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A SPECTRAL GALERKIN EXPONENTIAL EULER TIME-STEPPING SCHEME FOR PARABOLIC SPDES ON TWO-DIMENSIONAL DOMAINS WITH A C2 BOUNDARY

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arXiv 2023年

We consider the numerical approximation of second-order semi-linear parabolic stochastic partial differential equations interpreted in the mild sense which we solve on general two-dimensional domains with a C2 boundary with homogeneous Dirichlet boundary conditions. The equations are driven by Gaussian additive noise, and several Lipschitz-like conditions are imposed on the nonlinear function. We discretize in space with a spectral Galerkin method and in time using an explicit Euler-like scheme. For irregular shapes, the necessary Dirichlet eigenvalues and eigenfunctions are obtained from a boundary integral equation method. This yields a nonlinear eigenvalue problem, which is discretized using a boundary element collocation method and is solved with the Beyn contour integral algorithm. We present an error analysis as well as numerical results on an exemplary asymmetric shape, and point out limitations of the approach. © 2023, CC BY.

关键词： Eigenvalues and eigenfunctions

High-resolution simulations of nonlinear electromagnetic turbulence in tokamak devices

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The European Physical Journal Special Topics 2025年 1-14页

作者： Liu, Yanjun Liu, Zixi Liu, Jian Gao, Baofeng Zhang, Wei Xia, Tianyang Liu, Haiqing Zhuang, Ge Gao, Xiang Institute of Plasma Physics Chinese Academy of Sciences Anhui China KTX Laboratory and Department of Engineering and Applied Physics University of Science and Technology of China Anhui China Weihai Institute for Interdisciplinary Research Shandong University Shandong China SDU-ANU Joint Science College Shandong University Shandong China Shandong Computer Science Center (National Supercomputing Center in Jinan) Qilu University of Technology (Shandong Academy of Sciences) Shandong China Advanced Algorithm Joint Lab National Supercomputing Center in Jinan Shandong China

To achieve long-pulse steady operation, the physical mechanisms of boundary turbulence need further investigation. We employ the two-fluid model with flute reduction on BOUT++ to simulate the boundary plasma in Tokamaks. The space and time scales of turbulence reproduced by our simulations closely relate to the spatial mesh size and time step size, respectively. As an inherent time scale, the Alfven time is sufficient to resolve MHD instabilities. The spatial scale can be refined by increasing mesh resolutions, which necessitates larger scale parallel computing resources. We have conducted nonlinear simulations using more than 33 million spatial meshes with 16,384 CPU processors in parallel. The results indicate that while the decrease in parallel efficiency with an increase in core numbers does not necessarily lead to shorter runtimes, higher computational complexity improves parallel efficiency for the same number of cores. In addition, the mesh resolution required for convergence conditions differs between linear and nonlinear simulations, with nonlinear simulations demanding higher resolution. Besides finer structure obtained, the fluctuation characteristic of density similar to WCM, which is more consistent with the experimental observation, also shows the requirement for high-resolution meshes and large-scale computing in the future.

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Learning-Driven Annealing with Adaptive Hamiltonian Modification for Solving Large-Scale Problems on Quantum Devices

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arXiv 2025年

作者： Schulz, Sebastian Willsch, Dennis Michielsen, Kristel Jülich Supercomputing Centre Institute for Advanced Simulation Forschungszentrum Jülich Jülich52425 Germany Faculty of Medical Engineering and Technomathematics University of Applied Sciences Aachen Jülich52428 Germany AIDAS Jülich52425 Germany RWTH Aachen University Aachen52056 Germany

We present Learning-Driven Annealing (LDA), a framework that links individual quantum annealing evolutions into a global solution strategy to mitigate hardware constraints such as short annealing times and integrated control errors. Unlike other iterative methods, LDA does not tune the annealing procedure (e.g. annealing time or annealing schedule), but instead learns about the problem structure to adaptively modify the problem Hamiltonian. By deforming the instantaneous energy spectrum, LDA suppresses transitions into high-energy states and focuses the evolution into low-energy regions of the Hilbert space. We demonstrate the efficacy of LDA by developing a hybrid quantum-classical solver for large-scale spin glasses. The hybrid solver is based on a comprehensive study of the internal structure of spin glasses, outperforming other quantum and classical algorithms (e.g., reverse annealing, cyclic annealing, simulated annealing, Gurobi, Toshiba’s SBM, VeloxQ and D-Wave hybrid) on 5580-qubit problem instances in both runtime and lowest energy. LDA is a step towards practical quantum computation that enables today’s quantum devices to compete with classical solvers. Copyright © 2025, The Authors. All rights reserved.

关键词： Quantum computers

Parallelization of Particle-Mass-Transfer Algorithms on Shared-Memory, Multi-Core Cpus

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SSRN

SSRN 2024年

作者： Benson, David A. Pribec, Ivan Engdahl, Nicholas B. Pankavich, Stephen D. Schauer, Lucas Hydrologic Science and Engineering Colorado School of Mines GoldenCO80401 United States Leibniz Supercomputing Center Boltzmannstraße 1 Garching bei MünchenD-85748 Germany Department of Civil and Environmental Engineering Washington State University PullmanWA United States Department of Applied Mathematics and Statistics Colorado School of Mines GoldenCO80401 United States

Simulating the transfer of mass between particles is not straightforwardly parallelized because it involves the calculation of the influence of many particles on each other. Engdahl et al. (2019) intuited that the number of matrix operations used for mass transfer grows quadratically with the number of particles, so that dividing the domain geometrically into sub-domains will give speed and memory advantages, even on a single processing thread. Those authors also showed the speed scalability of several one-dimensional examples on multiple cores. Here, we extend those results for more general cases, both in terms of spatial dimensions and algorithmic implementation. We show that there is an optimal subdivision scheme for naive, full-matrix calculations on a multi- processor, or multi-threading shared-memory machine. A similar sparse-matrix implementation that also uses row-and-column-sum normalization often greatly reduces the memory requirements. We also introduce a new mass transfer algorithm that uses a non-geometric domain decomposition and only matrix row-sum normalization. This allows the mass-transfer "matrix" to be constructed and solved one row at a time in parallel, so it is faster and vastly more memory efficient than previous methods, but requires more care for suitable accuracy. © 2024, The Authors. All rights reserved.

关键词： Hydrodynamics

On the transmission eigenvalues for scattering by a clamped planar region

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arXiv 2025年

作者： Harris, Isaac Lee, Heejin Kleefeld, Andreas Department of Mathematics Purdue University West LafayetteIN47907 United States Forschungszentrum Jülich GmbH Jülich Supercomputing Centre Wilhelm-Johnen-Straße Jülich52425 Germany University of Applied Sciences Aachen Faculty of Medical Engineering and Technomathematics Heinrich-Mußmann-Str. 1 Jülich52428 Germany

In this paper, we consider a new transmission eigenvalue problem derived from the scattering by a clamped cavity in a thin elastic material. Scattering in a thin elastic material can be modeled by the Kirchhoff–Love infinite plate problem. This results in a biharmonic scattering problem that can be handled by operator splitting. The main novelty of this transmission eigenvalue problem is that it is posed in all of R2. This adds analytical and computational difficulties in studying this eigenvalue problem. Here, we prove that the eigenvalues can be recovered from the far field data as well as discreteness of the transmission eigenvalues. We provide some numerical experiments via boundary integral equations to demonstrate the theoretical results. We also conjecture monotonicity with respect to the measure of the scatterer from our numerical experiments. © 2025, CC BY.

关键词： Choquet integral

Sampling methods for recovering buried corroded boundaries from partial electrostatic Cauchy data

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arXiv 2024年

We consider the inverse shape and parameter problem for detecting corrosion from partial boundary measurements. This problem models the non-destructive testing for a partially buried object from electrostatic measurements on the accessible part of the boundary. The main novelty is the extension of the linear sampling and factorization methods to an electrostatic problem with partial measurements. These methods so far have only mainly applied to recovering interior defects which is a simpler problem. Another important aspect of this paper is in our numerics, where we derive a system of boundary integral equations to recover the mixed Green’s function which is needed for our inversion. With this, we are able to analytically and numerically solve the inverse shape problem. For the inverse parameter problem, we prove uniqueness and Lipschitz-stability (in a finite dimensional function space) assuming that one has the associated Neumann-to-Dirichlet operator on the accessible part of the boundary. © 2024, CC BY.

关键词： Boundary integral equations

A fourth-order exponential time differencing scheme with dimensional splitting for non-linear reaction-diffusion systems

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arXiv 2024年

作者： Asante-Asamani, E.O. Kleefeld, A. Wade, B.A. Department of Mathematics Clarkson University PotsdamNY13676 United States Jülich Supercomputing Centre Forschungszentrum Jülich GmbH Jülich52425 Germany Faculty of Medical Engineering and Technomathematics University of Applied Sciences Aachen Jülich52428 Germany Department of Mathematics University of Louisiana at Lafayette LafayetteLA70504 United States

A fourth-order exponential time differencing (ETD) Runge-Kutta scheme with dimensional splitting is developed to solve multidimensional non-linear systems of reaction-diffusion equations (RDE). By approximating the matrix exponential in the scheme with the A-acceptable Padé (2,2) rational function, the resulting scheme (ETDRK4P22-IF) is verified empirically to be fourth-order accurate for several RDE. The scheme is shown to be more efficient than competing fourth-order ETD and IMEX schemes, achieving up to 20X speedup in CPU time. Inclusion of up to three pre-smoothing steps of a lower order L-stable scheme facilitates efficient damping of spurious oscillations arising from problems with non-smooth initial/boundary conditions. © 2024, CC BY.

关键词： Rational functions

Two direct sampling methods for an anisotropic scatterer with a conductive boundary

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arXiv 2024年

作者： Hughes, Victor Harris, Isaac Kleefeld, Andreas Department of Mathematics Purdue University West Lafayette47907 United States Forschungszentrum Jülich GmbH Jülich Supercomputing Centre Wilhelm-Johnen-Straße Jülich52425 Germany University of Applied Sciences Aachen Faculty of Medical Engineering Technomathematics Heinrich-Mußmann-Str. 1 Jülich52428 Germany

In this paper, we consider the inverse scattering problem associated with an anisotropic medium with a conductive boundary condition. We will assume that the corresponding far–field pattern or Cauchy data is either known or measured. The conductive boundary condition models a thin coating around the boundary of the scatterer. We will develop two direct sampling methods to solve the inverse shape problem by numerically recovering the scatterer. To this end, we study direct sampling methods by deriving that the corresponding imaging functionals decay as the sampling point moves away from the scatterer. These methods have been applied to other inverse shape problems, but this is the first time they will be applied to an anisotropic scatterer with a conductive boundary condition. These methods allow one to recover the scatterer by considering an inner–product of the far–field data or the Cauchy data. Here, we will assume that the Cauchy data is known on the boundary of a region Ω that completely encloses the scatterer D. We present numerical reconstructions in two dimensions to validate our theoretical results for both circular and non-circular scatterers. © 2024, CC BY.

关键词： Anisotropic media

Inverse parameter and shape problem for an isotropic scatterer with two conductivity coefficients

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arXiv 2024年

作者： Ayala, Rafael Ceja Harris, Isaac Kleefeld, Andreas Department of Mathematics Purdue University West LafayetteIN47907 United States Forschungszentrum Jülich GmbH Jülich Supercomputing Centre Wilhelm-Johnen-Straße Jülich52425 Germany University of Applied Sciences Aachen Faculty of Medical Engineering Technomathematics Heinrich-Mußmann-Str. 1 Jülich52428 Germany

In this paper, we consider the direct and inverse problem for isotropic scatterers with two conductive boundary conditions. First, we show the uniqueness for recovering the coefficients from the known far-field data at a fixed incident direction for multiple frequencies. Then, we address the inverse shape problem for recovering the scatterer for the measured far-field data at a fixed frequency. Furthermore, we examine the direct sampling method for recovering the scatterer by studying the factorization for the far-field operator. The direct sampling method is stable with respect to noisy data and valid in two dimensions for partial aperture data. The theoretical results are verified with numerical examples to analyze the performance by the direct sampling method. © 2024, CC BY.

关键词： Inverse problems