2,994 research outputs found
Parallel sparse interpolation using small primes
To interpolate a supersparse polynomial with integer coefficients, two
alternative approaches are the Prony-based "big prime" technique, which acts
over a single large finite field, or the more recently-proposed "small primes"
technique, which reduces the unknown sparse polynomial to many low-degree dense
polynomials. While the latter technique has not yet reached the same
theoretical efficiency as Prony-based methods, it has an obvious potential for
parallelization. We present a heuristic "small primes" interpolation algorithm
and report on a low-level C implementation using FLINT and MPI.Comment: Accepted to PASCO 201
Multivariate sparse interpolation using randomized Kronecker substitutions
We present new techniques for reducing a multivariate sparse polynomial to a
univariate polynomial. The reduction works similarly to the classical and
widely-used Kronecker substitution, except that we choose the degrees randomly
based on the number of nonzero terms in the multivariate polynomial, that is,
its sparsity. The resulting univariate polynomial often has a significantly
lower degree than the Kronecker substitution polynomial, at the expense of a
small number of term collisions. As an application, we give a new algorithm for
multivariate interpolation which uses these new techniques along with any
existing univariate interpolation algorithm.Comment: 21 pages, 2 tables, 1 procedure. Accepted to ISSAC 201
To be or not to be intrusive? The solution of parametric and stochastic equations - the "plain vanilla" Galerkin case
In parametric equations - stochastic equations are a special case - one may
want to approximate the solution such that it is easy to evaluate its
dependence of the parameters. Interpolation in the parameters is an obvious
possibility, in this context often labeled as a collocation method. In the
frequent situation where one has a "solver" for the equation for a given
parameter value - this may be a software component or a program - it is evident
that this can independently solve for the parameter values to be interpolated.
Such uncoupled methods which allow the use of the original solver are classed
as "non-intrusive". By extension, all other methods which produce some kind of
coupled system are often - in our view prematurely - classed as "intrusive". We
show for simple Galerkin formulations of the parametric problem - which
generally produce coupled systems - how one may compute the approximation in a
non-intusive way
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