48,180 research outputs found
Symmetry groups, semidefinite programs, and sums of squares
We investigate the representation of symmetric polynomials as a sum of
squares. Since this task is solved using semidefinite programming tools we
explore the geometric, algebraic, and computational implications of the
presence of discrete symmetries in semidefinite programs. It is shown that
symmetry exploitation allows a significant reduction in both matrix size and
number of decision variables. This result is applied to semidefinite programs
arising from the computation of sum of squares decompositions for multivariate
polynomials. The results, reinterpreted from an invariant-theoretic viewpoint,
provide a novel representation of a class of nonnegative symmetric polynomials.
The main theorem states that an invariant sum of squares polynomial is a sum of
inner products of pairs of matrices, whose entries are invariant polynomials.
In these pairs, one of the matrices is computed based on the real irreducible
representations of the group, and the other is a sum of squares matrix. The
reduction techniques enable the numerical solution of large-scale instances,
otherwise computationally infeasible to solve.Comment: 38 pages, submitte
A micro/macro parallel-in-time (parareal) algorithm applied to a climate model with discontinuous non-monotone coefficients and oscillatory forcing
We present the application of a micro/macro parareal algorithm for a 1-D
energy balance climate model with discontinuous and non-monotone coefficients
and forcing terms. The micro/macro parareal method uses a coarse propagator,
based on a (macroscopic) 0-D approximation of the underlying (microscopic) 1-D
model. We compare the performance of the method using different versions of the
macro model, as well as different numerical schemes for the micro propagator,
namely an explicit Euler method with constant stepsize and an adaptive library
routine. We study convergence of the method and the theoretical gain in
computational time in a realization on parallel processors. We show that, in
this example and for all settings, the micro/macro parareal method converges in
fewer iterations than the number of used parareal subintervals, and that a
theoretical gain in performance of up to 10 is possible
Mathematics Is Biology's Next Microscope, Only Better; Biology Is Mathematics' Next Physics, Only Better
Joel Cohen offers a historical and prospective analysis of the relationship between mathematics and biolog
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