Remarks on parameter identification. I
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.
scientific article; zbMATH DE number 4008986
scientific article