摘要
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.
Consider an inverse using the final problem for recovering q（x） in the nent data u（x, T） = ZT（X）. By equation ut-a^2uxx＋q（x）u=0, introducing a function transform v = ut/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 inversion 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.
出处
《计算数学》
CSCD
北大核心
2008年第1期99-112,共14页
Mathematica Numerica Sinica
基金
国家自然科学基金(10771033)
江苏省自然科学基金(BK2007101)的资助
关键词
反问题
抛物型方程
迭代方法
Sweep方法
正则化
数值解
Inverse problem
parabolic equation
iteration algorithm
sweep method
regularization
numerics
