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Data completion for linear symmetric operators as a cauchy problem: An efficient method via energy-like error minimization

Thus the aim is to determine the solution of a known PDE defined throughout the domain, which satisfies the superabundant data and then identifies the missing ones. For linear symmetric operators, we propose a general method to solve the data completion problem as a Cauchy problem. Various applications are described for stationary conduction and elastostatic problems. | Vietnam Journal of Mechanics, VAST, Vol. 31, No. 3 &4 (2009), pp. 247 – 261 DATA COMPLETION FOR LINEAR SYMMETRIC OPERATORS AS A CAUCHY PROBLEM: AN EFFICIENT METHOD VIA ENERGY-LIKE ERROR MINIMIZATION Thouraya N. Baranger1 and Stéphane Andrieux2 1 Université de Lyon, CNRS LaMCoS, UMR5259, INSA-Lyon, F-69621, Villeurbanne; Université Lyon 1, F-69622, Villeurbanne, France 2 Mechanics of Sustainable Industrial Structures Laboratory, UMR CNRS-EDF 2832, Clamart, France Abstract. Data completion is a problem in which known or measured superabundant data exist for part of the boundaries of a domain, whereas the data for the rest of the boundaries are unknown. Thus the aim is to determine the solution of a known PDE defined throughout the domain, which satisfies the superabundant data and then identifies the missing ones. For linear symmetric operators, we propose a general method to solve the data completion problem as a Cauchy problem. Various applications are described for stationary conduction and elastostatic problems. 1. INTRODUCTION Given the growth of development in measurement and data imaging technologies [11.], data completion problems can arise in a large range of applications. These problems occur when dealing with PDE known to hold true in a solid for which data is lacking on a part of its boundary but with superabundant boundary data on another part of it. Thus data completion problems consist in recovering the lacking data. It is an inverse, then an ill-posed problem. Examples can be found in thermal and electric conduction problems or linear elasticity, in saturated porous media, for linear fracture mechanics applications. The data completion problem has been addressed by many authors, firstly for the Laplace operator and mainly in two dimensions. Various approaches have been proposed using boundary element techniques [2,3], fundamental solutions [4,5], regularized least squares methods [6], moment methods associated with the Backus–Gilbert procedure .