A weighted singular value decomposition for the discrete inverse problems. Issue 1 (6th August 2017) - Jozi, Meisam
Summary: In this paper, we introduce and analyze a new singular value decomposition (SVD) called weighted SVD (WSVD) using a new inner product instead of the Euclidean one. We use the WSVD to approximate the singular values and the singular functions of the Fredholm integral operators. In this case, the new inner product arises from the numerical integration used to discretize the operator. Then, the truncated WSVD (TWSVD) is used to regularize the Nyström discretization of the first‐kind Fredholm integral equations. Also, we consider the weighted LSQR (WLSQR) to approximate the solution obtained by the TWSVD method for large problems. Numerical experiments on a few problems are used to illustrate that the TWSVD can perform better than the TSVD.