Reciprocal-Log Approximation and Planar PDE Solvers

Nakatsukasa, Yuji; Trefethen, Lloyd N.

doi:10.1137/20m1369555

Cited by 8 publications

(3 citation statements)

References 18 publications

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“…In the context of the global representations explored in this paper, a natural idea for such problems would be to combine a general purpose set of basis functions to capture the "smooth part" of the solution with additional singular terms near the corners. For Laplace Dirichlet or Neumann problems, representations of this kind led to the lightning and log-lightning solvers introduced in [16] and [26]. Here we illustrate that such an approach may be effective for eigenvalue problems too.…”

Section: Two-dimensional Examples (Pdes)mentioning

confidence: 87%

“…In the finite elements literature there are Least-Squares Finite Element Methods [6,20,24]. With expansion functions that satisfy the differential equation but not the boundary conditions, one gets series methods [33] or the Method of Fundamental Solutions (MFS) [3,13] or lightning or log-lightning methods for PDE problems with corner singularities [16,26]. Related expansions that do not satisfy the differential equation and hence need fitting in the interior of a domain, not just on the boundary, lead to least-squares methods for radial basis functions (RBFs) or other kernels [10,14,21,29].…”

Section: Introductionmentioning

confidence: 99%

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Rectangular eigenvalue problems

2022

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Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m ≫ n collocation points. We show how eigenvalue problems can be solved in this setting by QR reduction to square matrix generalized eigenvalue problems. The method applies equally in the limit “$m=\infty $ m = ∞ ” of eigenvalue problems for quasimatrices. Numerical examples are presented as well as pointers to related literature.

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Section: Two-dimensional Examples (Pdes)mentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Rectangular eigenvalue problems

2022

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“…AAA converges without computing spurious poles, so the cleanup step proposed in [29] is not performed. The application of AAA to functions with branch cut singularities, as the FD model function, has been recently examined in [58].…”

Section: Viscoelastic Sandwich Beam Modelmentioning

confidence: 99%

Automatic model order reduction for systems with frequency-dependent material properties

Aumann

Deckers

Jonckheere

et al. 2022

Computer Methods in Applied Mechanics and Engineering

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Numerical analytic continuation

Trefethen

2023

Japan J. Indust. Appl. Math.

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Let f be an analytic function on a simply-connected compact continuum E of the complex z-plane. This might be an interval of the real line, where f might be real analytic. How can we calculate good estimates of the analytic continuation of f to other points $$z\in {\mathbb C}$$ z ∈ C ? How can we estimate the locations of real or complex singularities of f? We review both the theory and the practice of some existing methods for these problems and propose that excellent results can be obtained from the computation of rational approximations of f by the AAA algorithm. In the case of analytic functions of two or more variables, the rational approximations are applied along line segments or other analytic arcs.

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Reciprocal-Log Approximation and Planar PDE Solvers

Cited by 8 publications

References 18 publications

Rectangular eigenvalue problems

Rectangular eigenvalue problems

Automatic model order reduction for systems with frequency-dependent material properties

Numerical analytic continuation

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