Multiscale scattering in nonlinear Kerr-type media

Maier, Roland; Verfürth, Barbara

doi:10.1090/mcom/3722

Cited by 7 publications

(1 citation statement)

References 54 publications

(56 reference statements)

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“…The LOD was introduced in [46] and theoretically and practically works for very general coefficients. It has also been successfully applied to other problem classes, for instance, wave propagation problems in the context of Helmholtz and Maxwell equations [26,27,45,57,61] or the wave equation [4,29,44,59], eigenvalue problems [47,48], and in connection with time-dependent nonlinear Schrödinger equations [37]. However, it requires a slight deviation from locality.…”

Section: Compression By Numerical Homogenizationmentioning

confidence: 99%

Operator compression with deep neural networks

2022

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This paper studies the compression of partial differential operators using neural networks. We consider a family of operators, parameterized by a potentially high-dimensional space of coefficients that may vary on a large range of scales. Based on the existing methods that compress such a multiscale operator to a finite-dimensional sparse surrogate model on a given target scale, we propose to directly approximate the coefficient-to-surrogate map with a neural network. We emulate local assembly structures of the surrogates and thus only require a moderately sized network that can be trained efficiently in an offline phase. This enables large compression ratios and the online computation of a surrogate based on simple forward passes through the network is substantially accelerated compared to classical numerical upscaling approaches. We apply the abstract framework to a family of prototypical second-order elliptic heterogeneous diffusion operators as a demonstrating example.

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Section: Compression By Numerical Homogenizationmentioning

confidence: 99%

Operator compression with deep neural networks

2022

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Higher-Order Finite Element Methods for the Nonlinear Helmholtz Equation

Verfürth

2024

J Sci Comput

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In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and error estimates of the finite element solution under a resolution condition between the wave number k, the mesh size h and the polynomial degree p of the form “$$k(kh)^{p+1}$$ k ( k h ) p + 1 sufficiently small” and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in h from the case $$p=1$$ p = 1 in Wu and Zou (SIAM J Numer Anal 56(3):1338–1359, 2018) can be removed for $$p\ge 2$$ p ≥ 2 . We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data.

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A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity

Angermann

2023

Communications in Nonlinear Science and Numerical Simulation

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Multiscale scattering in nonlinear Kerr-type media

Cited by 7 publications

References 54 publications

Operator compression with deep neural networks

Operator compression with deep neural networks

Higher-Order Finite Element Methods for the Nonlinear Helmholtz Equation

A radiation and propagation problem for a Helmholtz equation with a compactly supported nonlinearity

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