Non&war Analysis. Theory . .clerhods & Applicmom. Vol. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDC 10. No. 2. pp. 121-146. 1986. 0362-546X 86 53 00 + .&I Prmted m Great Britain. 0 1986 Pergamon Press Ltd. PARAMETER ESTIMATION FOR ELLIPTIC EQUATIONS IN MULTIDIMENSIONAL DOMAINS WITH POINT AND FLUX OBSERVATIONS K. KUNISCH* Institut fur Mathematik, Technische Universitlt Graz, A-8010 Graz, Austria and Department of Mathematics, The University of Oklahoma, Norman, Oklahoma 73019, U.S.A. and L. WHITE: Department of Mathematics, The University of Oklahoma. Norman, Oklahoma 73019. U.S..-! (Received 15 April 1984; received for publication 9 October 1981) Key words and phrases: Elliptic equations in multidimensional domains, point and flux observation. 1. INTRODUCTION IN THIS paper we study the approximation of parameter identification problems of uniformly strongly elliptic boundary value problems with Dirichlet boundary conditions. The fit-to-data criteria that we consider are motivated by certain physical problems arising in elastic systems and flow in porous media and involve point observations or observation of flux at the boundary or in the interior. We describe the problems here, and in later sections we specify more precisely our assumptions. Let R be a bounded open domain in R”, n = 2 or 3. We consider equations of the form L(C?)” = - X& zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFED (aij(x>uxi) + bi(x)ux, +c(x)U=f inS2 I (1.1) with boundary conditions u=OonaS2 (1.2) where q denotes the triple (Sa, 6, c) in which Sp = &4(x) is the n x n matrix with entries (d(x))ij = ali and 6 = d(x) is the n-vector with entries (6(x)), = b;(x). The solution of (1.1) we denote by u(q) or also u(. ; q). The identification problem is to determine a parameter q within some suitable set of admissible elements QOdthat produces an observation of the system (l.l)-(1.2) which is closest to the given information z. We view z as an element in an observation space. The effect of any given admissible parameter q may be seen only through an observation operator C that maps solutions u(q) of (l.l)-(1.2) into the observation space. The comparison of zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA G(q) with z is made through a fit-to-data criterion defined on the l This author gratefully acknowledges support from the Max Kade Foundation. Research supported in part by the Fonds zur Forderune der Wissenschaftlichen Forschuna, AUSTRIA. No. P4534. and bv the Steiermlrkischen Wissenschafts und F&schungsfGrderungsfonds. -. t This author was supported in part by NSF Grant No. MCS-7902037 and by the Cooperative Institute for Mesoscale Meteorological Studies. 121