{'fraction_non_alphanumeric': 0.013921113689095127, 'fraction_numerical': 0.0, 'mean_word_length': 5.638461538461539, '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}}
Meshfree based multiresolution solver for PDEs with large gradients

We introduce a multilevel PDE solver for equations whose solutions exhibit large gradients. Expanding on Ami Harten's ideas, we construct an alternative to wavelet-based grid refinement, a multiresolution coarsening method capable of capturing sharp gradients across different scales and thus improving PDE-based simulations by concentrating computational resources in places where the solution varies sharply. Our scheme is akin to Finite Differences in that it computes derivatives explicitly and then uses the derivative information to march the solution in time. However, we utilize meshfree methods to compute derivatives and integrals in space-time to increase the robustness of our solver and tailor the basis functions to the Kd-tree structure provided by the multiresolution analysis.