论文标题
反电位问题的galerkin近似的收敛速率分析
Convergence Rate Analysis of Galerkin Approximation of Inverse Potential Problem
论文作者
论文摘要
在这项工作中，我们分析了从分布式观察结果中恢复椭圆 /抛物线问题中空间依赖性潜在系数的反问题。我们在问题数据上非常温和的条件下建立了新颖的（加权）条件稳定性估计。然后，我们根据标准输出最小二乘形式进行了标准重建方案的错误分析（通过tikhonov正则化（通过$ h^1 $ - 巨型惩罚），然后由Galerkin有限元方法与连续的分段线性有限元在太空中（以及向后的Euler euler方法，以及用于Parabolic问题的后退欧拉方法）的离散化。我们提供了离散方案的详细分析，并在加权$ l^2（ω）$中提供了收敛速率，以相对于确切的电位。误差边界明确取决于噪声水平，正则参数和离散参数。在适当的条件下，我们还会在标准$ l^2（ω）$和内部$ l^2 $规范中得出错误估计。该分析采用尖锐的先验误差估计和非标准测试功能。进行了几个数值实验以补充理论分析。
In this work we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very mild conditions on the problem data. Then we provide an error analysis of a standard reconstruction scheme based on the standard output least-squares formulation with Tikhonov regularization (by an $H^1$-seminorm penalty), which is then discretized by the Galerkin finite element method with continuous piecewise linear finite elements in space (and also backward Euler method in time for parabolic problems). We present a detailed analysis of the discrete scheme, and provide convergence rates in a weighted $L^2(Ω)$ for discrete approximations with respect to the exact potential. The error bounds are explicitly dependent on the noise level, regularization parameter and discretization parameter(s). Under suitable conditions, we also derive error estimates in the standard $L^2(Ω)$ and interior $L^2$ norms. The analysis employs sharp a priori error estimates and nonstandard test functions. Several numerical experiments are given to complement the theoretical analysis.
