Optimization method for the inverse problem of reconstructing the source term in a parabolic equation

Yang, Liu; Deng, Zui‐Cha; Jiang, Yu; Luo, Guoxi

doi:10.1016/j.matcom.2009.06.031

Cited by 56 publications

(21 citation statements)

References 15 publications

(18 reference statements)

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“…Several regularization effects of the present LGAM stem from the DQ, GPS, Lie-group shooting method and iterations, of which the convergence criterion with ε = 10 −2 is taken the noisy intensity into account according to the Morozov discrepancy principle. [26,28,29] Although the problem is more ill-posed, the present result for the recovery of H (x) is as accurate as those obtained by Farcas and Lesnic [8], and Yang et al [17], even the noise we imposed is large up to 10%.…”

Section: Letting V = U T : a Numerical Differential Methodssupporting

confidence: 77%

“…There were many researches of the inverse heat source problems to determine the unknown heat source terms since the works by Cannon [2] in the 1960s, to name a few, Cannon and Duchateau [3], Savateev [4], Borukhov and Vabishchevich [5], Dehghan [6], Yi and Murio [7], Farcas and Lesnic [8], Ling et al [9], Johansson and Lesnic [10,11], Yan et al [12], Ahmadabadi et al [13], Liu [14], Yeih and Liu [15], Yang et al [16,17], Yan et al [18] and Ahmadabadi et al [13]. Many researchers were sought the heat source as a function of only space or time, and only a few was to recover the space-time-dependent heat source.…”

Section: Introductionmentioning

confidence: 99%

“…For example, in order to identify H (x), Johansson and Lesnic [10] and Yang et al [17] required an overspecified data u(x, t f ) measured at a final time t = t f ; in order to identify H (t), Liu [14] and Yeih and Liu [15] required an overspecified data u(x m , t) measured at an internal point x = x m . Moreover, in order to identify H (x), Yang et al [16] required many overspecified data u(x i , t j ) measured at many space points and times.…”

Section: Introductionmentioning

confidence: 99%

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An iterative algorithm for identifying heat source by using a DQ and a Lie-group method

Liu

2014

Inverse Problems in Science and Engineering

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We develop a novel yet simple iterative algorithm to solve a mixed-type inverse heat conduction problem, which is a combination of the Cauchy problem, the backward heat conduction problem (BHCP) and a heat source recovery problem. First, we transform the heat conduction equation into a new one, including a spring term and a damping term, which can raise the accuracy of numerical solution. Second, we use the differential quadrature method to semi-discretize the differential term, and then the group-preserving scheme leads to an accurate solution of the mixing Cauchy and BHCP. Third, with the aid of an overspecified rightend boundary condition, an iterative Lie-group adaptive method is developed to recover the space-dependent heat source function through a few iterations. The heat source term recovered is quite smooth, even the input data are polluted by a random noise, being large up to 10%. The accuracy and efficiency of the new methods are assessed by comparing the estimated results with exact solutions from several numerical examples.

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Section: Letting V = U T : a Numerical Differential Methodssupporting

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An iterative algorithm for identifying heat source by using a DQ and a Lie-group method

Liu

2014

Inverse Problems in Science and Engineering

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“…But an inevitable reality is that the inverse problem of identifying the coefficients in parabolic equations is often ill-posed, which leads to many difficulties on the study of the original inverse problem. For a helpful conversion from the original inverse problem to a tractable one, optimal control methods have been accepted by increasing researchers recently (for a survey of the studies see, for example [7][8][9][10][11][12]). Here, we mostly recall the studies of inverse problems on reaction-diffusion equations to state our main motivation in the study from the aspect of theory research.…”

Section: Introductionmentioning

confidence: 99%

Solving the inverse problem of an SIS epidemic reaction–diffusion model by optimal control methods

Xiang

Liu

2015

Computers & Mathematics with Applications

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a b s t r a c tIn this contribution, a novel inverse problem on an SIS epidemic reaction-diffusion model is investigated by employing the optimal control methods. By setting a proper regular cost functional and choosing the coefficients to be identified as control variables, we embed the original inverse problem which is very difficult to solve into an optimal control problem. The existence of the optimal controls and the first order necessary optimality condition satisfied by the optimal controls are established, respectively. In particular, the local uniqueness and stability of the identification of the coefficients are proved by using analytical techniques.

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“…(1). The inverse source problem has been studied in different cases, for example, the recovering of the location and time-dependent intensity of point sources in [4,13,20,34], the piecewise-constant sources in [3,37] and Gaussian concentrated sources in [2,3]. Among these different approaches, the Tikhonov optimization method is most popular [4,13,19,34], which reformulates the original inverse source problem into an output least-squares optimization problem with PDE-constraints, by introducing some appropriate regularizations to ensure the stability of the resulting optimization problem with respect to the change of noise in the observation data [14,35].…”

Section: Introductionmentioning

confidence: 99%

Two-Level Space–Time Domain Decomposition Methods for Three-Dimensional Unsteady Inverse Source Problems

2015

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As the number of processor cores on supercomputers becomes larger and larger, algorithms with high degree of parallelism attract more attention. In this work, we propose a two-level space-time domain decomposition method for solving an inverse source problem associated with the time-dependent convection-diffusion equation in three dimensions. We introduce a mixed finite element/finite difference method and a one-level and a two-level space-time parallel domain decomposition preconditioner for the Karush-Kuhn-Tucker system induced from reformulating the inverse problem as an output least-squares optimization problem in the entire space-time domain. The new full space-time approach eliminates the sequential steps in the optimization outer loop and the inner forward and backward time marching processes, thus achieves high degree of parallelism. Numerical experiments validate that this approach is effective and robust for recovering unsteady moving sources. We will present strong scalability results obtained on a supercomputer with more than 1000 processors.

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Optimization method for the inverse problem of reconstructing the source term in a parabolic equation

Cited by 56 publications

References 15 publications

An iterative algorithm for identifying heat source by using a DQ and a Lie-group method

An iterative algorithm for identifying heat source by using a DQ and a Lie-group method

Solving the inverse problem of an SIS epidemic reaction–diffusion model by optimal control methods

Two-Level Space–Time Domain Decomposition Methods for Three-Dimensional Unsteady Inverse Source Problems

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