Remarks on parameter identification. I
type: Item
instance of: scholarly article
title: Remarks on parameter identification. I
zbMATH Open document ID: 0622.35077
DOI: 10.1007/BF01389584
author: Ronald B. Guenther
author: William McDougal
author: Jürgen Gerlach
author: Robert T. Hudspeth
published in: Numerische Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133038
review text: To determine the diffusion coefficient a(x) in the equation \[ (1)\quad u_ t-(a(x)u_ x)_ x = f(x,t) \] supposed to hold on \(\{0<t\leq T\), \(0<x<1\}\) with Dirichlet and initial conditions \(u(0,t)=\phi (t)\), \(u(1,t)=\psi (t)\), \(u(x,0)=g(x)\), a uniqueness theorem is proven and an approximation algorithm is suggested. The scheme is based on minimizing the cost functional \[ I=(1/2)\int^{T}_{0}\int^{1}_{0}[u_ n(x,t)-\tilde u(x,t)]^ 2 dx dt+ \] \[ (\lambda /2)\int^{1}_{0}[u_ n(x,T)-\tilde u(x,T)]^ 2 dt+(1/2)\sum^{\ell}_{j=1}\mu_ j[p_ n(x_ j)-\tilde a_ j]^ 2 \] where \(p_ n\) is taken from an n- dimensional approximation space to a; \(u_ n\) is the direct solution to (1) corresponding to \(p_ n\); \(\tilde a_ j\) are measured values for \(a(x_ j)\); \(\tilde u\) is a measure for u; and \(\mu_ j\), \(\lambda\) are weights. Some numerical evidence of the scheme behavior is presented, while convergence results are postponed to Part II of the paper.
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 65Z05
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 93B30
Mathematics Subject Classification ID: 35K20
zbMATH DE Number: 4008986
zbMATH Keywords: parameter identification
zbMATH Keywords: minimization numerical scheme
zbMATH Keywords: diffusion coefficient
zbMATH Keywords: Dirichlet
zbMATH Keywords: initial conditions
zbMATH Keywords: uniqueness
zbMATH Keywords: approximation algorithm
zbMATH Keywords: numerical evidence
MaRDI profile type: Publication
cites work: Q3799606
cites work: Q4062302
cites work: On the identification of coefficients of elliptic problems by asymptotic regularization
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description: scientific article; zbMATH DE number 4008986