Conjugate gradient method for the Robin inverse problem associated with the Laplace equation

Abstract

SUMMARYThis paper studies a non-linear inverse problem associated with the Laplace equation of identifying the Robin coefficient from boundary measurements. A variational formulation of the problem is suggested, thereby transforming it into an optimization problem. Mathematical properties relevant to its numerical computation are established. The optimization problem is solved using the conjugate gradient method in conjunction with the discrepancy principle, and the algorithm is implemented using the boundary element method. Numerical results are presented for several benchmark problems with both exact and noisy data, and the convergence of the algorithm with respect to mesh refinement and decreasing the amount of noise in the data is investigated.

Inversion of Robin coefficient by a spectral stochastic finite element approach

“…M for all the examples. The conventional CGM is known to converge slowly, and it might stagnate after a few iterations [11]. To accelerate the convergence of the CGM, a preconditioner via the Sobolev inner product (see [35,12] for more details) is applied to the mean c 0 ðxÞ of the Robin coefficient cðx; yÞ.…”

“…Several numerical methods [6,[8][9][10][11][12]14] have been proposed for the Robin inverse problem, in particular in the context of corrosion detection, among which the least-squares method [9][10][11][12] has received intensive investigations and it has been implemented in the boundary integral equation method [9,10], the boundary element method [11] and the finite element method [12]. The Robin inverse problem studied in the literature so far is deterministic in the sense that the stochastic nature of the measurement errors is not rigorously considered and analyzed, and these deterministic inverse techniques yield only a single estimate of the large ensemble of solutions that are consistent with the given data without quantifying the uncertainties in the inverse solution.…”

Recovery of an interface from boundary measurement in an elliptic differential equation

“…The usual treatment is to select the "optimal" iteration count to stop the process by a discrepancy principle (e.g. [9]), but for this particular problem, this treatment is unable to yield satisfactory results for us. On the other hand, with the regularization we included in the cost functional, both the CG and the GN methods perform well.…”

Inverse estimation for unknown fouling-layer profiles with arbitrary geometries on the inner wall of a forced-convection duct

“…Theoretically, the current inverse method is able to cope with arbitrary fouling profiles. Here, we employ the conjugate gradient method (CGM) [14][15][16] and the discrepancy principle [17] to the inverse geometry problem to determine the fouling-layer configuration in the duct system. The conjugate gradient method with an adjoint equation, also called Alifanov's iterative regularization method, belongs to a class of iterative regularization techniques, which mean the regularization procedure is performed during the iterative processes, thus the determination of optimal regularization conditions is not needed.…”