Numerical identifications of parameters in parabolic systems

Abstract

In this paper, we investigate the numerical identifications of physical parameters in parabolic initial-boundary value problems. The identifying problem is first formulated as a constrained minimization one using the output least squares approach with the H 1-regularization or BV-regularization. Then a simple finite element method is used to approximate the constrained minimization problem and the convergence of the approximation is shown for both regularizations. The discrete constrained problem can be reduced to a sequence of unconstrained minimization problems. Numerical experiments are presented to show the efficiency of the proposed method, even for identifying highly discontinuous and oscillating parameters.

An Efficient Linear Solver for Nonlinear Parameter Identification Problems

“…For parameter identifications, Itô and Kunisch proposed a hybrid method in [11,12,13] which combines the output least squares and the equation error formulation within the mathematical framework given by the augmented Lagrangian technique and incorporates a regularization term of the H 2 -seminorm of the parameters to be recovered. Chen and Zou [5] and Keung and Zou [14] generalized the method to the case which allows the identifying coefficients to be discontinuous by using the regularization of bounded variations and they provided the rigorous theoretical justifications of the method and its finite element approximation. Independently, Chan and Tai [3,4,18] considered also the regularization of bounded variations and did numerous experiments on the performance of the augmented Lagrangian method for identifying highly discontinuous parameters.…”

“…at each iteration, where ε is the smoothing parameter introduced to smooth the BVnorm term in numerical implementations and c(q) is a linear function of q, which causes the indefiniteness of the system; see [3,4,5,14,18]. It seems there are very few iterative methods which are known to be globally convergent for solving such a troublesome system.…”

Global superconvergence of finite element methods for parabolic inverse problems

“…Recently, the study of parabolic inverse problems has received much attention. For example, finite element methods and finite difference methods have been investigated in [10], [11], and [19], respectively. Here and throughout this paper "parabolic inverse problem" means that an unknown coefficient that is assumed to be a function of only the time variable and the solution of a parabolic equation subject to suitable initial-boundary conditions is to be determined.…”

Identification of diffusion parameters in a nonlinear convection–diffusion equation using the augmented Lagrangian method

“…Recovery of q(x) in (1.6) using the augmented Lagrangian method is investigated in [9,13,17], and other methods are studied in [5,12,15]. In [16] the augmented Lagrangian method for recovery of q(x) within the nonlinear parabolic equation (1.7) u t − ∇ · (q(x)N (∇u, u)∇u) = f (x, t), is studied.…”