A Comparison of Two Solution Techniques for the Inverse Problem of Simultaneously Estimating the Spatial Variations of Diffusion Coefficients and Source Terms

Abstract

This work deals with the simultaneous estimation of the spatially varying diffusion coefficient and of the source term distribution in a one-dimensional nonlinear diffusion problem. This work can be physically associated with the detection of material non-homogeneities such as inclusions, obstacles or cracks, heat conduction, groundwater flow detection, and tomography. Two solution techniques are applied in this paper to the inverse problem under consideration, namely: the conjugate gradient method with adjoint problem and a hybrid optimization algorithm. The hybrid optimization technique incorporates several of the most popular optimization modules; the Davidon-Fletcher-Powell (DFP) gradient method, a genetic algorithm (GA), the Nelder-Mead (NM) simplex method, quasi-Newton algorithm of Pshenichny-Danilin (LM), differential evolution (DE), and sequential quadratic programming (SQP). The accuracy of the two solution approaches was examined by using simulated transient measurements containing random errors in the inverse analysis.

Inverse and optimization problems in heat transfer

“…Indeed, the stochastic algorithms can require thousands of evaluations of the objective functions and, in some cases, become non-practical. In order to overcome these difficulties, we will also discuss the so-called hybrid algorithm, which takes advantage of the robustness of the stochastic methods and of the fast convergence of the deterministic methods (Colaço et al, 2003a;Colaço et al, 2003b;Dulikravich et al, 2003c;Dulikravich et al, 2004;Colaço et al, 2003c). Each technique provides a unique approach with varying degrees of convergence, reliability and robustness at different cycles during the iterative minimization procedure.…”

Determination of space‐dependent coefficients from temperature measurements using the conjugate gradient method

“…The adjoint problem for the Lagrange multiplier remains the same as (10)–(12), and the gradients with respect to and are still given by (15) and (46), respectively. The Subsections 3.3–3.5 remain the same but in the recurrences we have that the search step sizes are given by, , where , …”

“…we have that the search step sizes are given by, [13,14], where ΔT n q,i (t) := ΔT q (x i , t; q n , k n ), ΔT n k,i (t) := ΔT k (x i , t; q n , k n ) and…”

“…Also, the identifiability of the space‐dependent thermal conductivity from a single point temperature observation in the one‐dimensional heat equation was investigated in as an inverse Sturm‐Liouville problem. In , both the space‐dependent perfusion coefficient and thermal conductivity have been determined simultaneously using the CGM from boundary and interior temperature measurements. The identification of the perfusion coefficient, being constant or dependent on time only, has been investigated in .…”

Incremental Identification of Transport Coefficients in Convection-Diffusion Systems

“…Quite some literature is available on the subject (cf. [3,11,28] and the references therein); the treatment, however, is typically restricted to one or two space dimensions. Furthermore, these studies do not aim at the reconstruction of a suitable transport model (structure and parameters) for transport coefficients.…”