Efficient computer manipulation of tensor products with applications to multidimensional approximation

Abstract

Abstract. The objective of this paper is twofold :(a) To make it possible to perform matrix-vector operations in tensor product spaces, using only the factors (n-p1 words of information for ®"=1 A¡, A¡ £ £(£", E")) instead of the tensor-product operators themselves ((/>2)" words of information).(b) To produce efficient algorithms for solving systems of linear equations with coefficient matrices being tensor products of nonsingular matrices, with special application to the approximation of multidimensional linear functionals.

Tensor methods for parameter estimation and bifurcation analysis of stochastic reaction networks

“…Specifically, we keep all the objects in the separated form of (2) during the computations, such that exponential scaling in complexity does not apply during any step of the TPA. We refer to the representation (2) as tensor-structured, because computations are performed on p(x|k) as multidimensional arrays of real numbers, which we call tensors [20]. The (canonical) tensor decomposition [21], as a discrete counterpart of (2), then allows a multidimensional array to be approximated as a sum of tensor products of one-dimensional vectors.…”

“…We refer to the representation (2.2) as tensor-structured, because computations are performed on p(xjk) as multidimensional arrays of real numbers, which we call tensors [20]. The (canonical) tensor decomposition [21], as a discrete counterpart of (2.2), then allows a multi-dimensional array to be approximated as a sum of tensor products of one-dimensional vectors.…”

Spline approximation, Kronecker products and multilinear forms

“…To make the long story short: in the tensor product spline approach the high dimensional approximation problem from (11) is equivalent to solving a linear problem of the form (13), where the first matrix is a sum of #X Kronecker products and the second one is also a finite combination of Kronecker products (all of the same dimension), but the number of terms depends on the differential operator in the smoothing term. Usually, those are homogeneous operators of some order d , thus contributing pCd 1 d terms in G each of which requires a storage of P n 2 j .…”

“…where y j 2 R n=n 1 , j D 1; : : : ; m 1 . The products A 2 y j can then be computed recursively by the same method, yielding the following algorithm, very similar to the one given in [13].…”

TWO‐POINT RAY TRACING IN GENERAL 3D MEDIA1

“…Fast algorithms to solve such multidimensional tensor-product least-squares problems have been developed earlier. The speed-up over a conventional, direct approach, is achieved by reducing the multidimensional problem to a cascade of 1D ones (Pereyra and Scherer 1973;Grosse 1980). These algorithms work in any dimension, and they provide even more striking savings when fitting 3D material properties by tricubic tensor products of Bsplines, as is shown below.…”