We propose an innovative method, at the interface between statistics and numerical analysis, for the estimation of surfaces and spatial fields. Specifically, we consider non-parametric regression models for the estimation of spatial fields from pointwise and noisy observations, when a prior knowledge on the phenomenon under study is available. The prior knowledge included in the model derives from physics, physiology or mechanics of the problem at hand, and is formalized in terms of a partial differential equation (PDE) governing the phenomenon behavior, as well as conditions that the phenomenon has to satisfy at the boundary of the problem domain. This prior information is incorporated in the model via a roughness term using a penalized regression framework. The proposed method allows for important modeling flexibility, accounting for space anisotropy and non-stationarity in a straightforward way, as well as unidirectional smoothing effects.
Spatial Regression with PDE penalization (SR-PDE) has very broad applicability since PDEs are commonly used to describe phenomena behavior in many fields of physical and biological sciences. The proposed technique is in fact particularly well suited for applications in physics, engineering, biomedicine, etc., where a prior knowledge on spatial field might be available from physical principles and should be taken into account in the field estimation or smoothing process. Specifically we focus on phenomena that are well described by linear second order elliptic PDEs.
The proposed models exploit advanced scientific computing techniques and specifically make use of the Finite Element method. The proposed mixed Finite Element method provides a good approximation of the spatial field but also of its first and second order derivatives that can be useful in order to compute physical quantities of interest. The resulting estimators have a typical penalized regression form, they are linear in the observed data values and classical inferential tools can be derived.
The smoothing technique is also extended to the case of areal data, particularly interesting for the driving application that concerns the estimation of the blood-flow velocity field in a section of a carotid artery, using data provided by echo-color doppler; this applied problem arises within a research project that aims at studying atherosclerosis pathogenesis.