Espen Sande: Best approximations of matrices and differential operators

It is well known that if the singular values of a matrix are distinct, then its best rank-n approximation in the Frobenius norm is uniquely determined and given by the truncated singular value decomposition. On the other hand, this uniqueness is in general not true for best rank-n approximations in the spectral norm. In this talk we relate the problem of finding best rank-n approximations in the spectral norm to Kolmogorov n-widths and corresponding optimal spaces. By providing new criteria for optimality of subspaces with respect to the n-width, we describe a large family of best rank-n approximations to a given matrix. This results in a variety of solutions to the best low-rank approximation problem and provides alternatives to the truncated singular value decomposition. This variety can be exploited to obtain best low-rank approximations with problem-oriented properties.
We further discuss the generalization of these results to compact operators in L2, and explain how they can be used to both describe the out-performance of smooth spline approximations of solutions to differential equations when compared to classical finite element methods, and to solve the outlier-problem in isogeometric analysis.
This talk is based on work done in collaboration with Michael Floater, Carla Manni and Hendrik Speleers.