We present an efficient matrix-free point spread function (PSF) method for approximating operators that have locally supported non-negative integral kernels. The method computes impulse responses of the operator at scattered points, and interpolates these impulse responses to approximate integral kernel entries. Impulse responses are computed by applying the operator to Dirac comb batches of point sources, which are chosen by solving an ellipsoid packing problem. Evaluation of kernel entries allows us to construct a hierarchical matrix (H-matrix) approximation of the operator. Further matrix computations are performed with H-matrix methods. We use the method to build preconditioners for the Hessian operator in two inverse problems governed by partial differential equations (PDEs): inversion for the basal friction coefficient in an ice sheet flow problem and for the initial condition in an advective-diffusive transport problem. While for many ill-posed inverse problems the Hessian of the data misfit term exhibits a low rank structure, and hence a low rank approximation is suitable, for many problems of practical interest the numerical rank of the Hessian is still large. But Hessian impulse responses typically become more local as the numerical rank increases, which benefits the PSF method. Numerical results reveal that the PSF preconditioner clusters the spectrum of the preconditioned Hessian near one, yielding roughly 5x-10x reductions in the required number of PDE solves, as compared to regularization preconditioning and no preconditioning. We also present a numerical study for the influence of various parameters (that control the shape of the impulse responses) on the effectiveness of the advection-diffusion Hessian approximation. The results show that the PSF-based preconditioners are able to form good approximations of high-rank Hessians using a small number of operator applications.

翻译：我们提出了一种高效的无矩阵点扩散函数方法，用于逼近具有局部支撑非负积分核的算子。该方法在散点处计算算子的脉冲响应，并通过插值这些脉冲响应来近似积分核的矩阵元素。脉冲响应通过将算子作用于由求解椭球填充问题选定的狄拉克梳状点源批次来计算。核元素的评估使我们能够构建算子的层级矩阵逼近。后续矩阵计算采用层级矩阵方法。我们将该方法用于构建两个由偏微分方程控制的反问题中Hessian算子的预条件子：冰盖流动问题中的基底摩擦系数反演和对流扩散输运问题中的初始条件反演。尽管许多病态反问题中数据失配项的Hessian矩阵呈现低秩结构，因此低秩逼近是合适的，但在许多实际感兴趣的问题中，Hessian矩阵的数值秩仍然较大。然而，随着数值秩增大，Hessian脉冲响应通常变得更加局部化，这有利于点扩散函数方法。数值结果表明，点扩散函数预条件子能将预条件后Hessian矩阵的特征值聚类在1附近，与正则化预条件和无预条件相比，所需的偏微分方程求解次数减少约5-10倍。我们还数值研究了控制脉冲响应形状的各种参数对对流扩散Hessian逼近有效性的影响。结果表明，基于点扩散函数的预条件子能够仅通过少量算子应用即可形成对高秩Hessian矩阵的良好逼近。