一类Robin反问题的数值解法

Robin反问题是来自偏微分方程的非线性反问题，有广泛的应用背景，例如非破坏性估测技术的定量分析、温度与热流量的确定以及冰原预报的初始化等。由于Robin反问题应用广泛且求解难度大，其数值解法一直是一个研究热点。对该反问题的数值解法作深入系统的研究，具有重要的科学意义和应用价值。.本项目以边界积分方程为工具研究一类Robin反问题的数值解法。基础研究内容是构造核函数有弱奇性的积分算子的高精度离散公式、矩阵-向量快速相乘新算法和新的罚函数。这些基础研究内容有很好的可移植性，预期成果可以推广应用于有弱奇异核的积分方程和一些类型的不适定问题。我们计划把Robin反问题转化为极小化问题，并根据Robin系数的光滑性质提出不同的罚函数，得到不同的正则化极小化问题，然后研究这些极小化问题的快速数值求解方法。.预期分别得到求解连续、分段/分块常数和分段/分块连续Robin系数的有效数值方法。

Robin inverse problems are nonlinear inverse problems arising from partial differential equations. This kind of inverse problems arise in a variety of applications, including various nondestructive evaluation methods where an unknown material profile in a non-accessible part of the boundary is to be recovered from a partial boundary measurement made on an accessible part of the boundary, determination of temperatures and heat fluxes, and initialization of ice-sheet forecasts, etc. Robin inverse problems are very difficult to be solved numerically due to the ill-posedness nature of the problem. In recent years, numerical solution methods for the problem have been attracting a lot of attentions. A systematic research on numerical solution methods for Robin inverse problems has significant scientific meanings and great potential in applications..In this project, we are concerned with numerical solution methods for a type of Robin inverse problems with boundary integral equation as a tool. The basic contents of this project are construction of highly accurate discretization methods for integral operators with weak singular kernel functions, new algorithms for fast matrix-vector multiplications, and new penalty functions, which are capable of being transplanted to solve other problems. For example, the expected achievements of this project can be applied to integral equations with weak singular kernels and some other ill-posed problems..We will reformulate the Robin inverse problems as minimization problems and introduce different penalty functions for the minimization problems according to the smoothness of the Robin coefficients. We will then study fast numerical solution methods for the relevant regularized minimization problems. We expect to construct suitable penalty functions for continuous Robin coefficients, piecewise constant Robin coefficients, and piecewise continuous Robin coefficients, respectively, and derive efficient numerical solution methods for the regularized minimization problems.