55,411 research outputs found
Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions: Error bounds and tractability
We study multivariate integration of functions that are invariant under
permutations (of subsets) of their arguments. We find an upper bound for the
th minimal worst case error and show that under certain conditions, it can
be bounded independent of the number of dimensions. In particular, we study the
application of unshifted and randomly shifted rank- lattice rules in such a
problem setting. We derive conditions under which multivariate integration is
polynomially or strongly polynomially tractable with the Monte Carlo rate of
convergence . Furthermore, we prove that those tractability
results can be achieved with shifted lattice rules and that the shifts are
indeed necessary. Finally, we show the existence of rank- lattice rules
whose worst case error on the permutation- and shift-invariant spaces converge
with (almost) optimal rate. That is, we derive error bounds of the form
for all , where denotes
the smoothness of the spaces.
Keywords: Numerical integration, Quadrature, Cubature, Quasi-Monte Carlo
methods, Rank-1 lattice rules.Comment: 26 pages; minor changes due to reviewer's comments; the final
publication is available at link.springer.co
A quasi-Monte Carlo Method for an Optimal Control Problem Under Uncertainty
We study an optimal control problem under uncertainty, where the target
function is the solution of an elliptic partial differential equation with
random coefficients, steered by a control function. The robust formulation of
the optimization problem is stated as a high-dimensional integration problem
over the stochastic variables. It is well known that carrying out a
high-dimensional numerical integration of this kind using a Monte Carlo method
has a notoriously slow convergence rate; meanwhile, a faster rate of
convergence can potentially be obtained by using sparse grid quadratures, but
these lead to discretized systems that are non-convex due to the involvement of
negative quadrature weights. In this paper, we analyze instead the application
of a quasi-Monte Carlo method, which retains the desirable convexity structure
of the system and has a faster convergence rate compared to ordinary Monte
Carlo methods. In particular, we show that under moderate assumptions on the
decay of the input random field, the error rate obtained by using a specially
designed, randomly shifted rank-1 lattice quadrature rule is essentially
inversely proportional to the number of quadrature nodes. The overall
discretization error of the problem, consisting of the dimension truncation
error, finite element discretization error and quasi-Monte Carlo quadrature
error, is derived in detail. We assess the theoretical findings in numerical
experiments
Simple Monte Carlo and the metropolis algorithm
We study the integration of functions with respect to an unknown
density. Information is available as oracle calls to the integrand and to the
non-normalized density function. We are interested in analyzing the
integration error of optimal algorithms (or the complexity of the problem)
with emphasis on the variability of the weight function. For a corresponding
large class of problem instances we show that the complexity grows linearly
in the variability, and the simple Monte Carlo method provides an almost
optimal algorithm. Under additional geometric restrictions (mainly
log-concavity) for the density functions, we establish that a suitable
adaptive local Metropolis algorithm is almost optimal and outperforms any
non-adaptive algorithm
BER Performance of IM/DD FSO System with OOK using APD Receiver
In this paper, the performance of intensity-modulated with direct detection (IM/DD) free space optical (FSO) system using the on-off keying (OOK) and avalanche photodiode (APD) receiver is observed. The gamma-gamma model is used to describe the effect of atmospheric turbulence since it provides good agreement in the wide range of atmospheric conditions. In addition, the same FSO system with equal gain combining applied at the reception is analyzed. After theoretical derivation of the expression for the bit error rate (BER), the numerical integration with previously specified relative calculation error is performed. Numerical results are presented and confirmed by Monte Carlo simulations. The effects of the FSO link and receiver parameters on the BER performance are discussed. The results illustrate that the optimal APD gain in the minimum BER sense depends considerably on the link distance, atmospheric turbulence strength and receiver temperature. In addition, the value of this optimal gain is slightly different in the case of spatial diversity application compared with single channel reception
Multilevel Double Loop Monte Carlo and Stochastic Collocation Methods with Importance Sampling for Bayesian Optimal Experimental Design
An optimal experimental set-up maximizes the value of data for statistical
inferences and predictions. The efficiency of strategies for finding optimal
experimental set-ups is particularly important for experiments that are
time-consuming or expensive to perform. For instance, in the situation when the
experiments are modeled by Partial Differential Equations (PDEs), multilevel
methods have been proven to dramatically reduce the computational complexity of
their single-level counterparts when estimating expected values. For a setting
where PDEs can model experiments, we propose two multilevel methods for
estimating a popular design criterion known as the expected information gain in
simulation-based Bayesian optimal experimental design. The expected information
gain criterion is of a nested expectation form, and only a handful of
multilevel methods have been proposed for problems of such form. We propose a
Multilevel Double Loop Monte Carlo (MLDLMC), which is a multilevel strategy
with Double Loop Monte Carlo (DLMC), and a Multilevel Double Loop Stochastic
Collocation (MLDLSC), which performs a high-dimensional integration by
deterministic quadrature on sparse grids. For both methods, the Laplace
approximation is used for importance sampling that significantly reduces the
computational work of estimating inner expectations. The optimal values of the
method parameters are determined by minimizing the average computational work,
subject to satisfying the desired error tolerance. The computational
efficiencies of the methods are demonstrated by estimating the expected
information gain for Bayesian inference of the fiber orientation in composite
laminate materials from an electrical impedance tomography experiment. MLDLSC
performs better than MLDLMC when the regularity of the quantity of interest,
with respect to the additive noise and the unknown parameters, can be
exploited
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