A unified approach to identifying an unknown spacewise dependent source in a variable coefficient parabolic equation from final and integral overdeterminations. (English) Zbl 1284.65123
Summary: An adjoint problem approach with subsequent conjugate gradient algorithm (CGA) for a class of problems of identification of an unknown spacewise dependent source in a variable coefficient parabolic equation \(u_t = (k(x)u_x)_x + F(x)H(t)\), \((x, t) \in (0, l){\times}(0, T]\) is proposed. The cases of final time and time-average, i.e. integral type, temperature observations are considered. We use the well-known Tikhonov regularization method and show that the adjoint problems, corresponding to inverse problems ISPF1 and ISPF2 can uniquely be derived by the Lagrange multiplier method. This result allows us to obtain a representation formula for the unique solutions of each regularized inverse problem. Using standard Fourier analysis, we show that series solutions for the case in which the governing parabolic equation has constant coefficient, coincide with Picard’s singular value decomposition. It is shown that the use of these series solutions in CGA as an initial guess substantially reduces the number of iterations. A comparative numerical analysis between the proposed version of CGA and the Fourier method is performed using typical classes of sources, including oscillating and discontinuous functions. Numerical experiments for the variable coefficient parabolic equation with different smoothness properties show the effectiveness of the proposed version of CGA.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 35R30 | Inverse problems for PDEs |
| 65F10 | Iterative numerical methods for linear systems |
Keywords:
inverse source problem; parabolic equation; final and time-average temperature overdeterminations; integral representation formula; filter function; conjugate gradient algorithm; Tikhonov regularization; Lagrange multiplier method; singular value decomposition; Fourier method; numerical experimentsReferences:
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