Stability results for backward parabolic equations with time-dependent coefficients

Hào, Ðinh Nho; Duc, Nguyen Van

doi:10.1088/0266-5611/27/2/025003

Cited by 28 publications

(15 citation statements)

References 18 publications

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“…It would take us too far to quote the large amount of work on the backward uniqueness in more loosely connected situations, often adopting the log-convexity method (if |u(t)| ≤ |u(T)| t/T |u(0)| 1−t/T then u(T) = 0 implies u(t) = 0 for all t > 0, hence u(0) = 0 by continuity) attributed to Krein, Agmon and Nirenberg. Instead we refer the reader to [38][39][40] and the references therein.…”

Section: Notesmentioning

confidence: 99%

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

Christensen

Johnsen

2018

Axioms

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This article concerns the basic understanding of parabolic final value problems, and a large class of such problems is proved to be well posed. The clarification is obtained via explicit Hilbert spaces that characterise the possible data, giving existence, uniqueness and stability of the corresponding solutions. The data space is given as the graph normed domain of an unbounded operator occurring naturally in the theory. It induces a new compatibility condition, which relies on the fact, shown here, that analytic semigroups always are invertible in the class of closed operators. The general set-up is evolution equations for Lax-Milgram operators in spaces of vector distributions. As a main example, the final value problem of the heat equation on a smooth open set is treated, and non-zero Dirichlet data are shown to require a non-trivial extension of the compatibility condition by addition of an improper Bochner integral.

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Section: Notesmentioning

confidence: 99%

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

Christensen

Johnsen

2018

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“…In the case of constant or time-dependent diffusion coefficient, the classical backward heat conduction equation has been studied in many works, see for example, other studies. [8][9][10] However, to the best of our knowledge, there are not any result on (1)(2)(3). Our paper is the first study on this direction for parabolic systems with random model.…”

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confidence: 82%

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

Triet

Binh²,

Phương

et al. 2020

Math Methods in App Sciences

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In this work, we study the final value problem for a system of parabolic diffusion equations. In which, the final value functions are derived from a random model. This problem is severely ill‐posed in the sense of Hadamard. By nonparametric estimation and truncation methods, we offer a new regularized solution. We also investigate an estimate of the error and a convergence rate between a mild solution and its regularized solutions. Finally, some numerical experiments are constructed to confirm the efficiency of the proposed method.

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“…Hào and Duc [22] suggest a mollification method where stability for the inverse diffusion problem follows from a convolution with the Dirichlet kernel. 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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Stability results for backward parabolic equations with time-dependent coefficients

Cited by 28 publications

References 18 publications

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

Final Value Problems for Parabolic Differential Equations and Their Well-Posedness

Recovering the initial value for a system of nonlocal diffusion equations with random noise on the measurements

Stable Backward Diffusion Models that Minimise Convex Energies

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