A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements

Abstract

We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of 2 . We then solve this system by a predictor-corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows. Problem 1.1 (Coefficient Inverse Problem (CIP)). Assume that ( ) ≠ 0 for all ∈ Ω. Given a time > 0 and the lateral Cauchy data ( , ) = ( , ) and ( , ) = ( , )for all ∈ Ω and ∈ [0, ], determine the coefficient ( ), ∈ Ω. Here is the outward normal vector of Ω.

Convexification for an inverse parabolic problem

“…Both convergence analysis and numerical results are presented. The CIP, which is considered here, has applications in heat conduction [1], diffusion theory [39] and in medical optical imaging using the diffuse infrared light [12]. In addition, this CIP has applications in financial mathematics in the search of the volatility coefficient in the Black-Scholes equation using the market data [8,30].…”

Convexification of a 3-D coefficient inverse scattering problem

“…Both convergence analysis and numerical results are presented. The CIP, which is considered here, has applications in heat conduction [1], diffusion theory [33] and in medical optical imaging using the diffuse infrared light [10]. In addition, this CIP has applications in financial mathematics in the search of the volatility coefficient in the Black-Scholes equation using the market data [7,25].…”

The quasi-reversibility method to numerically solve an inverse source problem for hyperbolic equations

“…Remark 5.1. The basis {Ψ n } n≥1 was successfully used very often in our research group to solve a long list of inverse problems including the nonlinear coefficient inverse problems for elliptic equations [50] and parabolic equations [51,43,44,52], and ill-posed inverse source problems for elliptic equations [41] and parabolic equations [42], transport equations [45] and full transfer equations [53]. Another reason for us to employ this basis rather than the well-known basis of the Fourier series is that the first elements of this basis is a constant.…”