3,808 research outputs found
Random Bit Multilevel Algorithms for Stochastic Differential Equations
We study the approximation of expectations \E(f(X)) for solutions of
SDEs and functionals by means of restricted
Monte Carlo algorithms that may only use random bits instead of random numbers.
We consider the worst case setting for functionals from the Lipschitz class
w.r.t.\ the supremum norm. We construct a random bit multilevel Euler algorithm
and establish upper bounds for its error and cost. Furthermore, we derive
matching lower bounds, up to a logarithmic factor, that are valid for all
random bit Monte Carlo algorithms, and we show that, for the given quadrature
problem, random bit Monte Carlo algorithms are at least almost as powerful as
general randomized algorithms
Central limit theorems for multilevel Monte Carlo methods
In this work, we show that uniform integrability is not a necessary condition
for central limit theorems (CLT) to hold for normalized multilevel Monte Carlo
(MLMC) estimators and we provide near optimal weaker conditions under which the
CLT is achieved. In particular, if the variance decay rate dominates the
computational cost rate (i.e., ), we prove that the CLT applies
to the standard (variance minimizing) MLMC estimator.
For other settings where the CLT may not apply to the standard MLMC
estimator, we propose an alternative estimator, called the mass-shifted MLMC
estimator, to which the CLT always applies.
This comes at a small efficiency loss: the computational cost of achieving
mean square approximation error is at worst a factor
higher with the mass-shifted estimator than
with the standard one
Optimization of mesh hierarchies in Multilevel Monte Carlo samplers
We perform a general optimization of the parameters in the Multilevel Monte
Carlo (MLMC) discretization hierarchy based on uniform discretization methods
with general approximation orders and computational costs. We optimize
hierarchies with geometric and non-geometric sequences of mesh sizes and show
that geometric hierarchies, when optimized, are nearly optimal and have the
same asymptotic computational complexity as non-geometric optimal hierarchies.
We discuss how enforcing constraints on parameters of MLMC hierarchies affects
the optimality of these hierarchies. These constraints include an upper and a
lower bound on the mesh size or enforcing that the number of samples and the
number of discretization elements are integers. We also discuss the optimal
tolerance splitting between the bias and the statistical error contributions
and its asymptotic behavior. To provide numerical grounds for our theoretical
results, we apply these optimized hierarchies together with the Continuation
MLMC Algorithm. The first example considers a three-dimensional elliptic
partial differential equation with random inputs. Its space discretization is
based on continuous piecewise trilinear finite elements and the corresponding
linear system is solved by either a direct or an iterative solver. The second
example considers a one-dimensional It\^o stochastic differential equation
discretized by a Milstein scheme
Random Bit Quadrature and Approximation of Distributions on Hilbert Spaces
We study the approximation of expectations \E(f(X)) for Gaussian random
elements with values in a separable Hilbert space and Lipschitz
continuous functionals . We consider restricted Monte Carlo
algorithms, which may only use random bits instead of random numbers. We
determine the asymptotics (in some cases sharp up to multiplicative constants,
in the other cases sharp up to logarithmic factors) of the corresponding -th
minimal error in terms of the decay of the eigenvalues of the covariance
operator of . It turns out that, within the margins from above, restricted
Monte Carlo algorithms are not inferior to arbitrary Monte Carlo algorithms,
and suitable random bit multilevel algorithms are optimal. The analysis of this
problem leads to a variant of the quantization problem, namely, the optimal
approximation of probability measures on by uniform distributions supported
by a given, finite number of points. We determine the asymptotics (up to
multiplicative constants) of the error of the best approximation for the
one-dimensional standard normal distribution, for Gaussian measures as above,
and for scalar autonomous SDEs
Multilevel Monte Carlo simulation for Levy processes based on the Wiener-Hopf factorisation
In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was
introduced for a large family of Levy processes that is based on the
Wiener-Hopf decomposition. We pursue this idea further by combining their
technique with the recently introduced multilevel Monte Carlo methodology.
Moreover, we provide here for the first time a theoretical analysis of the new
Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its
multilevel variant for computing expectations of functions depending on the
historical trajectory of a Levy process. We derive rates of convergence for
both methods and show that they are uniform with respect to the "jump activity"
(e.g. characterised by the Blumenthal-Getoor index). We also present a modified
version of the algorithm in Kuznetsov et al. (2011) which combined with the
multilevel methodology obtains the optimal rate of convergence for general Levy
processes and Lipschitz functionals. This final result is only a theoretical
one at present, since it requires independent sampling from a triple of
distributions which is currently only possible for a limited number of
processes
Estimating expected first passage times using multilevel Monte Carlo algorithm
In this paper we devise a method of numerically estimating the expected first passage times of stochastic processes. We use Monte Carlo path simulations with Milstein discretisation scheme to approximate the solutions of scalar stochastic differential equations. To further reduce the variance of the estimated expected stopping time and improve computational efficiency, we use the multi-level Monte Carlo algorithm, recently developed by Giles (2008a), and other variance-reduction techniques. Our numerical results show significant improvements over conventional Monte Carlo techniques
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