Regularization Methods of the Continuation Problem for the Parabolic Equation

Belonosov, A. S.; Shishlenin, Maxim A.

doi:10.1007/978-3-319-57099-0_22

Cited by 10 publications

(3 citation statements)

References 14 publications

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“…In the formulations of inverse problems for partial differential equations additional information about the solution on a part of the boundary is often used (see, for example, [21][22][23][24][25][26][27][28][29][30]). For example, in formulation of so-called inverse backward problem (or retrospective inverse problems) it is required to find a solution at the initial time from a known solution at the final time [31][32][33].…”

Section: Introductionmentioning

confidence: 99%

Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

et al. 2021

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In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is the use of additional information about the value of the solution of the equation at the known position of a reaction front, measured experimentally with a delay relative to the initial moment of time. In this case, for the numerical solution of the inverse problem, the gradient method of minimizing the cost functional is applied. In the case when only the position of the reaction front is known, the method of deep machine learning is applied. Numerical experiments demonstrated the possibility of solving such kinds of considered inverse problems.

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Section: Introductionmentioning

confidence: 99%

Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…To formulate the inverse problem, additional information is required, which is usually measured in an experiment. Often, in the formulation of inverse problems for partial differential equations, additional information about the solution on a part of the boundary of the domain is used (see, for example, [24,[28][29][30][31][32][33][34][35][36][37]). However, one of the possible statements of inverse problems for equations of the type under consideration is a statement with additional information about the dynamics of the reaction front motion (see, for example, [38][39][40]).…”

Section: Introductionmentioning

confidence: 99%

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

et al. 2021

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The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.

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“…To solve inverse problems, it is necessary to have additional information about the solution of the equation. Most often, such additional information is information about the solution on a part of the boundary of the domain on which the equation is considered (see, for example, [25][26][27][28][29][30][31][32][33][34][35]). However, recently, the formulation of inverse problems for the equation of this type with additional information on the dynamics of the reaction front has become relevant because the reaction front is an experimentally easily distinguishable contrast structure (see, for example, [36,37]).…”

Section: Introductionmentioning

confidence: 99%

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front

et al. 2021

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The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.

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Regularization Methods of the Continuation Problem for the Parabolic Equation

Cited by 10 publications

References 14 publications

Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front

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