发表机构
Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种利用局部行空间信息进行无伴随、查询高效的线性算子近似方法，适用于渐近光滑核的积分算子，并证明其误差随查询次数指数衰减。
AI 中文摘要
仅使用前向算子-向量乘积查询来近似高维空间上的线性算子是科学计算和机器学习中常见但具有挑战性的任务，当伴随作用的访问在实验上不可用或计算上难以构造时，就会出现这种情况。现有的查询高效方法在误差容限下具有多对数查询复杂度，但依赖于自伴性或对伴随算子的访问，而现有的基于全局低阶正则性的无伴随方法具有多项式查询复杂度。我们证明，先验的局部行空间信息可以用于查询高效、无伴随的算子近似，并且这些信息可以通过局部高阶正则性估计来提供。我们的方法首先将(Amsel et al., SIMAX, 2026)的固定稀疏矩阵近似算法扩展到用行位于指定子空间中的矩阵来近似矩阵，为仅从前向查询中利用局部行空间结构提供了一个通用模板。然后，我们将此框架应用于具有渐近光滑核的Hilbert-Schmidt积分算子。我们建立了在$d\leq 3$维中典型二阶椭圆偏微分方程（包括球中的泊松方程和环面上的常系数方程）的格林函数的渐近光滑性。在$d$空间维度中，我们的算法使用$\mathcal{O}\big(\log(\varepsilon^{-1})\log^{d}(\varepsilon^{-1}\delta^{-1/2})\big)$次非自适应随机查询，以概率$1-\delta$产生Hilbert-Schmidt范数下的$\varepsilon$精确近似。这意味着误差随查询次数$q$以$\mathcal{O}\big( \exp(- c q^{1/(d+1)}) \big)$的速度减小，我们在使用数十到数百次查询来近似一维变系数二阶椭圆方程的解算子的数值实验中观察到了这一点。
英文摘要
Approximating linear operators on high-dimensional spaces using only forward operator-vector product queries is a common but challenging task in scientific computing and machine learning arising when access to the adjoint action is experimentally unavailable or difficult to construct computationally. Existing query-efficient methods with polylogarithmic query complexity in the error tolerance rely on self-adjointness or access to the adjoint operator, while existing adjoint-free methods based on global low-order regularity have polynomial query complexity. We show that a priori local row-space information can be exploited for query-efficient, adjoint-free operator approximation, and that this information can be supplied by local high-order regularity estimates. Our approach first extends the fixed-sparsity matrix approximation algorithm of (Amsel et al., SIMAX, 2026) to approximating a matrix by one whose rows lie in prescribed subspaces, providing a general template for exploiting local row-space structure from forward queries alone. We then apply this framework to Hilbert-Schmidt integral operators with asymptotically smooth kernels. We establish asymptotic smoothness for Green's functions of canonical second-order elliptic PDEs in $d\leq 3$ dimensions including Poisson's equation in balls and constant coefficient equations on tori. In $d$ spatial dimensions, our algorithm uses $\mathcal{O}\big(\log(\varepsilon^{-1})\log^{d}(\varepsilon^{-1}δ^{-1/2})\big)$ non-adaptive randomized queries to produce an $\varepsilon$-accurate approximation in Hilbert-Schmidt norm with probability $1-δ$. This implies the error decreases as $\mathcal{O}\big( \exp(- c q^{1/(d+1)}) \big)$ in the query count $q$, which we observe in numerical experiments using tens to hundreds of queries to approximate solution operators of one-dimensional variable-coefficient second-order elliptic equations.
