A Discrepancy Principle for the Source Points Location in Using the MFS for Solving the BHCP

Abstract

Based on the discrepancy principle, we develop in this paper a new method of choosing the location of source points to solve the backward heat conduction problem (BHCP) by using the method of fundamental solutions (MFS). The standard Tikhonov regularization technique with the L curve method for an optimal regularized parameter is adopted for solving the resultant highly ill-conditioned system of linear equations. Numerical verifications of the proposed computational method are presented for both the one-dimensional and the two-dimensional BHCP.

A method of fundamental solutions for two-dimensional heat conduction

“…To show that {v m (x, t)} is dense we have to show that f (x, t) ≡ 0 in (14). From definition (6), equation (14) can be rewritten as where the Heaviside function in (5) has been used to reduce the range of integration with respect to the time variable.…”

“…From definition (6), equation (14) can be rewritten as where the Heaviside function in (5) has been used to reduce the range of integration with respect to the time variable. We introduce an equivalent form of the classical single-layer heat potential given by…”

“…Also, it might be more practical to take the sources on the interval (−ǫ, T ), where ǫ > 0 is small, instead of the full interval (−T, T ). However, some preliminary numerical investigations in [14] for the backward heat conduction problem showed that ǫ cannot be chosen too small if no loss in accuracy and stability is to be secured. …”

A State Feedback Controller Used to Solve an Ill-posed Linear System by a GL(n, R) Iterative Algorithm

“…In a practical application of MFS to solve the BHCP, the source points are uniformly located on two vertical straight lines parallel to the t-axis not over the final time, which was adopted by Hon and Li [26] and Liu [27], showing a large improvement than the line location of source points below the initial time. After imposing the boundary conditions and the final time condition on Eq.…”

Inverse source identification by Green's function

“…In particular, placement of source points depending on the shape of contours in using the MFS for solving elastic rod problem has been investigated in [20] conclusion that smaller error can be obtained if the source points are geometrically located along with the boundary shape of the domain. More recently, an optimal choice strategy based on discrepancy principle has been given in [28]. Applications of the MFS for solving inverse source identification problems can be found in [4,32,57,58].…”