A wavelet multiscale iterative regularization method for the parameter estimation problems of partial differential equations
A wavelet multiscale iterative regularization method for the parameter estimation problems of partial differential equations
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DOI：
10.1016/j.neucom.2012.10.007
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发表时间：
2013-03
期刊：
影响因子：
6
通讯作者：
H. Fu;B. Han;Hongbo Liu
中科院分区：
文献类型：
--
作者：
H. Fu;B. Han;Hongbo Liu
针对偏微分方程的参数估计问题，提出了一种小波多尺度迭代正则化方法。引入小波分析，基于层次逼近的思想，构造了小波多尺度方法。反问题被分解为一系列依赖于尺度变量的反问题序列，并按照尺度的大小从最长到最短依次求解。通过结合多尺度近似，克服了局部最小化问题，降低了计算成本。在每个尺度上，基于小波近似，将参数反演问题转化为估计尺度空间中有限小波系数的问题。构建了一种新颖的迭代正则化方法。通过求解一维和二维椭圆偏微分方程的系数反问题说明了该方法的有效性。
A wavelet multiscale iterative regularization method is proposed for the parameter estimation problems of partial differential equations. The wavelet analysis is introduced and a wavelet multiscale method is constructed based on the idea of hierarchical approximation. The inverse problem is decomposed into a sequence of inverse problems which rely on the scale variables and are solved successively according to the size of scale from the longest to the shortest. By combining multiscale approximations, the problem of local minimization is overcomed, and the computational cost is reduced. At each scale, based on the wavelet approximation, the problem of inverting the parameter is transformed into the problem of estimating the finite wavelet coefficients in the scale space. A novel iterative regularization method is constructed. The efficiency of the method is illustrated by solving the coefficient inverse problems of one- and two-dimensional elliptical partial differential equations.