Bi-quadratic polynomial approach for global convergent algorithm in high dimensions coefficient inverse problems

Abstract

Bi-quadratic polynomial approach for global convergent algorithm in high dimensions coefficient inverse problems Quan-Fang WangMechanical and Automation Engineering, Chinese University of Hong Kong, Shatin, N.T., Hong Kong E-mail: quanfangwang@hotmail.com Abstract. Sequential minimization algorithm in convexification approach established a stable approximate solution via minimizing a finite sequence of strictly convex objective function. Its application to 1D/2D dimension cases is reported in a great deal literatures. An interesting question should be asked is how about the development of the convex approach to high dimension case. That is to say, whether the approach could be applied to realistic three dimensions issue by a routing way. It becomes a much more challenging topic in present research. In this work, a newly modified global convergent algorithm for recovering the profile of coefficient inverse problems will be constructed by using so-called bi-quadratic polynomial (BQP) in 2D/3D cases. Hope the developed paradigm would be applied in a broad class application.

A triply cubic polynomials approach for globally convergent algorithm in coefficient inverse problems

“…Next, we intend to deduce equivalent equation without the unknown coefficient k(x) from the nonlinear parabolic equation (2.1). Cite [3] for two dimension case, and refer Lemma 2 of [11] for 1D case. Actually, whether the source function S 0 (x, t) exist or not, it is convenient to neglect this term in following deduces.…”

“…With the results in [3] on two dimension parabolic coefficient inverse problems for space variables (x 2 , x 3 ) and time variable s, suppose…”

“…Citing [3] to do the minimization for a proper objective function given in the form of quadratic criteria as…”

“…Worldwidelly, current BQP (Bi-quadratic polynomials) approach is used to solve the globally convergent algorithm of coefficient inverse problems in the spatial dimension of two. Since it had been proposed in [1,2], recognized and named as BQP approach in article [3], fortunately, it already had been adopted in numerous publications such as [4], [5], [6,7,8,9], and so forth. Actually, the methodology is worked to recover the profile of realistic systems (parabolic, hyperbolic, elliptic type) in industrial, electricmagnetic, optical frequency sounding, engineer, medical tomography, physical as well as chemical fields.…”

“…The purpose of the article is to address the triple cubic polynomials approach for solving coefficient inverse problems in three dimension at the globally convergent algorithm, upon with previous bi-quadratic polynomials methodology given in [3] for two dimension case. The systematical theoretical proposals and computational procedure are feasible in 3-dimensional case.…”