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A DISCONTINUOUS PLANE WAVE NEURAL NETWORK METHOD FOR HELMHOLTZ EQUATION AND TIME-HARMONIC MAXWELL'S EQUATIONS

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arXiv 2023年

作者： Yuan, Long Hu, Qiya College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao266590 China LSEC Institute of Computational Mathematics and Scientic/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this paper we propose a discontinuous plane wave neural network (DPWNN) method with hp−refinement for approximately solving Helmholtz equation and time-harmonic Maxwell equations. In this method, we define a quadratic functional as in the plane wave least square (PWLS) method with h−refinement and introduce new discretization sets spanned by element-wise neural network functions with a single hidden layer, where the activation function on each element is chosen as a complex-valued exponential function like the plane wave function. The desired approximate solution is recursively generated by iteratively solving a quasi-minimization problem associated with the functional and the sets described above, which is defined by a sequence of approximate minimizers of the underlying residual functionals, where plane wave direction angles and activation coefficients are alternatively computed by iterative algorithms. For the proposed DPWNN method, the plane wave directions are adaptively determined in the iterative process, which is different from that in the standard PWLS method (where the plane wave directions are preliminarily given). Numerical experiments will confirm that this DPWNN method can generate approximate solutions with higher accuracy than the PWLS method. Copyright © 2023, The Authors. All rights reserved.

关键词： Wave propagation

ON A CLASSIFICATION OF STEADY SOLUTIONS TO TWO-DIMENSIONAL EULER EQUATIONS

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arXiv 2024年

作者： Gui, Changfeng Xie, Chunjing Xu, Huan Department of Mathematics University of Macau China School of Mathematical Sciences Institute of Natural Sciences Ministry of Education Key Laboratory of Scientific and Engineering Computing CMA-Shanghai Shanghai Jiao Tong University Shanghai200240 China Department of Mathematics The University of Texas at San Antonio San AntonioTX78249 United States

In this paper, we provide a classification of steady solutions to two-dimensional incompressible Euler equations in terms of the set of flow angles. The first main result asserts that the set of flow angles of any bounded steady flow in the whole plane must be the whole circle unless the flow is a parallel shear flow. In an infinitely long horizontal strip or the upper half-plane supplemented with slip boundary conditions, besides the two types of flows appeared in the whole space case, there exists an additional class of steady flows for which the set of flow angles is either the upper or lower closed semicircles. This type of flows is proved to be the class of non-shear flows that have the least total curvature. As applications, we obtain Liouville-type theorems for two-dimensional semilinear elliptic equations with only bounded and measurable nonlinearity, and the structural stability of shear flows whose all stagnation points are not inflection points, including Poiseuille flow as a special case. Our proof relies on the analysis of some quantities related to the curvature of the streamlines. Copyright © 2024, The Authors. All rights reserved.

关键词： Stability

A high-order unfitted finite element method for moving interface problems

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arXiv 2021年

作者： Ma, Chuwen Zheng, Weiying School of Mathematical Science University of Chinese Academy of Sciences Institute of Computational Mathematics and Scientific Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China LSEC Institute of Computational Mathematics and Scientific Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing100190 China School of Mathematical Science University of Chinese Academy of Sciences China

We propose a kth-order unfitted finite element method (2 ≤ k ≤ 4) to solve moving interface problem of the Oseen equations. Thorough error estimates for the discrete solutions are presented by considering errors from interface-tracking, time integration, and spatial discretization. In literatures on time-dependent Stokes interface problems, error estimates for the discrete pressure are usually sub-optimal, namely, (k−1)th-order, under the L2-norm. We have obtained a (k − 1)th-order error estimate for the discrete pressure under the H1-norm. Numerical experiments for a severely deforming interface show that optimal convergence orders are obtained for k = 3 and 4. Copyright © 2021, The Authors. All rights reserved.

关键词： Convolution

Smoothness and continuity of cost functionals for ECG mismatch computation

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IFAC-PapersOnLine 2022年第20期55卷 181-186页

作者： Thomas Grandits Simone Pezzuto Gernot Plank Institute of Mathematics and Scientific Computing University of Graz Graz Austria Gottfried Schatz Research Center - Division of Biophysics Medical University of Graz Graz Austria Center for Computational Medicine in Cardiology Euler Institute Università della Svizzera italiana Lugano Switzerland BioTechMed-Graz Graz Austria

The field of cardiac electrophysiology tries to abstract, describe and finally model the electrical characteristics of a heartbeat. With recent advances in cardiac electrophysiology, models have become more powerful and descriptive as ever. However, to advance to the field of inverse electrophysiological modeling, i.e. creating models from electrical measurements such as the ECG, the less investigated field of smoothness of the simulated ECGs w.r.t. model parameters need to be further explored. The present paper discusses smoothness in terms of the whole pipeline which describes how from physiological parameters, we arrive at the simulated ECG. Employing such a pipeline, we create a test-bench of a simplified idealized left ventricle model and demonstrate the most important factors for efficient inverse modeling through smooth cost functionals. Such knowledge will be important for designing and creating inverse models in future optimization and machine learning methods.

关键词： ECG Cardiac modeling Physiology Parameter estimation Inverse problems Smoothness

Hessian Riemannian Flow For Multi-Population Wardrop Equilibrium

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arXiv 2025年

作者： Bakaryan, Tigran Aoun, Christoph Lima Ribeiro, Ricardo de Hovakimyan, Naira Gomes, Diogo Center for Scientific Innovation and Education Institute of Mathematics NAS RA Yerevan Armenia Department of Aerospace Engineering University of Illinois at Urbana-Champaign UrbanaIL61801 United States Applied Mathematics and Computational Sciences King Abdullah University of Science and Technology Thuwal23955 Saudi Arabia Department of Mechanical Science and Engineering University of Illinois at Urbana-Champaign UrbanaIL61801 United States

In this paper, we address the problem of optimizing flows on generalized graphs that feature multiple entry points and multiple populations, each with varying cost structures. We tackle this problem by considering the multi-population Wardrop equilibrium, defined through variational inequalities. We rigorously analyze the existence and uniqueness of the Wardrop equilibrium. Furthermore, we introduce an efficient numerical method to find the solution. In particular, we reformulate the equilibrium problem as a distributed optimization problem over subgraphs and introduce a novel Hessian Riemannian flow method—a Riemannian-manifold-projected Hessian flow—to efficiently compute a solution. Finally, we demonstrate the effectiveness of our approach through examples in urban traffic management, including routing for diverse vehicle types and strategies for minimizing emissions in congested environments. © 2025, CC BY-NC-ND.

关键词： Highway administration

HYDRODYNAMICS FOR ASYMMETRIC SIMPLE EXCLUSION ON A FINITE SEGMENT WITH GLAUBER-TYPE SOURCE

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arXiv 2024年

作者： Xu, Lu Zhao, Linjie Gran Sasso Science Institute AQ L'Aquila67100 Italy School of Mathematics and Statistics Huazhong University of Science & Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

We consider an open interacting particle system on a finite lattice. The particles perform asymmetric simple exclusion and are randomly created or destroyed at all sites, with rates that grow rapidly near the boundaries. We study the hydrodynamic limit for the particle density at the hyperbolic space-time scale and obtain the entropy solution to a boundary-driven quasilinear conservation law with a source term. Different from the usual boundary conditions introduced in [3, 20], discontinuity (boundary layer) does not formulate at the boundaries due to the strong relaxation scheme. Copyright © 2024, The Authors. All rights reserved.

关键词： Hydrodynamics

A mathematical aspect of Hohenberg-Kohn theorem

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Science China mathematics 2019年第1期62卷 63-68页

作者： Aihui Zhou LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences

The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become the most popular and powerful computational approach to study the electronic structure of *** this article, we study t... 详细信息

关键词： density functional theory electronic structure unique continuation principle Hohenberg-Kohn theorem

Stochastic Three-Operator Splitting Algorithms for Nonconvex and Nonsmooth Optimization Arising from FLASH Radiotherapy

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arXiv 2023年

作者： Bian, Fengmiao Liu, Jiulong Zhang, Xiaoqun Gao, Hao Cai, Jian-Feng Department of Mathematics The Hong Kong University of Science and Technology Hong Kong Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing China School of Mathematical Sciences MOE-LSC Institute of Natural Sciences Shanghai Jiao Tong University Shanghai China Department of Radiation Oncology University of Kansas Medical Center Kansas CityKS United States HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute Futian Shenzhen China

Radiation therapy (RT) aims to deliver tumoricidal doses with minimal radiation-induced normal-tissue toxicity. Compared to conventional RT (of conventional dose rate), FLASH-RT (of ultra-high dose rate) can provide additional normal tissue sparing, which however has created a new nonconvex and nonsmooth optimization problem that is highly challenging to solve. In this paper, we propose a stochastic three-operator splitting (STOS) algorithm to address the FLASH optimization problem. We establish the convergence and convergence rates of the STOS algorithm under the nonconvex framework for both unbiased gradient estimators and variance-reduced gradient estimators. These stochastic gradient estimators include the most popular ones, such as SGD, SAGA, SARAH, and SVRG, among others. The effectiveness of the STOS algorithm is validated using FLASH radiotherapy planning for patients. Copyright © 2023, The Authors. All rights reserved.

关键词： Radiotherapy

NeurAM: nonlinear dimensionality reduction for uncertainty quantification through neural active manifolds

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arXiv 2024年

作者： Zanoni, Andrea Geraci, Gianluca Salvador, Matteo Marsden, Alison L. Schiavazzi, Daniele E. Institute for Computational and Mathematical Engineering Stanford University StanfordCA United States Center for Computing Research Sandia National Laboratories AlbuquerqueNM United States Stanford University StanfordCA United States Stanford University StanfordCA United States Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN United States

We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the model output variability, plus a simultaneously learnt surrogate model with inputs on this manifold. The proposed dimensionality reduction framework can then be applied to perform outer loop many-query tasks, like sensitivity analysis and uncertainty propagation. In particular, we prove, both theoretically under idealized conditions, and numerically in challenging test cases, how NeurAM can be used to obtain multifidelity sampling estimators with reduced variance by sampling the models on the discovered low-dimensional and shared manifold among models. Several numerical examples illustrate the main features of the proposed dimensionality reduction strategy, and highlight its advantages with respect to existing approaches in the literature. Copyright © 2024, The Authors. All rights reserved.

关键词： Dimensionality reduction