一类抛物型方程系数反问题的分裂算法

考虑利用终端时刻的温度u(x,T)=Z_T(x)反演热传导方程u_t-a~2u_(xx) q(x)u=0,x∈(0,1)中的未知系数q(x)的反问题.通过引进变换v(x,t)=(u_t(x,t)/u(x,t))将此非线性不适定问题的求解分解为两步.首先利用输入数据迭代求解一个非线性的正问题(该过程独立于未知系数),得到其迭代解v~(k)(x,t).其次利用q(x)与v(x,t)的关系式求出q(x)的近似解.对提出的反演方法,证明了采用的变换的可行性,得到了原反问题与由变换后的非线性正问题反演q(x)的等价性并且证明了迭代解的收敛性,给出了收敛速度.数值结果表明了该方法的有效性.

ON THE ITERATION ALGORITHM FOR AN INVERSE PROBLEM OF PARABOLIC EQUATION

Consider an inverse problem for recovering q(x) in the equation u_t-a~2u_(xx) q(x)u=0, using the final measurement data u(x,T)=Z_T(x).By introducing a function transform v=u_t/u,the solution to this nonlinear ill-posed inverse problem is obtained by two steps. We firstly solve a non-linear direct problem for v(x,t) by iteration algorithm from the inver- sion input data.This procedure is independent of the unknown coefficient q(x).Then the reconstruction of q(x) is finished from a differential relation between v and q approximately. Using this scheme,the original problem is decomposed as a nonlinear well-posed problem and an ill-posed problem.The convergence result is proven.Numerical results are given to show the validity of this decomposition scheme.