Computationally efficient techniques for spatial regression with differential regularization

We investigate some computational aspects of an innovative class of PDE-regularized statistical models: Spatial Regression with Partial Differential Equation regularization (SR-PDE). These physics-informed regression methods can account for the physics of the underlying phenomena and handle data observed over spatial domains with nontrivial shapes, such as domains with concavities and holes or curved domains. The computational bottleneck in SR-PDE estimation is the solution of a computationally demanding linear system involving a low-rank but dense block. We address this aspect by innovatively using Sherman-Morrison-Woodbury identity. We also investigate the efficient selection of the smoothing parameter in SR-PDE estimates. Specifically, we propose ad hoc optimization methods to perform Generalized Cross-Validation, coupling suitable reformulation of key matrices, e.g. those based on Sherman-Morrison-Woodbury formula, with stochastic trace estimation, to approximate the equivalent degrees of freedom of the problem. These solutions permit high computational efficiency also in the context of massive data.