Antireflective Boundary Conditions for Deblurring Problems

We propose a numerical algorithm for backward stochastic differential equations based on time discretization and trigonometric wavelets. This method combines the effectiveness of Fourier-based methods and the simplicity of a wavelet-based formula, resulting in an algorithm that is both accurate and easy to implement. Furthermore, we mitigate the problem of errors near the computation boundaries by means of an antireflective boundary technique, giving an improved approximation. We test our algorithm with different numerical experiments.In this article, we shall adopt the quick variant of the SWIFT formula proposed in [13].The reason behind this is left for the error section, but this gives an approximation for E x p [v(t p+1 , X ∆ t p+1 )], which we see in the discrete-time approximation of FBSDE,

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“…(3.6b) Therefore, we have an approximation for h: 7) and the functionŷ ∆ p is defined implicitly by:…”

Section: Quick Swift Approximation Of Function Y ∆ P (X)mentioning

confidence: 99%

“…Antireflective boundary conditions form a popular technique for extrapolation in image deblurring methods. For its applications in image deblurring, the reader may refer to [7] and the references therein.…”

Section: Antireflective Boundarymentioning

confidence: 99%

On the wavelet-based SWIFT method for backward stochastic differential equations

Chau

Oosterlee

2017

IMA Journal of Numerical Analysis

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“…In general, the exact structure of them depends on the imposed boundary conditions. In practice, boundary conditions are often chosen for algebraic and computational convenience [55][56][57] [58,59] . For simplicity, we use the periodic boundary conditions to construct H , ∇ x and ∇ y in this work.…”

Section: Algorithm For Poisson Noisy Image Deblurringmentioning

confidence: 99%

Deblurring Poisson noisy images by total variation with overlapping group sparsity

Jiang

Liu

2016

Applied Mathematics and Computation

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“…For a recent survey on antireflective boundary conditions; see [12]. These boundary conditions give rise to the second difference matrix…”

Section: Antireflective Boundary Conditionsmentioning

confidence: 99%

“…The matrix L D (W ) improves the properties of L D defined in (12) by prescribing that the null space contains the range of W . Specifically,…”

Section: Regularization Matrixmentioning

confidence: 99%

Square smoothing regularization matrices with accurate boundary conditions

Donatelli

Reichel

2014

Journal of Computational and Applied Mathematics

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This paper is concerned with the solution of large-scale linear discrete ill-posed problems. The determination of a meaningful approximate solution of these problems requires regularization. We discuss regularization by the Tikhonov method and by truncated iteration. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. The present paper describes the construction of square regularization matrices from finite difference equations with a focus on the boundary conditions. The regularization matrices considered have a structure that makes them easy to apply in iterative methods, including methods based on the Arnoldi process. Numerical examples illustrate the properties and effectiveness of the regularization matrices described.

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Antireflective Boundary Conditions for Deblurring Problems

Cited by 16 publications

References 47 publications

On the wavelet-based SWIFT method for backward stochastic differential equations

On the wavelet-based SWIFT method for backward stochastic differential equations

Deblurring Poisson noisy images by total variation with overlapping group sparsity

Square smoothing regularization matrices with accurate boundary conditions

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