AI 中文总结
针对大型稀疏非埃尔米特矩阵函数计算问题，提出草图与重启框架，结合求积重启和类阿诺尔迪分解，开发两类重启算法，建立收敛性，经数值实验验证该框架在计算节省、存储减少和加速方面的有效性。
AI 中文摘要
我们开发了一种草图与重启框架，用于计算矩阵函数对向量的作用$f(A)b$，其中$A$是大型、稀疏且非埃尔米特矩阵。该框架将基于求积的重启与由草图或截断阿诺尔迪过程生成的类阿诺尔迪分解相结合。在此框架内，我们开发了两类重启算法。第一类使用固定的克里洛夫子空间维度，基于草图阿诺尔迪过程或本文提出的新草图调和阿诺尔迪过程。第二类通过运行截断阿诺尔迪过程自适应选择克里洛夫子空间维度，直到从其草图估计的生成基的条件数超过规定阈值。我们还在$A$为正实矩阵的假设下，建立了重启草图调和阿诺尔迪方法对斯蒂尔杰斯函数的收敛性。数值实验证明了所提框架的有效性，包括通过草图实现的计算节省、自适应截断带来的存储减少以及厚重启获得的加速。
英文摘要
We develop a sketch-and-restart framework for computing the action of a matrix function on a vector, $f(A) b$, where $A$ is large, sparse, and non-Hermitian. The framework combines quadrature-based restarting with Arnoldi-like decompositions generated by sketched or truncated Arnoldi processes. Within this framework, we develop two classes of restarted algorithms. The first uses a fixed Krylov subspace dimension and is based either on the sketched Arnoldi process or on a new sketched harmonic Arnoldi process proposed in this work. The second class chooses the Krylov subspace dimension adaptively by running the truncated Arnoldi process until the condition number of the generated basis, estimated from its sketch, exceeds a prescribed threshold. We also establish the convergence of the restarted sketched harmonic Arnoldi method for Stieltjes functions under the assumption that $A$ is positive real. Numerical experiments demonstrate the effectiveness of the proposed framework, including the computational savings achieved through sketching, the storage reduction enabled by adaptive truncation, and the acceleration obtained from thick restarting.
Comments23 pages, 5 figures
