62,848 research outputs found
Nonlinear tensor product approximation of functions
We are interested in approximation of a multivariate function
by linear combinations of products
of univariate functions , . In the case it is a
classical problem of bilinear approximation. In the case of approximation in
the space the bilinear approximation problem is closely related to the
problem of singular value decomposition (also called Schmidt expansion) of the
corresponding integral operator with the kernel . There are known
results on the rate of decay of errors of best bilinear approximation in
under different smoothness assumptions on . The problem of multilinear
approximation (nonlinear tensor product approximation) in the case is
more difficult and much less studied than the bilinear approximation problem.
We will present results on best multilinear approximation in under mixed
smoothness assumption on
A continuous analogue of the tensor-train decomposition
We develop new approximation algorithms and data structures for representing
and computing with multivariate functions using the functional tensor-train
(FT), a continuous extension of the tensor-train (TT) decomposition. The FT
represents functions using a tensor-train ansatz by replacing the
three-dimensional TT cores with univariate matrix-valued functions. The main
contribution of this paper is a framework to compute the FT that employs
adaptive approximations of univariate fibers, and that is not tied to any
tensorized discretization. The algorithm can be coupled with any univariate
linear or nonlinear approximation procedure. We demonstrate that this approach
can generate multivariate function approximations that are several orders of
magnitude more accurate, for the same cost, than those based on the
conventional approach of compressing the coefficient tensor of a tensor-product
basis. Our approach is in the spirit of other continuous computation packages
such as Chebfun, and yields an algorithm which requires the computation of
"continuous" matrix factorizations such as the LU and QR decompositions of
vector-valued functions. To support these developments, we describe continuous
versions of an approximate maximum-volume cross approximation algorithm and of
a rounding algorithm that re-approximates an FT by one of lower ranks. We
demonstrate that our technique improves accuracy and robustness, compared to TT
and quantics-TT approaches with fixed parameterizations, of high-dimensional
integration, differentiation, and approximation of functions with local
features such as discontinuities and other nonlinearities
Bivariate Segment Approximation
In this note we state some problems on approximation by univariate splines with free knots, bivariate segment approximation and tensor product splines with variable knot lines. There is a vast literature on approximation and interpolation by univariate splines with fixed knots (see e.g. the books of de Boor [1], Braess [2], DeVore & Lorentz [4], Powell [20], Schumaker [21], Nürnberger [13] and the book of Chui [3] on multivariate splines). On the other hand, numerical examples show that in general, the error is much smaller if variable knots are used for the approximation of functions instead of fixed knots. This is true for univariate splines as well as for bivariate splines. But approximation by splines with free knots leads to rather difficult nonlinear problems.[...
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
- …