Two stable methods with numerical experiments for solving the backward heat equation

Ternat, Fabien; Orellana, Oscar; Daripa, Prabir

doi:10.1016/j.apnum.2010.09.006

Cited by 16 publications

(7 citation statements)

References 21 publications

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“…while the DIRK scheme is: Ternat et al (2011) is done for both the DIRK and the CN schemes. For brevity we skip the details of this analysis and only compare their performance via an example.…”

Section: Numerical Scheme For the Linear Equationmentioning

confidence: 99%

On a fractional step-splitting scheme for the Cahn-Hilliard equation

Aderogba

Chapwanya

Djoko

2014

Engineering Computations

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For a partial differential equation with a fourth order derivative such as the CahnHilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. In this work, a fractional splitting method is employed to isolate the convective, the nonlinear second order and the fourth order differential terms. The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second order accuracy, we will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzburg-Landau energy and the coarsening properties of the solution.

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Section: Numerical Scheme For the Linear Equationmentioning

confidence: 99%

On a fractional step-splitting scheme for the Cahn-Hilliard equation

Aderogba

Chapwanya

Djoko

2014

Engineering Computations

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“…In order to deal with the backward heat system, Chugunova-Karabash-Pyatkov ( [7]), Cheng-Fu-Qin ( [6]), Boulton-Marletta-Rule ( [4]) applied their spectrum methods. Lu and Maubach ([15]), Ternat-Orellana-Daripa ( [20]), Douglas and Gallie ( [9]) used numerical methods to analyze the backward heat equation. Payne summarized various methods to treat the backward system into their paper ( [18]).…”

Section: Introductionmentioning

confidence: 99%

Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian

Koba¹,

Matsuoka

2015

Analysis

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This paper considers a backward problem on a heat equation with a fractional Laplacian. It is not easy to solve a backward heat equation directly. This problem is a well-known ill-posed problem. In order to consider a backward heat equation with a fractional Laplacian, we apply the -th power of the Dirichlet-Laplacian and small parameters to regularize the equation. This method is called a quasi-reversibility method. We use the generalized quasi-reversibility method to change the backward heat system into another system. This paper shows the existence of a strong solution of the modi ed backward heat system, and derives 2 -estimates of the di erence between a solution of the heat equation with the fractional Laplacian and a solution of our system.

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“…In [23], the same authors provide a regularisation method for backward parabolic equations with time-dependent coefficients. Ternat et al [51] suggest low-pass filters and fourth-order regularisation terms for stabilisation.…”

Section: Introductionmentioning

confidence: 99%

Stable Backward Diffusion Models that Minimise Convex Energies

et al. 2020

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The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a backward diffusion model which implements a smart stabilisation approach that can be used in combination with an easy-to-handle numerical scheme. So far, existing stabilisation strategies in the literature require sophisticated numerics to solve the underlying initial value problem. We derive a class of space-discrete one-dimensional backward diffusion as gradient descent of energies where we gain stability by imposing range constraints. Interestingly, these energies are even convex. Furthermore, we establish a comprehensive theory for the time-continuous evolution and we show that stability carries over to a simple explicit time discretisation of our model. Finally, we confirm the stability and usefulness of our technique in experiments in which we enhance the contrast of digital greyscale and colour images.

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Two stable methods with numerical experiments for solving the backward heat equation

Cited by 16 publications

References 21 publications

On a fractional step-splitting scheme for the Cahn-Hilliard equation

On a fractional step-splitting scheme for the Cahn-Hilliard equation

Generalized quasi-reversibility method for a backward heat equation with a fractional Laplacian

Stable Backward Diffusion Models that Minimise Convex Energies

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