37,325 research outputs found
Discontinuous collocation methods and gravitational self-force applications
Numerical simulations of extereme mass ratio inspirals, the mostimportant
sources for the LISA detector, face several computational challenges. We
present a new approach to evolving partial differential equations occurring in
black hole perturbation theory and calculations of the self-force acting on
point particles orbiting supermassive black holes. Such equations are
distributionally sourced, and standard numerical methods, such as
finite-difference or spectral methods, face difficulties associated with
approximating discontinuous functions. However, in the self-force problem we
typically have access to full a-priori information about the local structure of
the discontinuity at the particle. Using this information, we show that
high-order accuracy can be recovered by adding to the Lagrange interpolation
formula a linear combination of certain jump amplitudes. We construct
discontinuous spatial and temporal discretizations by operating on the
corrected Lagrange formula. In a method-of-lines framework, this provides a
simple and efficient method of solving time-dependent partial differential
equations, without loss of accuracy near moving singularities or
discontinuities. This method is well-suited for the problem of time-domain
reconstruction of the metric perturbation via the Teukolsky or
Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and
GPU architectures are discussed.Comment: 29 pages, 5 figure
Computing derivative-based global sensitivity measures using polynomial chaos expansions
In the field of computer experiments sensitivity analysis aims at quantifying
the relative importance of each input parameter (or combinations thereof) of a
computational model with respect to the model output uncertainty. Variance
decomposition methods leading to the well-known Sobol' indices are recognized
as accurate techniques, at a rather high computational cost though. The use of
polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to
alleviate the computational burden though. However, when dealing with large
dimensional input vectors, it is good practice to first use screening methods
in order to discard unimportant variables. The {\em derivative-based global
sensitivity measures} (DGSM) have been developed recently in this respect. In
this paper we show how polynomial chaos expansions may be used to compute
analytically DGSMs as a mere post-processing. This requires the analytical
derivation of derivatives of the orthonormal polynomials which enter PC
expansions. The efficiency of the approach is illustrated on two well-known
benchmark problems in sensitivity analysis
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