Towards Data-Driven Inference of Stencils for Discrete Differential Operators
Presenter
DescriptionFinite element, finite difference or similar methods allow to numerically solve ordinary (ODEs) and partial differential equations (PDEs). The backbone of all these discrete approaches are matrices with non-zero coefficients for close-by degrees of freedom —the so called stencil—which approximate the underlying differential operators. However, knowledge of the symbolic differential equation is required to compute the stencil coefficients. We introduce comprehensive regression techniques, that allow to infer stencil coefficients from (experimental) data for linear, oneand two-dimensional PDEs. Starting with the 1D case, we discuss how mathematically meaningful stencil coefficients can be obtained via an ordinary least-squares (OLS) linear regression if a full-rank matrix of explanatory variables is given. We continue by demonstrating how regularization techniques allow for the recovery of the correct stencil coefficients from data even for singular matrices. In the considered settings, the measurement error might affect explanatory and response variable alike, such that we discuss the impact of different levels of noise in both of them and compare different errors-in-variables approaches to mitigate the inherent consequences of noisy independent variables in OLS regression. We extend our considerations to the two-dimensional case and higher-order expansions of the differential operator and compare the stencils obtained from our regression approach with their analytic solutions. Finally, we demonstrate both the relevance as well as the applicability of the presented technique by applying it to two scenarios from physics and hydrogeology.
TimeTuesday, June 2815:00 - 15:30 CEST
LocationSamarkand Room
SessionAP2D - ACM Papers Session 2D
Session Chair
Event Type
Paper
Computer Science and Applied Mathematics
