38,537 research outputs found
MDFEM: Multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficients using higher-order QMC and FEM
We introduce the multivariate decomposition finite element method for
elliptic PDEs with lognormal diffusion coefficient where is a
Gaussian random field defined by an infinite series expansion
with and a given sequence of functions . We
use the MDFEM to approximate the expected value of a linear functional of the
solution of the PDE which is an infinite-dimensional integral over the
parameter space. The proposed algorithm uses the multivariate decomposition
method to compute the infinite-dimensional integral by a decomposition into
finite-dimensional integrals, which we resolve using quasi-Monte Carlo methods,
and for which we use the finite element method to solve different instances of
the PDE.
We develop higher-order quasi-Monte Carlo rules for integration over the
finite-dimensional Euclidean space with respect to the Gaussian distribution by
use of a truncation strategy. By linear transformations of interlaced
polynomial lattice rules from the unit cube to a multivariate box of the
Euclidean space we achieve higher-order convergence rates for functions
belonging to a class of anchored Gaussian Sobolev spaces while taking into
account the truncation error.
Under appropriate conditions, the MDFEM achieves higher-order convergence
rates in term of error versus cost, i.e., to achieve an accuracy of
the computational cost is where and
are respectively the cost of the quasi-Monte Carlo
cubature and the finite element approximations, with
for some and the physical dimension, and is a parameter representing the sparsity of .Comment: 48 page
Quasi-geometric integration of guiding-center orbits in piecewise linear toroidal fields
A numerical integration method for guiding-center orbits of charged particles
in toroidal fusion devices with three-dimensional field geometry is described.
Here, high order interpolation of electromagnetic fields in space is replaced
by a special linear interpolation, leading to locally linear Hamiltonian
equations of motion with piecewise constant coefficients. This approach reduces
computational effort and noise sensitivity while the conservation of total
energy, magnetic moment and phase space volume is retained. The underlying
formulation treats motion in piecewise linear fields exactly and thus preserves
the non-canonical symplectic form. The algorithm itself is only quasi-geometric
due to a series expansion in the orbit parameter. For practical purposes an
expansion to the fourth order retains geometric properties down to computer
accuracy in typical examples. When applied to collisionless guiding-center
orbits in an axisymmetric tokamak and a realistic three-dimensional stellarator
configuration, the method demonstrates stable long-term orbit dynamics
conserving invariants. In Monte Carlo evaluation of transport coefficients, the
computational efficiency of quasi-geometric integration is an order of
magnitude higher than with a standard fourth order Runge-Kutta integrator.Comment: 38 pages, 11 figure
Application of quasi-Monte Carlo methods to PDEs with random coefficients -- an overview and tutorial
This article provides a high-level overview of some recent works on the
application of quasi-Monte Carlo (QMC) methods to PDEs with random
coefficients. It is based on an in-depth survey of a similar title by the same
authors, with an accompanying software package which is also briefly discussed
here. Embedded in this article is a step-by-step tutorial of the required
analysis for the setting known as the uniform case with first order QMC rules.
The aim of this article is to provide an easy entry point for QMC experts
wanting to start research in this direction and for PDE analysts and
practitioners wanting to tap into contemporary QMC theory and methods.Comment: arXiv admin note: text overlap with arXiv:1606.0661
Pricing and Risk Management with High-Dimensional Quasi Monte Carlo and Global Sensitivity Analysis
We review and apply Quasi Monte Carlo (QMC) and Global Sensitivity Analysis
(GSA) techniques to pricing and risk management (greeks) of representative
financial instruments of increasing complexity. We compare QMC vs standard
Monte Carlo (MC) results in great detail, using high-dimensional Sobol' low
discrepancy sequences, different discretization methods, and specific analyses
of convergence, performance, speed up, stability, and error optimization for
finite differences greeks. We find that our QMC outperforms MC in most cases,
including the highest-dimensional simulations and greeks calculations, showing
faster and more stable convergence to exact or almost exact results. Using GSA,
we are able to fully explain our findings in terms of reduced effective
dimension of our QMC simulation, allowed in most cases, but not always, by
Brownian bridge discretization. We conclude that, beyond pricing, QMC is a very
promising technique also for computing risk figures, greeks in particular, as
it allows to reduce the computational effort of high-dimensional Monte Carlo
simulations typical of modern risk management.Comment: 43 pages, 21 figures, 6 table
Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules
We show how to obtain a fast component-by-component construction algorithm
for higher order polynomial lattice rules. Such rules are useful for
multivariate quadrature of high-dimensional smooth functions over the unit cube
as they achieve the near optimal order of convergence. The main problem
addressed in this paper is to find an efficient way of computing the worst-case
error. A general algorithm is presented and explicit expressions for base~2 are
given. To obtain an efficient component-by-component construction algorithm we
exploit the structure of the underlying cyclic group.
We compare our new higher order multivariate quadrature rules to existing
quadrature rules based on higher order digital nets by computing their
worst-case error. These numerical results show that the higher order polynomial
lattice rules improve upon the known constructions of quasi-Monte Carlo rules
based on higher order digital nets
Estimation with Numerical Integration on Sparse Grids
For the estimation of many econometric models, integrals without analytical solutions have to be evaluated. Examples include limited dependent variables and nonlinear panel data models. In the case of one-dimensional integrals, Gaussian quadrature is known to work efficiently for a large class of problems. In higher dimensions, similar approaches discussed in the literature are either very specific and hard to implement or suffer from exponentially rising computational costs in the number of dimensions - a problem known as the "curse of dimensionality" of numerical integration. We propose a strategy that shares the advantages of Gaussian quadrature methods, is very general and easily implemented, and does not suffer from the curse of dimensionality. Monte Carlo experiments for the random parameters logit model indicate the superior performance of the proposed method over simulation techniques
Evaluation of advanced optimisation methods for estimating Mixed Logit models
The performances of different simulation-based estimation techniques for mixed logit modeling are evaluated. A quasi-Monte Carlo method (modified Latin hypercube sampling) is compared with a Monte Carlo algorithm with dynamic accuracy. The classic Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization algorithm line-search approach and trust region methods, which have proved to be extremely powerful in nonlinear programming, are also compared. Numerical tests are performed on two real data sets: stated preference data for parking type collected in the United Kingdom, and revealed preference data for mode choice collected as part of a German travel diary survey. Several criteria are used to evaluate the approximation quality of the log likelihood function and the accuracy of the results and the associated estimation runtime. Results suggest that the trust region approach outperforms the BFGS approach and that Monte Carlo methods remain competitive with quasi-Monte Carlo methods in high-dimensional problems, especially when an adaptive optimization algorithm is used
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