Spectral Approximation of Fractional PDEs in Image Processing and Phase Field Modeling

Antil, Harbir; Bartels, Sören

doi:10.1515/cmam-2017-0039

Cited by 80 publications

(102 citation statements)

References 42 publications

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“…Among others, we mention applications in turbulence, anomalous transport and diffusion, elasticity, image processing, porous media flow, wave propagation in heterogeneous high contrast media (see e.g. [1,8,36,34] and their references). Also, it is well known that the fractional Laplace operator is the generator of the so-called s-stable Lévy process, and it is often used in stochastic models with applications, for instance, in mathematical finance (see e.g.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Analysis of the controllability from the exterior of strong damping nonlocal wave equations

Warma

Zamorano

2020

ESAIM: COCV

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We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval (−1, 1) associated with the fractional Laplace operator (−∂ 2x ) s , where 0 < s < 1. We show that there is a control function which is localized in a nonempty open set O ⊂ (R \ (−1, 1)), that is, at the exterior of the interval (−1, 1), such that the system is null controllable at any time T > 0 if and only if 1 2 < s < 1.2010 Mathematics Subject Classification. 35R11, 35S11, 35K20, 93B05, 93B07.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Analysis of the controllability from the exterior of strong damping nonlocal wave equations

Warma

Zamorano

2020

ESAIM: COCV

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“…Such a problem has a unique solution in the fractional order Sobolev space X = H s (Ω) [30]. This regularization has been applied successfully in image denoising [2] (with K = I, but with u ∈ X, instead of X ad , as a result (2.6) becomes an equality). The success of this regularizer can be attributed to the fact that when s < 1 2 , the fractional Sobolev-space H s (Ω) is larger than BV (Ω) ∩ L ∞ (Ω), see [2].…”

Section: Fractional Laplacianmentioning

confidence: 99%

Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography

Antil

Khatri

2020

Inverse Problems

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In this work we consider a generalized bilevel optimization framework for solving inverse problems. We introduce fractional Laplacian as a regularizer to improve the reconstruction quality, and compare it with the total variation regularization. We emphasize that the key advantage of using fractional Laplacian as a regularizer is that it leads to a linear operator, as opposed to the total variation regularization which results in a nonlinear degenerate operator. Inspired by residual neural networks, to learn the optimal strength of regularization and the exponent of fractional Laplacian, we develop a dedicated bilevel optimization neural network with a variable depth for a general regularized inverse problem. We also draw some parallels between an activation function in a neural network and regularization. We illustrate how to incorporate various regularizer choices into our proposed network. As an example, we consider tomographic reconstruction as a model problem and show an improvement in reconstruction quality, especially for limited data, via fractional Laplacian regularization. We successfully learn the regularization strength and the fractional exponent via our proposed bilevel optimization neural network. We observe that the fractional Laplacian regularization outperforms total variation regularization. This is specially encouraging, and important, in the case of limited and noisy data.

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“…where µ := −ε∆u + 1 ε f (u) is often called the chemical potential. On the other hand, equation (16) in Example 5 is the L 2 -gradient flow of the Willmore energy functional (17). The energy law for the coupled Navier-Stokes and Cahn-Hilliard system in Example 8 can also be formulated using the idea of flow map for the total energy that is the sum of the phase field interfacial energy and the fluid kinetic energy.…”

Section: Phase Field Formulations Of Other Moving Interface Problemsmentioning

confidence: 99%

“…Algorithmic development and numerical analysis concerning these nonlocal models (including the fractional ones in either space or time or both ) can be found in [6,49,159,150,238,260,394,445]. Related algorithmic studies with respect to different applications were presented in [17].…”

Section: Fluid and Solid Mechanicsmentioning

confidence: 99%

The phase field method for geometric moving interfaces and their numerical approximations

Feng

2020

Handbook of Numerical Analysis

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This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state-of-the-art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modeling and their numerical methods in materials science, fluid mechanics, biology and image science. Key words and phrases. Phase field method, geometric law, curvature-driven flow, geometric nonlinear PDEs, finite difference methods, finite element methods, spectral methods, discontinuous Galerkin methods, adaptivity, coarse and fine error estimates, convergence of numerical interfaces, nonlocal and stochastic phase field models, microstructure evolution, biology and image science applications.

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Spectral Approximation of Fractional PDEs in Image Processing and Phase Field Modeling

Cited by 80 publications

References 42 publications

Analysis of the controllability from the exterior of strong damping nonlocal wave equations

Analysis of the controllability from the exterior of strong damping nonlocal wave equations

Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography

The phase field method for geometric moving interfaces and their numerical approximations

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