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UID:submissions.pasc-conference.org_PASC22_sess178_pap106@linklings.com
SUMMARY:Towards Data-Driven Inference of Stencils for Discrete Differentia
 l Operators
DESCRIPTION:Paper\n\nTowards Data-Driven Inference of Stencils for Discret
 e Differential Operators\n\nSchumann, Neumann\n\nFinite element, finite di
 fference 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 o
 f freedom —the so called stencil—which approximate the underlying differen
 tial operators. However, knowledge of the symbolic differential equation i
 s required to compute the stencil coefficients. We introduce comprehensive
  regression techniques, that allow to infer stencil coefficients from (exp
 erimental) data for linear, oneand two-dimensional PDEs. Starting with the
  1D case, we discuss how mathematically meaningful stencil coefficients ca
 n be obtained via an ordinary least-squares (OLS) linear regression if a f
 ull-rank matrix of explanatory variables is given. We continue by demonstr
 ating how regularization techniques allow for the recovery of the correct 
 stencil coefficients from data even for singular matrices. In the consider
 ed settings, the measurement error might affect explanatory and response v
 ariable alike, such that we discuss the impact of different levels of nois
 e in both of them and compare different errors-in-variables approaches to 
 mitigate the inherent consequences of noisy independent variables in OLS r
 egression. We extend our considerations to the two-dimensional case and hi
 gher-order expansions of the differential operator and compare the stencil
 s obtained from our regression approach with their analytic solutions. Fin
 ally, we demonstrate both the relevance as well as the applicability of th
 e presented technique by applying it to two scenarios from physics and hyd
 rogeology.\n\nDomain: Computer Science and Applied Mathematics
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