Iterative Solution of the Retrospective Inverse Problem for a Parabolic Equation Using the Conjugate Gradient Method

Васильев, В. И.; Kardashevsky, A. M.

doi:10.1007/978-3-319-57099-0_80

Cited by 4 publications

(5 citation statements)

References 6 publications

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“…) otherwise. For σ < 1/2, the numerical discretization in Equation ( 14) is stable for a sufficiently small time step [33]. Now, we consider the RIP.…”

Section: Stability Discussionmentioning

confidence: 99%

“…The retrospective inverse problem for parabolic equation consists of reconstruction of the unknown initial condition from a given final time observation. It is not well posed, as a small perturbation of the input data can produce large perturbations in the solution (see, for example, [29][30][31][32][33][34][35]). The main difficulty in the construction of a numerical approximation of the solution comes from the strong ill-posedness of the differential problem and the poor conditioning of the corresponding algebraic equations.…”

Section: Introductionmentioning

confidence: 99%

“…The RIP for heat conduction was solved as an optimal control problem of an object with distributed parameters in [29]. In [31,33,34], the authors constructed an iterative numerical method for the solution of an RIP for one-and multi-dimensional parabolic equations. An efficient numerical method for solving the RIP for a time-fractional parabolic equation was developed in [32].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we unfold the finite-difference method proposed in [31,33,34] and solve numerically the nonstandard RIP for recovering the initial condition in a parabolic equation in disjoint domains.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals

Koleva,

Vulkov

2023

Computation

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The retrospective inverse problem for evolution equations is formulated as the reconstruction of unknown initial data by a given solution at the final time. We consider the inverse retrospective problem for a one-dimensional parabolic equation in two disconnected intervals with weak solutions in weighted Sobolev spaces. The two solutions are connected with nonstandard interface conditions, and thus this problem is solved in the whole spatial region. Such a problem, as with other inverse problems, is ill-posed, and for its numerical solution, specific techniques have to be used. The direct problem is first discretized by a difference scheme which provides a second order of approximation in space. For the resulting ordinary differential equation system, the positive coerciveness is established. Next, we develop an iterative conjugate gradient method to solve the ill-posed systems of the difference equations, which are obtained after weighted time discretization, of the inverse problem. Test examples with noisy input data are discussed.

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“…) otherwise. For σ < 1/2, the numerical discretization in Equation ( 14) is stable for a sufficiently small time step [33]. Now, we consider the RIP.…”

Section: Stability Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals

Koleva,

Vulkov

2023

Computation

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“…We consider a one-sided free domain problem in one-dimensional space for the parabolic heat equation, with non-homogenous Dirichlet boundary condition when the thermal conductivity is equal to unity. This problem contains free boundary depending on time only [2,3]. Therefore, the shape of the problem varies with time step marching.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution to Recover Time-dependent Coefficient and Free Boundary from Nonlocal and Stefan Type Overdetermination Conditions in Heat Equation

Qassim

Hussein

2021

eijs

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This paper investigates the recovery for time-dependent coefficient and free boundary for heat equation. They are considered under mass/energy specification and Stefan conditions. The main issue with this problem is that the solution is unstable and sensitive to small contamination of noise in the input data. The Crank-Nicolson finite difference method (FDM) is utilized to solve the direct problem, whilst the inverse problem is viewed as a nonlinear optimization problem. The latter problem is solved numerically using the routine optimization toolbox lsqnonlin from MATLAB. Consequently, the Tikhonov regularization method is used in order to gain stable solutions. The results were compared with their exact solution and tested via root mean squares error (RMSE). We found that the numerical results are accurate and stable.

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Conjugate Gradient Method for Identification of a Spacewise Heat Source

Васильев

Popov

Kardashevsky

2018

Large-Scale Scientific Computing

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Iterative Solution of the Retrospective Inverse Problem for a Parabolic Equation Using the Conjugate Gradient Method

Cited by 4 publications

References 6 publications

Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals

Numerical Solution of the Retrospective Inverse Parabolic Problem on Disjoint Intervals

Numerical Solution to Recover Time-dependent Coefficient and Free Boundary from Nonlocal and Stefan Type Overdetermination Conditions in Heat Equation

Conjugate Gradient Method for Identification of a Spacewise Heat Source

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