395 research outputs found
A Dynamically Adaptive Sparse Grid Method for Quasi-Optimal Interpolation of Multidimensional Analytic Functions
In this work we develop a dynamically adaptive sparse grids (SG) method for
quasi-optimal interpolation of multidimensional analytic functions defined over
a product of one dimensional bounded domains. The goal of such approach is to
construct an interpolant in space that corresponds to the "best -terms"
based on sharp a priori estimate of polynomial coefficients. In the past, SG
methods have been successful in achieving this, with a traditional construction
that relies on the solution to a Knapsack problem: only the most profitable
hierarchical surpluses are added to the SG. However, this approach requires
additional sharp estimates related to the size of the analytic region and the
norm of the interpolation operator, i.e., the Lebesgue constant. Instead, we
present an iterative SG procedure that adaptively refines an estimate of the
region and accounts for the effects of the Lebesgue constant. Our approach does
not require any a priori knowledge of the analyticity or operator norm, is
easily generalized to both affine and non-affine analytic functions, and can be
applied to sparse grids build from one dimensional rules with arbitrary growth
of the number of nodes. In several numerical examples, we utilize our
dynamically adaptive SG to interpolate quantities of interest related to the
solutions of parametrized elliptic and hyperbolic PDEs, and compare the
performance of our quasi-optimal interpolant to several alternative SG schemes
An implementation of radiative transfer in the cosmological simulation code GADGET
We present a novel numerical implementation of radiative transfer in the
cosmological smoothed particle hydrodynamics (SPH) simulation code {\small
GADGET}. It is based on a fast, robust and photon-conserving integration scheme
where the radiation transport problem is approximated in terms of moments of
the transfer equation and by using a variable Eddington tensor as a closure
relation, following the `OTVET'-suggestion of Gnedin & Abel. We derive a
suitable anisotropic diffusion operator for use in the SPH discretization of
the local photon transport, and we combine this with an implicit solver that
guarantees robustness and photon conservation. This entails a matrix inversion
problem of a huge, sparsely populated matrix that is distributed in memory in
our parallel code. We solve this task iteratively with a conjugate gradient
scheme. Finally, to model photon sink processes we consider ionisation and
recombination processes of hydrogen, which is represented with a chemical
network that is evolved with an implicit time integration scheme. We present
several tests of our implementation, including single and multiple sources in
static uniform density fields with and without temperature evolution, shadowing
by a dense clump, and multiple sources in a static cosmological density field.
All tests agree quite well with analytical computations or with predictions
from other radiative transfer codes, except for shadowing. However, unlike most
other radiative transfer codes presently in use for studying reionisation, our
new method can be used on-the-fly during dynamical cosmological simulation,
allowing simultaneous treatments of galaxy formation and the reionisation
process of the Universe.Comment: 21 pages, 17 figures, published in MNRA
A Discrete Adapted Hierarchical Basis Solver For Radial Basis Function Interpolation
In this paper we develop a discrete Hierarchical Basis (HB) to efficiently
solve the Radial Basis Function (RBF) interpolation problem with variable
polynomial order. The HB forms an orthogonal set and is adapted to the kernel
seed function and the placement of the interpolation nodes. Moreover, this
basis is orthogonal to a set of polynomials up to a given order defined on the
interpolating nodes. We are thus able to decouple the RBF interpolation problem
for any order of the polynomial interpolation and solve it in two steps: (1)
The polynomial orthogonal RBF interpolation problem is efficiently solved in
the transformed HB basis with a GMRES iteration and a diagonal, or block SSOR
preconditioner. (2) The residual is then projected onto an orthonormal
polynomial basis. We apply our approach on several test cases to study its
effectiveness, including an application to the Best Linear Unbiased Estimator
regression problem
Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws
We propose a predictor-corrector adaptive method for the study of hyperbolic
partial differential equations (PDEs) under uncertainty. Constructed around the
framework of stochastic finite volume (SFV) methods, our approach circumvents
sampling schemes or simulation ensembles while also preserving fundamental
properties, in particular hyperbolicity of the resulting systems and
conservation of the discrete solutions. Furthermore, we augment the existing
SFV theory with a priori convergence results for statistical quantities, in
particular push-forward densities, which we demonstrate through numerical
experiments. By linking refinement indicators to regions of the physical and
stochastic spaces, we drive anisotropic refinements of the discretizations,
introducing new degrees of freedom (DoFs) where deemed profitable. To
illustrate our proposed method, we consider a series of numerical examples for
non-linear hyperbolic PDEs based on Burgers' and Euler's equations
Lowest order Virtual Element approximation of magnetostatic problems
We give here a simplified presentation of the lowest order Serendipity
Virtual Element method, and show its use for the numerical solution of linear
magneto-static problems in three dimensions. The method can be applied to very
general decompositions of the computational domain (as is natural for Virtual
Element Methods) and uses as unknowns the (constant) tangential component of
the magnetic field on each edge, and the vertex values of the
Lagrange multiplier (used to enforce the solenoidality of the magnetic
induction ). In this respect the method can be seen
as the natural generalization of the lowest order Edge Finite Element Method
(the so-called "first kind N\'ed\'elec" elements) to polyhedra of almost
arbitrary shape, and as we show on some numerical examples it exhibits very
good accuracy (for being a lowest order element) and excellent robustness with
respect to distortions
Comparison of some Reduced Representation Approximations
In the field of numerical approximation, specialists considering highly
complex problems have recently proposed various ways to simplify their
underlying problems. In this field, depending on the problem they were tackling
and the community that are at work, different approaches have been developed
with some success and have even gained some maturity, the applications can now
be applied to information analysis or for numerical simulation of PDE's. At
this point, a crossed analysis and effort for understanding the similarities
and the differences between these approaches that found their starting points
in different backgrounds is of interest. It is the purpose of this paper to
contribute to this effort by comparing some constructive reduced
representations of complex functions. We present here in full details the
Adaptive Cross Approximation (ACA) and the Empirical Interpolation Method (EIM)
together with other approaches that enter in the same category
Efficient Methods for Multidimensional Global Polynomial Approximation with Applications to Random PDEs
In this work, we consider several ways to overcome the challenges associated with polynomial approximation and integration of smooth functions depending on a large number of inputs. We are motivated by the problem of forward uncertainty quantification (UQ), whereby inputs to mathematical models are considered as random variables. With limited resources, finding more efficient and accurate ways to approximate the multidimensional solution to the UQ problem is of crucial importance, due to the “curse of dimensionality” and the cost of solving the underlying deterministic problem.
The first way we overcome the complexity issue is by exploiting the structure of the approximation schemes used to solve the random partial differential equations (PDE), thereby significantly reducing the overall cost of the approximation. We do this first using multilevel approximations in the physical variables, and second by exploiting the hierarchy of nested sparse grids in the random parameter space. With these algorithmic advances, we provably decrease the complexity of collocation methods for solving random PDE problems.
The second major theme in this work is the choice of efficient points for multidimensional interpolation and interpolatory quadrature. A major consideration in interpolation in multiple dimensions is the balance between stability, i.e., the Lebesgue constant of the interpolant, and the granularity of the approximation, e.g., the ability to choose an arbitrary number of interpolation points or to adaptively refine the grid. For these reasons, the Leja points are a popular choice for approximation on both bounded and unbounded domains. Mirroring the best-known results for interpolation on compact domains, we show that Leja points, defined for weighted interpolation on R, have a Lebesgue constant which grows subexponentially in the number of interpolation nodes. Regarding multidimensional quadratures, we show how certain new rules, generated from conformal mappings of classical interpolatory rules, can be used to increase the efficiency in approximating multidimensional integrals. Specifically, we show that the convergence rate for the novel mapped sparse grid interpolatory quadratures is improved by a factor that is exponential in the dimension of the underlying integral
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