Localized Reduced Basis Methods for PDE-constrained Parameter Optimization

本课题研究非线性椭圆型偏微分方程参数优化的模型约简问题。目标是开发一种基于自适应在线富集的pde约束优化新范式。其核心思想是设计一种局部化的约简基方法，称为局部化约简基方法。这使我们能够在应用优化算法的每次迭代中在线收紧降阶近似的质量。一个局部后验误差分析保证了基化后解收敛于无限维参数优化问题的解。在局部逼近质量不准确的情况下，RB离散化只在局部得到了非常有效的改进。该方法适用于数值多尺度方法、基于信任域的优化方法和迭代正则化Gauß-Newton算法。

This projects is concerned with model reduction for parameter optimization of nonlinear elliptic partial differential equations (PDEs). The goal is to develop a new paradigm for PDE-constrained optimization based on adaptive online enrichment. The essential idea is to design a localized version of the reduced basis (RB) method which is called Localized Reduced Basis Method (LRBM). This allows us to tighten the quality of the reduced order approximation online within each iteration of the applied optimization algorithms. A localized a posteriori error analysis ensures convergence of the reduced basis solution to the solution of the underlying infinite dimensional parameter optimization problem. In the case of a locally inaccurate approximation quality the RB discretization is improved only locally in a very efficient way. The approach is designed for numerical multiscale methods, trust region based optimization methods and for iteratively regularized Gauß-Newton algorithms.