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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

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

ERROR ESTIMATE OF MULTISCALE FINITE ELEMENT METHOD FOR PERIODIC MEDIA REVISITED

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arXiv 2022年

作者： Ming, Pingbing Song, Siqi LSEC Institute of Computational Mathematics and Scientific/Engineering Computing AMSS Chinese Academy of Sciences No. 55 East Road Zhong-Guan-Cun Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

We derive the optimal energy error estimate for multiscale finite element method with oversampling technique applying to elliptic systems with rapidly oscillating periodic coefficients that are bounded measurable, which may admit rough microstructures. As a by-product of the energy error estimate, we derive the rate of convergence in Ld/(d−1)−norm with d the dimensionality. © 2022, CC BY.

关键词： Finite element method

engineering quantum states with neutral atoms

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arXiv 2024年

作者： Balewski, Jan Kornjača, Milan Klymko, Katherine Darbha, Siva Hirsbrunner, Mark R. Lopes, Pedro L.S. Liu, Fangli Camps, Daan National Energy Research Scientific Computing Center Lawrence Berkeley National Laboratory BerkeleyCA United States QuEra Computing Inc. BostonMA United States Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA United States Dept. of Physics Institute for Condensed Matter Theory University of Illinois at Urbana-Champaign UrbanaIL United States

Aquila, an analog quantum simulation platform developed by QuEra computing, supports control of the position and coherent evolution of up to 256 neutral atoms. This study details novel experimental protocols designed for analog quantum simulators that generate Bell state entanglement far away from the blockade regime, construct a Z2 state with a defect induced by an ancilla, and optimize the driving fields schedule to prepare excited states with enhanced fidelity. We additionally evaluate the effectiveness of readout error mitigation techniques in improving the fidelity of measurement results. All experiments were executed on Aquila from QuEra and facilitated by the AWS Braket interface. Our experimental results closely align with theoretical predictions and numerical simulations. The insights gained from this study showcase Aquila’s capabilities in handling complex quantum simulations and computations, and also pave the way for new avenues of research in quantum information processing and physics that employ programmable analog hardware platforms. Copyright © 2024, The Authors. All rights reserved.

关键词： Excited states

Sparse Approximation of Data-Driven Polynomial Chaos Expansions: An Induced Sampling Approach

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Communications in Mathematical Research 2020年第2期36卷 128-153页

作者： Ling Guo Akil Narayan Yongle Liu Tao Zhou Department of Mathematics Shanghai Normal UniversityShanghai 200234China Department of Mathematics and Scientific Computing and Imaging InstituteUniversity of UtahSalt Lake CityUT 84112USA Department of Mathematics Southern University of Science and TechnologyShenzhen 518055China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China.

One of the open problems in the field of forward uncertainty quantification(UQ)is the ability to form accurate assessments of uncertainty having only incomplete information about the distribution of random *** challenge is to efficiently make use of limited training data for UQ predictions of complex engineering problems,particularly with high dimensional random *** address these challenges by combining data-driven polynomial chaos expansions with a recently developed preconditioned sparse approximation approach for UQ *** first task in this two-step process is to employ the procedure developed in[1]to construct an"arbitrary"polynomial chaos expansion basis using a finite number of statistical moments of the random *** second step is a novel procedure to effect sparse approximation via l1 minimization in order to quantify the forward *** enhance the performance of the preconditioned l1 minimization problem,we sample from the so-called induced distribution,instead of using Monte Carlo(MC)sampling from the original,unknown probability *** demonstrate on test problems that induced sampling is a competitive and often better choice compared with sampling from asymptotically optimal measures(such as the equilibrium measure)when we have incomplete information about the *** demonstrate the capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions,and on a Kirchoff plating bending problem with random Young’s modulus.

关键词： Uncertainty quantification data-driven polynomial chaos expansions sparse approximation equilibrium measure induced measure