高秩Hessian的局部支撑非负积分核点扩散函数逼近
Point spread function approximation of high rank Hessians with locally supported non-negative integral kernels
AI总结：
本文提出一种无矩阵点扩散函数方法，通过局部支撑非负积分核的脉冲响应插值构造分层矩阵逼近，用于为PDE反问题中的高秩Hessian构建高效预条件子，显著减少求解次数。
摘要：
我们提出了一种高效的无矩阵点扩散函数（PSF）方法，用于逼近具有局部支撑非负积分核的算子。该方法在散乱点上计算脉冲响应，并对这些脉冲响应进行插值以逼近积分核元素。脉冲响应通过将算子应用于Dirac梳状点源批次来计算，这些点源通过椭球填充过程选取。核元素的求值使我们能够构造算子的分层矩阵逼近，该逼近可用于进一步的矩阵计算。我们在一个模糊问题上展示了端到端方法，然后使用该方法为两个由偏微分方程（PDE）控制的反问题中的Hessian构建预条件子：冰盖流动问题中基底摩擦系数的反演，以及平流-扩散输运问题中初始条件的反演。虽然对于许多不适定反问题，数据失配项的Hessian表现出低秩结构，因此低秩逼近是合适的，但对于许多实际感兴趣的问题，Hessian的数值秩仍然很大。但Hessian脉冲响应通常随着数值秩的增加而变得更加局部化，这有利于PSF方法。数值结果表明，PSF预条件子使预条件后的Hessian谱聚集在1附近，与正则化预条件和不使用预条件相比，所需的PDE求解次数减少了约5倍到10倍。我们还给出了一项数值研究，研究各种参数（控制脉冲响应形状的参数）对平流-扩散Hessian逼近有效性的影响。结果表明，基于PSF的预条件子能够使用少量算子应用来形成高秩Hessian的良好逼近。
英文摘要：
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 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 via an ellipsoid packing procedure. Evaluation of kernel entries allows us to construct a hierarchical matrix approximation of the operator, which is used for further matrix computations. We illustrate the end-to-end method on a blur problem, then use the method to build preconditioners for the Hessian 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.
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