An Efficient Quasi-Newton Method with Tensor Product Implementation for Solving Quasi-Linear Elliptic Equations and Systems

In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of linear Laplacian and simplified nonlinear terms, our method redu...

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Vydané v:	Journal of scientific computing Ročník 103; číslo 3; s. 89
Hlavní autori:	Hao, Wenrui, Lee, Sun, Zhang, Xiangxiong
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.06.2025 Springer Nature B.V
Predmet:	Algorithms Boundary conditions Computational Mathematics and Numerical Analysis Convergence Efficiency Elliptic functions Exact solutions Finite volume method Jacobi matrix method Jacobian matrix Mathematical analysis Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Numerical analysis Partial differential equations Quasi Newton methods Regularization Sparse matrices Tensors Theoretical Quasi-Linear Elliptic equations and systems 90C53 Quasi-Newton method 65N35 35J62 Tensor product Kronecker product
ISSN:	0885-7474, 1573-7691
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Popis
Shrnutí:	In this paper, we introduce a quasi-Newton method optimized for efficiently solving quasi-linear elliptic equations and systems, with a specific focus on GPU-based computation. By approximating the Jacobian matrix with a combination of linear Laplacian and simplified nonlinear terms, our method reduces the computational overhead typical of traditional Newton methods while handling the large, sparse matrices generated from discretized PDEs. We also provide a convergence analysis demonstrating local convergence to the exact solution under optimal choices for the regularization parameter, ensuring stability and efficiency in each iteration. Numerical experiments in two- and three-dimensional domains validate the proposed method’s robustness and computational gains with tensor product implementation. This approach offers a promising pathway for accelerating quasi-linear elliptic equations and systems solvers, expanding the feasibility of complex simulations in physics, engineering, and other fields leveraging advanced hardware capabilities.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23
ISSN:	0885-7474 1573-7691
DOI:	10.1007/s10915-025-02897-y