24 research outputs found
Least-squares methods for nonnegative matrix factorization over rational functions
Nonnegative Matrix Factorization (NMF) models are widely used to recover
linearly mixed nonnegative data. When the data is made of samplings of
continuous signals, the factors in NMF can be constrained to be samples of
nonnegative rational functions, which allow fairly general models; this is
referred to as NMF using rational functions (R-NMF). We first show that, under
mild assumptions, R-NMF has an essentially unique factorization unlike NMF,
which is crucial in applications where ground-truth factors need to be
recovered such as blind source separation problems. Then we present different
approaches to solve R-NMF: the R-HANLS, R-ANLS and R-NLS methods. From our
tests, no method significantly outperforms the others, and a trade-off should
be done between time and accuracy. Indeed, R-HANLS is fast and accurate for
large problems, while R-ANLS is more accurate, but also more resources
demanding, both in time and memory. R-NLS is very accurate but only for small
problems. Moreover, we show that R-NMF outperforms NMF in various tasks
including the recovery of semi-synthetic continuous signals, and a
classification problem of real hyperspectral signals.Comment: 13 page
Conversions between barycentric, RKFUN, and Newton representations of rational interpolants
We derive explicit formulas for converting between rational interpolants in
barycentric, rational Krylov (RKFUN), and Newton form. We show applications of
these conversions when working with rational approximants produced by the AAA
algorithm [Y. Nakatsukasa, O. S\`ete, L. N. Trefethen, arXiv preprint
1612.00337, 2016] within the Rational Krylov Toolbox and for the solution of
nonlinear eigenvalue problems
Rational Krylov methods for functions of matrices with applications to fractional partial differential equations
In this paper, we propose a new choice of poles to define reliable rational
Krylov methods. These methods are used for approximating function of positive
definite matrices. In particular, the fractional power and the fractional
resolvent are considered because of their importance in the numerical solution
of fractional partial differential equations. The results of the numerical
experiments we have carried out on some fractional models confirm that the
proposed approach is promising
Adaptive rational Krylov methods for exponential Runge--Kutta integrators
We consider the solution of large stiff systems of ordinary differential
equations with explicit exponential Runge--Kutta integrators. These problems
arise from semi-discretized semi-linear parabolic partial differential
equations on continuous domains or on inherently discrete graph domains. A
series of results reduces the requirement of computing linear combinations of
-functions in exponential integrators to the approximation of the
action of a smaller number of matrix exponentials on certain vectors.
State-of-the-art computational methods use polynomial Krylov subspaces of
adaptive size for this task. They have the drawback that the required Krylov
subspace iteration numbers to obtain a desired tolerance increase drastically
with the spectral radius of the discrete linear differential operator, e.g.,
the problem size. We present an approach that leverages rational Krylov
subspace methods promising superior approximation qualities. We prove a novel
a-posteriori error estimate of rational Krylov approximations to the action of
the matrix exponential on vectors for single time points, which allows for an
adaptive approach similar to existing polynomial Krylov techniques. We discuss
pole selection and the efficient solution of the arising sequences of shifted
linear systems by direct and preconditioned iterative solvers. Numerical
experiments show that our method outperforms the state of the art for
sufficiently large spectral radii of the discrete linear differential
operators. The key to this are approximately constant rational Krylov iteration
numbers, which enable a near-linear scaling of the runtime with respect to the
problem size
Randomized sketching of nonlinear eigenvalue problems
Rational approximation is a powerful tool to obtain accurate surrogates for
nonlinear functions that are easy to evaluate and linearize. The interpolatory
adaptive Antoulas--Anderson (AAA) method is one approach to construct such
approximants numerically. For large-scale vector- and matrix-valued functions,
however, the direct application of the set-valued variant of AAA becomes
inefficient. We propose and analyze a new sketching approach for such functions
called sketchAAA that, with high probability, leads to much better approximants
than previously suggested approaches while retaining efficiency. The sketching
approach works in a black-box fashion where only evaluations of the nonlinear
function at sampling points are needed. Numerical tests with nonlinear
eigenvalue problems illustrate the efficacy of our approach, with speedups
above 200 for sampling large-scale black-box functions without sacrificing on
accuracy.Comment: 15 page
Rational minimax approximation via adaptive barycentric representations
Computing rational minimax approximations can be very challenging when there
are singularities on or near the interval of approximation - precisely the case
where rational functions outperform polynomials by a landslide. We show that
far more robust algorithms than previously available can be developed by making
use of rational barycentric representations whose support points are chosen in
an adaptive fashion as the approximant is computed. Three variants of this
barycentric strategy are all shown to be powerful: (1) a classical Remez
algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares,
and (3) a differential correction algorithm. Our preferred combination,
implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and
then switch to (1) for generically quadratic convergence. By such methods we
can calculate approximations up to type (80, 80) of on in
standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan,
and Carpenter required 200-digit extended precision.Comment: 29 pages, 11 figure
An algorithm for real and complex rational minimax approximation
Rational minimax approximation of real functions on real intervals is an
established topic, but when it comes to complex functions or domains, there
appear to be no algorithms currently in use. Such a method is introduced here,
the {\em AAA-Lawson algorithm,} available in Chebfun. The new algorithm solves
a wide range of problems on arbitrary domains in a fraction of a second of
laptop time by a procedure consisting of two steps. First, the standard AAA
algorithm is run to obtain a near-best approximation and a set of support
points for a barycentric representation of the rational approximant. Then a
"Lawson phase" of iteratively reweighted least-squares adjustment of the
barycentric coefficients is carried out to improve the approximation to
minimax
A convex dual programming for the rational minimax approximation and Lawson's iteration
Computing the discrete rational minimax approximation in the complex plane is
challenging. Apart from Ruttan's sufficient condition, there are few other
sufficient conditions for global optimality. The state-of-the-art rational
approximation algorithms, such as the adaptive Antoulas-Anderson (AAA),
AAA-Lawson, and the rational Krylov fitting (RKFIT) method, perform highly
efficiently, but the computed rational approximants may be near-best. In this
paper, we propose a convex programming approach, the solution of which is
guaranteed to be the rational minimax approximation under Ruttan's sufficient
condition. Furthermore, we present a new version of Lawson's iteration for
solving this convex programming problem. The computed solution can be easily
verified as the rational minimax approximant. Our numerical experiments
demonstrate that this updated version of Lawson's iteration generally converges
monotonically with respect to the objective function of the convex programming.
It is an effective competitive approach for the rational minimax problem,
compared to the highly efficient AAA, AAA-Lawson, and the stabilized
Sanathanan-Koerner iteration.Comment: 38 pages, 10 figure