伪微分算子学习的收敛速率
Convergence Rates for Learning Pseudo-Differential Operators
AI总结：
本文提出了一种高效估计器，用于学习椭圆伪微分算子并实现收敛速率分析，结合小波伽辽金框架和新型矩阵压缩方案，提升科学计算中的求解效率与稳定性。
摘要：
本文建立了学习椭圆伪微分算子的收敛速率，这是偏微分方程和数学物理中的基本算子类。在小波伽辽金框架下，我们将学习此类算子视为一个具有多尺度稀疏性的结构无穷维回归问题。在此基础上，我们提出了一种稀疏、数据和计算高效估计器，该估计器利用了针对学习任务定制的新型矩阵压缩方案和嵌套支持策略，以平衡近似误差和估计误差。除了获得估计器的收敛速率外，我们还证明所学算子诱导出一个高效且稳定的伽辽金求解器，其数值误差与其统计准确性相匹配。我们的结果因此促进了运算学习、数据驱动求解器和小波方法在科学计算中的结合。
英文摘要：
This paper establishes convergence rates for learning elliptic pseudo-differential operators, a fundamental operator class in partial differential equations and mathematical physics. In a wavelet-Galerkin framework, we formulate learning over this class as a structured infinite-dimensional regression problem with multiscale sparsity. Building on this structure, we propose a sparse, data- and computation-efficient estimator, which leverages a novel matrix compression scheme tailored to the learning task and a nested-support strategy to balance approximation and estimation errors. In addition to obtaining convergence rates for the estimator, we show that the learned operator induces an efficient and stable Galerkin solver whose numerical error matches its statistical accuracy. Our results therefore contribute to bringing together operator learning, data-driven solvers, and wavelet methods in scientific computing.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
