A weighted singular value decomposition for the discrete inverse problems

What is it about?

We have established and analyzed a new SVD, called WSVD, for the matrices obtained from the Nyström discretization of the first-kind linear Fredholm integral equations. The WSVD is based on a new inner product arisen from the approximation of L2-inner product by the quadrature rule. Also, the TWSVD has been used to regularize the final discrete problem. In addition, We have investigated the regularization properties of the WLSQR by comparing it to the TWSVD. Numerical experiments have shown that the TWSVD may yield satisfactory results when applied to the Nyström discretization of the first-kind Fredholm integral equations and the regularizing effects of the WLSQR algorithm are good enough to compute the TWSVD solution.

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Why is it important?

In this paper, we introduce a new singular value decomposition (SVD) of the matrix A called weighted SVD (WSVD) by using the D-inner product instead of the Euclidian one. In fact, we obtain the WSVD by discretization of the SVD of the integral operator.