Eugene E. Tyrtyshnikov

Breaking the Curse of Dimensionality, Or How to Use SVD in Many Dimensions

For

Incomplete cross approximation in the mosaic-skeleton method

d

Mosaic-Skeleton approximations

-dimensional tensors with possibly large

Tucker Dimensionality Reduction of Three-Dimensional Arrays in Linear Time

d>3

SPECTRA OF MULTILEVEL TOEPLITZ MATRICES : ADVANCED THEORY VIA SIMPLE MATRIX RELATIONSHIPS

, an hierarchical data structure, called the Tree-Tucker format, is presented as an alternative to the canonical decomposition. It has asymptotically the same (and often even smaller) number of representation parameters and viable stability properties. The approach involves a recursive construction described by a tree with the leafs corresponding to the Tucker decompositions of three-dimensional tensors, and is based on a sequence of SVDs for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial “spatial” dimensions. It is shown how this format can be applied to the problem of multidimensional convolution. Convincing numerical examples are given.

Any Circulant-Like Preconditioner for Multilevel Matrices Is Not Superlinear

Abstract The mosaic-skeleton method was bred in a simple observation that rather large blocks in very large matrices coming from integral formulations can be approximated accurately by a sum of just few rank-one matrices (skeletons). These blocks might correspond to a region where the kernel is smooth enough, and anyway it can be a region where the kernel is approximated by a short sum of separable functions (functional skeletons). Since the effect of approximations is like that of having small-rank matrices, we find it pertinent to say about mosaic ranks of a matrix which turn out to be pretty small for many nonsingular matrices.On the first stage, the method builds up an appropriate mosaic partitioning using the concept of a tree of clusters and some extra information rather than the matrix entries (related to the mesh). On the second stage, it approximates every allowed block by skeletons using the entries of some rather small cross which is chosen by an adaptive procedure. We focus chiefly on some aspects of practical implementation and numerical examples on which the approximation time was found to grow almost linearly in the matrix size.

Approximate iterations for structured matrices

If a matrix has a small rank then it can be multiplied by a vector with many savings in memory and arithmetic. As was recently shown by the author, the same applies to the matrices which might be of full classical rank but have a smallmosaic rank. The mosaic-skeleton approximations seem to have imposing applications to the solution of large dense unstructured linear systems. In this paper, we propose a suitable modification of brandts definition of an asymptotically smooth functionf(x,y). Then we considern×n matricesAn=[f(xi(n),yj(n))] for quasiuniform meshes {xi(n)} and {yj(n)} in some bounded domain in them-dimensional space. For such matrices, we prove that the approximate mosaic ranks grow logarithmically inn. From practical point of view, the results obtained lead immediately toO(n logn) matrix-vector multiplication algorithms.

Circulant preconditioners with unbounded inverses

We consider Tucker-like approximations with an

Multilevel Toeplitz Matrices Generated by Tensor-Structured Vectors and Convolution with Logarithmic Complexity

r \times r \times r

Some Remarks on the Elman Estimate for GMRES

core tensor for three-dimensional