Theoretical Numerical Analysis

Atkinson, Kendall; Han, Weimin

doi:10.1007/978-1-4419-0458-4

Cited by 92 publications

(16 citation statements)

References 130 publications

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“…First, note that these constants depend on . Using the Dirichlet kernel, one proves that I n is uniformly bounded from C./ to an L 2 -space with the Chebyshev weight [2]. The norm on this weighted space dominates the usual L 2 ./-norm, proving the first result.…”

Section: A Numerical Realizationmentioning

confidence: 95%

Nonlinear Steepest Descent and Numerical Solution of Riemann‐Hilbert Problems

Olver

Trogdon

2013

Comm Pure Appl Math

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The effective and efficient numerical solution of Riemann-Hilbert problems has been demonstrated in recent work. With the aid of ideas from the method of nonlinear steepest descent for RiemannHilbert problems, the resulting numerical methods have been shown numerically to retain accuracy as values of certain parameters become arbitrarily large. Remarkably, this numerical approach does not require knowledge of local parametrices; rather, the deformed contour is scaled near stationary points at a specific rate. The primary aim of this paper is to prove that this observed asymptotic accuracy is indeed achieved. To do so, we first construct a general theoretical framework for the numerical solution of Riemann-Hilbert problems. Second, we demonstrate the precise link between nonlinear steepest descent and the success of numerics in asymptotic regimes. In particular, we prove sufficient conditions for numerical methods to retain accuracy. Finally, we compute solutions to the homogeneous Painlevé II equation and the modified Korteweg-de Vries equations to explicitly demonstrate the practical validity of the theory.

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Section: A Numerical Realizationmentioning

confidence: 95%

Nonlinear Steepest Descent and Numerical Solution of Riemann‐Hilbert Problems

Olver

Trogdon

2013

Comm Pure Appl Math

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“…The following statement is well known (see [1]): suppose that g ∈ C 2 [a, b] and Π g is piecewise linear interpolating function of g. That is, Π g| [xi−1,xi] is linear function, which satisfies Π g(x i ) = g(x i ), then…”

Section: Error Order Estimationmentioning

confidence: 97%

“…More precisely, let R t (x) and r t (x) be the reproducing kernel functions of W 0 m [0, 1] and W 1 [0, 1], respectively. 1], and L * is the adjoint operator of L. Note that S n span{ψ i (x)} 1≤i≤n , and P n : W 0 m → S n is an orthogonal projection operator. It is easy to observe that…”

Section: Introductionmentioning

confidence: 99%

Convergence Order of the Reproducing Kernel Method for Solving Boundary Value Problems

Zhao

Yang

Jing

2016

Mathematical Modelling and Analysis

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In this paper, convergence rate of the reproducing kernel method for solving boundary value problems is studied. The equivalence of two reproducing kernel spaces and some results of adjoint operator are proved. Based on the classical properties of piecewise linear interpolating function, we provide the convergence rate analysis of at least second order. Moreover, some numerical examples showing the accuracy of the proposed estimations are also given.

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“…We remark that some of these assumptions can probably be weakened, nevertheless they are not uncommon for image processing purposes and ease the discussion on a few occasions. 1.…”

Section: Weighted Sobolev Spacesmentioning

confidence: 99%

“…The previous theorem can be seen as a generalisation to weighted spaces of a well-known theorem for constructing equivalent norms out of seminorms in regular Sobolev spaces. See Theorem 7.3.12 in [1]. Equation (2.26) can also be considered as a higher dimensional generalisation of the Hardy inequality.…”

Section: Weighted Sobolev Spacesmentioning

confidence: 99%

Theoretical foundation of the weighted Laplace inpainting problem

et al. 2019

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Laplace interpolation is a popular approach in image inpainting using partial differential equations. The classic approach considers the Laplace equation with mixed boundary conditions. Recently a more general formulation has been proposed where the differential operator consists of a point-wise convex combination of the Laplacian and the known image data. We provide the first detailed analysis on existence and uniqueness of solutions for the arising mixed boundary value problem. Our approach considers the corresponding weak formulation and aims at using the Theorem of Lax-Milgram to assert the existence of a solution. To this end we have to resort to weighted Sobolev spaces. Our analysis shows that solutions do not exist unconditionally. The weights need some regularity and fulfil certain growth conditions. The results from this work complement findings which were previously only available for a discrete setup.

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Theoretical Numerical Analysis

Cited by 92 publications

References 130 publications

Nonlinear Steepest Descent and Numerical Solution of Riemann‐Hilbert Problems

Nonlinear Steepest Descent and Numerical Solution of Riemann‐Hilbert Problems

Convergence Order of the Reproducing Kernel Method for Solving Boundary Value Problems

Theoretical foundation of the weighted Laplace inpainting problem

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