页

作者： Alexey Shcheglov Jingzhi Li Chao Wang Alexander Ilin Ye Zhang Faculty of Computational Mathematics and Cybernetics Shenzhen MSU-BIT UniversityShenzhenGuangdong 518172China Faculty of Computational Mathematics and Cybernetics Lomonosov Moscow State UniversityMoscowRussia Department of Mathematics&National Center for Applied Mathematics Shenzhen&SUSTech International Center for Mathematics Southern University of Science and TechnologyShenzhenGuangdong 518055China School of Mathematics and Statistics Beijing Institute of TechnologyBeijing 100081China

This study addresses the parameter identification problem in a system of time-dependent quasi-linear partial differential equations(PDEs).Using the integral equation method,we prove the uniqueness of the inverse problem in nonlinear ***,using the method of successive approximations,we develop a novel iterative algorithm to estimate sorption *** stability results of the algorithm are proven under both a priori and a posteriori stopping rules.A numerical example is given to show the efficiency and robustness of the proposed new approach.

关键词： Inverse problem quasi-linear dynamic model uniqueness method of successive approximations stability

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Inertial Proximal ADMM for Separable Multi-Block Convex Optimizations and Compressive Affine Phase Retrieval

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Acta Mathematica Sinica,English Series 2023年第8期39卷 1459-1496页

作者： Peng LI Wen Gu CHEN Qi Yu SUN School of Mathematics and Statistics Gansu Key Laboratory of Applied Mathematics and Complex SystemsLanzhou UniversityLanzhou 730000P.R.China Graduate School China Academy of Engineering PhysicsBeijing 100088P.R.China Institute of Applied Physics and Computational Mathematics Beijing 100088P.R.China Department of Mathematics University of Central FloridaOrlandoFL 32816USA

Separable multi-block convex optimization problem appears in many mathematical and engineering *** the first part of this paper,we propose an inertial proximal ADMM to solve a linearly constrained separable multi-block convex optimization problem,and we show that the proposed inertial proximal ADMM has global convergence under mild assumptions on the regularization *** phase retrieval arises in holography,data separation and phaseless sampling,and it is also considered as a nonhomogeneous version of phase retrieval,which has received considerable attention in recent *** by convex relaxation of vector sparsity and matrix rank in compressive sensing and by phase lifting in phase retrieval,in the second part of this paper,we introduce a compressive affine phase retrieval via lifting approach to connect affine phase retrieval with multi-block convex optimization,and then based on the proposed inertial proximal ADMM for 3-block convex optimization,we propose an algorithm to recover sparse real signals from their(noisy)affine quadratic *** numerical simulations show that the proposed algorithm has satisfactory performance for affine phase retrieval of sparse real signals.

关键词： Inertial proximal ADMM separable multi-block convex optimization affine phase retrieval

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