Recent Developments in Inverse Problems - Abstract
Uhlig, Dana
In principle, the determination of the copula density $operatornamec$ from a given copula $operatornameC$ of a $d$ dimensional random vector is a simple differentiation $operatornamec(u_1, ldots, u_d) = fracpartial^d operatornameCpartial u_1 cdots partial u_d$. In practical applications only the non smooth empirical copula beginequation* hatoperatornameC(rmbf u)= frac1T sumlimits_j=1^T 1hspace-0,9ex1_hatrmbf U_j le rm bf u = frac1T sumlimits_j=1^T prodlimits_k=1^d 1hspace-0,9ex1_hatU_kj le u_k endequation* is observable for samples $hatrmbf U_1, ldots, hatrmbf U_T$ and therefore we treat the integral equation beginequation* beginsplit intlimits_0^u_1 cdots intlimits_0^u_d operatornamec(s_1,ldots, s_d) rm d s_1 cdots rm d s_d &= operatornameC(u_1, ldots, u_d)
& qquad forall rmbf u=(u_1, ldots, u_d)^T in Omega = [0,1]^d ,, endsplit endequation* as a weak formulation of the differentiation. The corresponding linear Volterra integral operator $operatornameA$ with $operatornameA operatornamec = operatornameC$ is ill-posed and well studied in inverse problem theory, at least for the one dimensional case. vspace*1mm We discuss a particular discretization based on a Petrov--Galerkin projection and regularization of the discretized linear system, which allows us to compute also for higher dimensions numerical solutions. Furthermore, we consider the order of discretization and regularization and show that there are similarities between the Lavrientiev regularization of the discretized problem and the Tikhonov regularization $left( operatornameAhspace-0,6ex^star operatornameA + alpha I right) operatornamec = operatornameAhspace-0,6ex^star operatornameC$ of the linear integral equation $operatornameA operatornamec = operatornameC$ and their subsequent discretization by a Galerkin projection.