On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems

Hochman, Amit; Leviatan, Y.; White, Jacob K.

doi:10.1016/j.jcp.2012.08.015

Cited by 22 publications

(27 citation statements)

References 61 publications

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“…A second difference is that the MFS literature does not make the connection with rational approximation theory, and in particular, does not normally use exponentially clustered poles to achieve root-exponential convergence near singularities, though steps in this direction can be found in [33]. Another related method is that of Hochman, et al [23], involving rational functions with a nonlinear iteration.…”

Section: Variantsmentioning

confidence: 99%

“…However, virtually any exponential clustering is in fact sufficient, provided it scales with n −1/2 as n → ∞, and Section 2 is devoted to presenting theorems to establish this claim. Quite apart from their application to Laplace problems, we believe these results represent a significant addition to the approximation theory literature, as well as shedding light on the clustered poles observed experimentally in [15] and [23]. Section 3 describes our algorithm, which depends on placing sample points on the boundary with exponential clustering to match that of the poles outside.…”

mentioning

confidence: 98%

“…Computing an optimal rational function r is a difficult nonlinear problem in general [23,36,49], but our method is linear because it fixes the poles {z j } of the first sum in (1.3) a priori in a configuration with exponential clustering near each corner. We speak of this as the "Newman" part of (1.3), concerned with resolving singularities.…”

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confidence: 99%

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Solving Laplace Problems with Corner Singularities via Rational Functions

Gopal¹,

Trefethen²

2019

SIAM J. Numer. Anal.

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A new method is introduced for solving Laplace problems on 2D regions with corners by approximation of boundary data by the real part of a rational function with fixed poles exponentially clustered near each corner. Greatly extending a result of D. J. Newman in 1964 in approximation theory, we first prove that such approximations can achieve root-exponential convergence for a wide range of problems, all the way up to the corner singularities. We then develop a numerical method to compute approximations via linear least-squares fitting on the boundary. Typical problems are solved in < 1s on a desktop to 8-digit accuracy, with the accuracy guaranteed in the interior by the maximum principle. The computed solution is represented globally by a single formula, which can be evaluated in a few microseconds at each point.

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Section: Variantsmentioning

confidence: 99%

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confidence: 98%

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Solving Laplace Problems with Corner Singularities via Rational Functions

Gopal¹,

Trefethen²

2019

SIAM J. Numer. Anal.

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“…Recently, Hochman, Leviatan and White [33] also formulated rational least squares approximation using the information from the quadrature nodes. There, the problem is to find real valued potential U that satisfies Laplace equation in a simply connected domain Ω ⊂ R 3.5.…”

Section: Iss1r Modulementioning

confidence: 99%

Quadrature-Based Vector Fitting for Discretized $\mathcal{H}_2$ Approximation

Drmač¹,

Gugercin²,

Beattie³

2015

SIAM J. Sci. Comput.

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Abstract. Vector Fitting is a popular method of constructing rational approximants designed to fit given frequency response measurements. The original method, which we refer to as VF, is based on a least-squares fit to the measurements by a rational function, using an iterative reallocation of the poles of the approximant. We show that one can improve the performance of VF significantly, by using a particular choice of frequency sampling points and properly weighting their contribution based on quadrature rules that connect the least squares objective with an H 2 error measure. Our modified approach, designated here as QuadVF, helps recover the original transfer function with better global fidelity (as measured with respect to the H 2 norm), than the localized least squares approximation implicit in VF. We extend the new framework also to incorporate derivative information, leading to rational approximants that minimize system error with respect to a discrete Sobolev norm. We consider the convergence behavior of both VF and QuadVF as well, and evaluate potential numerical ill-conditioning of the underlying least-squares problems. We investigate briefly VF in the case of noisy measurements and propose a new formulation for the resulting approximation problem. Several numerical examples are provided to support the theoretical discussion.Key words. least squares, frequency response, model order reduction, vector fitting, transfer function AMS subject classifications. 34C20, 41A05, 49K15, 49M05, 93A15, 93C05, 93C151. Introduction. In many engineering applications, the dynamics that govern phenomenae of interest may be inaccessible to direct modeling, yet there may be an abundance of accurate frequency response measurements available. In such cases, one may build up an empirical dynamical system model that fits the measured frequency response data. This empirical system may then be used as a surrogate to predict behavior or derive control strategies.In other settings, one may have complete access to the underlying dynamical system of interest at least in principle (e.g., it may be an analytically derived computational model), however the full system may be a complex aggregate of many large subsystems, each perhaps representing diverse physics, and so it may be of such complexity that direct manipulation of the dynamical system is infeasible; potentially only simulation results would be available. Here, one may wish to capture the dominant dynamic features of the full aggregate system and realize them with a derived dynamical system (presumably of lower order) that can replicate the response characteristics of the full aggregate system. As before, this derived dynamical system may then be used as an efficient surrogate for the full system in contexts where performance is sensitive to model order.A natural formulation of this task leads one to a data fitting problem using rational functions and this ultimately is our principal focus. For convenience, we assume that the system of interest is a single-input/single-output...

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“…An orthonormal space for the Krylov subspace can thus be computed using the Arnoldi process [19, Sect. 10.5], as done for example in [24, App. A].…”

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confidence: 99%

Stable polefinding and rational least-squares fitting via eigenvalues

Ito

Nakatsukasa

2018

Numer. Math.

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A common way of finding the poles of a meromorphic function f in a domain, where an explicit expression of f is unknown but f can be evaluated at any given z, is to interpolate f by a rational function such that at prescribed sample points , and then find the roots of q. This is a two-step process and the type of the rational interpolant needs to be specified by the user. Many other algorithms for polefinding and rational interpolation (or least-squares fitting) have been proposed, but their numerical stability has remained largely unexplored. In this work we describe an algorithm with the following three features: (1) it automatically finds an appropriate type for a rational approximant, thereby allowing the user to input just the function f, (2) it finds the poles via a generalized eigenvalue problem of matrices constructed directly from the sampled values in a one-step fashion, and (3) it computes rational approximants in a numerically stable manner, in that with small at the sample points, making it the first rational interpolation (or approximation) algorithm with guaranteed numerical stability. Our algorithm executes an implicit change of polynomial basis by the QR factorization, and allows for oversampling combined with least-squares fitting. Through experiments we illustrate the resulting accuracy and stability, which can significantly outperform existing algorithms.

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On the use of rational-function fitting methods for the solution of 2D Laplace boundary-value problems

Cited by 22 publications

References 61 publications

Solving Laplace Problems with Corner Singularities via Rational Functions

Solving Laplace Problems with Corner Singularities via Rational Functions

Quadrature-Based Vector Fitting for Discretized $\mathcal{H}_2$ Approximation

Stable polefinding and rational least-squares fitting via eigenvalues

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