{'fraction_non_alphanumeric': 0.010922330097087379, 'fraction_numerical': 0.0, 'mean_word_length': 5.818181818181818, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
An Optimization-Based Approach for the Design of PDE Solution Algorithms

We develop and analyze an optimization-based approach for the robust and efficient solution of PDE problems consisting of multiple physics operators with fundamentally different mathematical properties. Our approach relies on three essential steps: decomposition of the original problem into subproblems for which robust solution algorithms are available; integration of the subproblems into an equivalent PDE-constrained optimization problem; and solution of the resulting optimization problem either directly as a fully coupled algebraic system or in the null space of the PDE constraints. This strategy gives rise to a general approach for synthesizing robust solvers for complex coupled problems from solvers for their simpler physics components.