{'fraction_non_alphanumeric': 0.01144578313253012, 'fraction_numerical': 0.0, 'mean_word_length': 5.04, '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}}
Information requirements and the implications for parallel computation

The best serial algorithms for numerically approximating the solution of a linear partial differential equation (PDE) exploit knowledge of the solution operator. This dissertation describes how the solution operator also influences the behavior of parallel algorithms. 
Approximating the solution at a single location in the problem domain is considered. We derive a lower bound on the error in the approximation that is a function of the amount of data used and the smoothness of the data functions. From this we derive a lower bound on the parallel complexity of algorithms that approximate the solution. The lower bound on the parallel complexity is a logarithmic function of the reciprocal of an upper bound on the error. Thus the parallel complexity increases as the error bound decreases, independent of the number of processors used. We also construct an algorithm whose parallel complexity is of this form, proving that the form of the bound is the best possible. 
The execution time of parallel algorithms is a function of both the communication costs and the parallel complexity. We describe bounds on the communication costs of parallel algorithms that are functions of the distance between collaborating processors. If the interconnection network of a given multiprocessor is a d dimensional grid, then we prove that the execution time for algorithms that approximate the solution is bounded from below by a negative power of the error bound. Thus, if the upper bound on the error is small, then the communication costs are the dominant constraints on the optimal performance.