A numerical solution of an inverse parabolic problem (original) (raw)
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A numerical solution of an inverse parabolic problem with unknown boundary conditions
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In this paper, we consider an inverse problem to determine a source term in a parabolic equation, where the data are obtained at a certain time. In general, this problem is ill-posed, therefore the Tikhonov regularization method is proposed to solve the problem. In the theoretical results, a priori error estimate between the exact solution and its regularized solution is obtained. We also propose both methods, a priori and a posteriori parameter choice rules. In addition, the proposed methods have been verified by numerical experiments to estimate the errors between the regularized solutions and exact solutions. Eventually, from the numerical results it shows that the a posteriori parameter choice rule method gives a better the convergence speed in comparison with the a priori parameter choice rule method in some specific applications.
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Inverse methods provide a good alternative to traditional trial-and-error methods for design of thermal systems. The inverse boundary condition estimation problem in radiating enclosures involves the solution of an ill-posed system that requires regularization to obtain a reasonable physical solution. This study compares three regularized solution techniques that can be used in the inverse boundary condition estimation problems in a three-dimensional radiating enclosure. The regularized solution techniques covered in this study are the conjugate gradient method, bi-conjugate gradient method and truncated singular value decomposition. ?
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Journal_Inverse_and_Ill-Posed_Problems_26(3)(2018).pdf
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This paper studies the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional J(k) := (1/2)‖u(0, ⋅ ; k) − f‖^2_{L^2(0,T)} corresponding to an inverse coefficient problem for the 1D parabolic equation u_t = (k(x)u_x)_x with the Neumann boundary conditions −k(0)u_x(0, t) = g(t) and u_x(l, t) = 0. In addition, compactness and Lipschitz continuity of the input-output operator Φ[k] := u(x, t; k)|_{x=0}+ , Φ[ ⋅ ] : K ⊂ H1(0, l) → H1(0, T), as well as solvability of the regularized inverse problem and the Lipschitz continuity of the Fréchet gradient of the Tikhonov functional are proved. Furthermore, relationships between the sufficient conditions for the Lipschitz continuity of the Fréchet gradient and the regularity of the weak solution of the direct problem as well as the measured output f(t) := u(0, t; k) are established. One of the derived lemmas also introduces a useful application of the Lipschitz continuity of the Fréchet gradient.
Regularization of Inverse Problems
The problem of determining the thermophysical properties by means of a discrete set of observations on the temperatures of the test object given with measurement errors is examined. The investigation of complex processes by using inverse problems has attracted considerable attention lately. Their solution is associated with certain singularities, particularly the influence of errors in the initial data on the desired solution. As is known from [i], in such cases it is necessary to limit the domain of the allowable solutions and to match the measurement errors. Since a number of stabilizing functionals with the same problem can be set in correspondence and different norms for the deviation from the quantities observed can be selected, then it is interesting to determine those among them which will permit, for sufficiently general assumptions about the desired quantities, obtaining the most exact solutions under conditions of unimprovable observations for a broad range of measurement errors. In addition, the question of selecting the method of matching the observations occurs in the solution of applied ill-posed problems. One condition that establishes a relation between the accuracy of the solution and the measurement error [2] is used in the widespread problem, in practice, of restoring the thermal flux. This condition expresses the total error in all observations for measurements executed at several points. However, one condition can turn out to be inadequate to determine several parameters of a model that is characteristic for the inverse coefficient problems, while taking total account of the errors results in a loss in accuracy of the solution of the inverse problem [3]. This paper is devoted to investigating the properties of the regularized solution of an inverse coefficient problem for the nonlinear heat-conduction equation as a function of the degree of limitation of the domain of admissible solutions, the form of the observation error estimate, and the methods of matching them. In the domain Q = {~, t):O < x < i, 0 < t < T} we examine the one-dimensional heat-conduction equation a~ O~-ox a~-~x + I (x, 0 (1) for which the initial and boundary conditions assuring uniqueness and stability in the determination of the function u(x, t) for given values of the specific heat a~(u) and the heat conductivity an(U) and any T > 0 are assumed known. Let us also assume that at m points of space, and for each of n times of the domain Q observation results are given u~l=u(x,, t~)+efy, i= 1, m, ]= 1, n, (2) with a known magnitude of the deviation norm 62=j-(l~y-uij)", i = i, ~, i=I (3) Balashikhinskoe NPO Kriogenmash.
Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations, Linköping studies
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Ranjbar, Z. (2010). Numerical Solution of Ill-posed Cauchy Problems for Parabolic Equations. Doctoral dissertation. Ill-posed mathematical problem occur in many interesting scientific and engineering applications. The solution of such a problem, if it exists, may not depend continuously on the observed data. For computing a stable approximate solution it is necessary to apply a regularization method. The purpose of this thesis is to investigate regularization approaches and develop numerical methods for solving certain ill-posed problems for parabolic partial differential equations. In thermal engineering applications one wants to determine the surface temperature of a body when the surface itself is inaccessible to measurements. This problem can be modelled by a sideways heat equation. The mathematical and numerical properties of the sideways heat equation with constant convection and diffusion coefficients is first studied. The problem is reformulated as a Volterra integral equati...
Journal of Inequalities and Applications, 2015
In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well-known to be ill-posed. We employ some elementary results of semi-group theory to present the formula of solution, then show the instability cause. Since the solution exhibits unstable dependence on the given data functions, we propose a regularization method to stabilize the solution. then obtain the error estimate. A numerical example shows that the method is efficient and feasible. This work slightly extends to the earlier results in Zouyed et al. (2014).