23 research outputs found
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
Statistical and numerical methods for diffusion processes with multiple scales
In this thesis we address the problem of data-driven coarse-graining, i.e. the process of inferring simplified models, which describe the evolution of the essential characteristics of a complex system, from available data (e.g. experimental observation or simulation data). Specifically, we consider the case where the coarse-grained model can be formulated as a stochastic differential equation. The main part of this work is concerned with data-driven coarse-graining when the underlying complex system is characterised by processes occurring across two widely separated time scales. It is known that in this setting commonly used statistical techniques fail to obtain reasonable estimators for parameters in the coarse-grained model, due to the multiscale structure of the data. To enable reliable data-driven coarse-graining techniques for diffusion processes with multiple time scales, we develop a novel estimation procedure which decisively relies on combining techniques from mathematical statistics and numerical analysis. We demonstrate, both rigorously and by means of extensive simulations, that this methodology yields accurate approximations of coarse-grained SDE models. In the final part of this work, we then discuss a systematic framework to analyse and predict complex systems using observations. Specifically, we use data-driven techniques to identify simple, yet adequate, coarse-grained models, which in turn allow to study statistical properties that cannot be investigated directly from the time series. The value of this generic framework is exemplified through two seemingly unrelated data sets of real world phenomena.Open Acces
A new framework for extracting coarse-grained models from time series with multiscale structure
In many applications it is desirable to infer coarse-grained models from
observational data. The observed process often corresponds only to a few
selected degrees of freedom of a high-dimensional dynamical system with
multiple time scales. In this work we consider the inference problem of
identifying an appropriate coarse-grained model from a single time series of a
multiscale system. It is known that estimators such as the maximum likelihood
estimator or the quadratic variation of the path estimator can be strongly
biased in this setting. Here we present a novel parametric inference
methodology for problems with linear parameter dependency that does not suffer
from this drawback. Furthermore, we demonstrate through a wide spectrum of
examples that our methodology can be used to derive appropriate coarse-grained
models from time series of partial observations of a multiscale system in an
effective and systematic fashion
Sequential Estimation using Hierarchically Stratified Domains with Latin Hypercube Sampling
Quantifying the effect of uncertainties in systems where only point
evaluations in the stochastic domain but no regularity conditions are available
is limited to sampling-based techniques. This work presents an adaptive
sequential stratification estimation method that uses Latin Hypercube Sampling
within each stratum. The adaptation is achieved through a sequential
hierarchical refinement of the stratification, guided by previous estimators
using local (i.e., stratum-dependent) variability indicators based on
generalized polynomial chaos expansions and Sobol decompositions. For a given
total number of samples , the corresponding hierarchically constructed
sequence of Stratified Sampling estimators combined with Latin Hypercube
sampling is adequately averaged to provide a final estimator with reduced
variance. Numerical experiments illustrate the procedure's efficiency,
indicating that it can offer a variance decay proportional to in some
cases
A Fully Parallelized and Budgeted Multi-Level Monte Carlo Method and the Application to Acoustic Waves
We present a novel variant of the multi-level Monte Carlo method that
effectively utilizes a reserved computational budget on a high-performance
computing system to minimize the mean squared error. Our approach combines
concepts of the continuation multi-level Monte Carlo method with dynamic
programming techniques following Bellman's optimality principle, and a new
parallelization strategy based on a single distributed data structure.
Additionally, we establish a theoretical bound on the error reduction on a
parallel computing cluster and provide empirical evidence that the proposed
method adheres to this bound. We implement, test, and benchmark the approach on
computationally demanding problems, focusing on its application to acoustic
wave propagation in high-dimensional random media
Plateau Proposal Distributions for Adaptive Component-wise Multiple-Try Metropolis
Markov chain Monte Carlo (MCMC) methods are sampling methods that have become
a commonly used tool in statistics, for example to perform Monte Carlo
integration. As a consequence of the increase in computational power, many
variations of MCMC methods exist for generating samples from arbitrary,
possibly complex, target distributions. The performance of an MCMC method is
predominately governed by the choice of the so-called proposal distribution
used. In this paper, we introduce a new type of proposal distribution for the
use in MCMC methods that operates component-wise and with multiple trials per
iteration. Specifically, the novel class of proposal distributions, called
Plateau distributions, do not overlap, thus ensuring that the multiple trials
are drawn from different regions of the state space. Furthermore, the Plateau
proposal distributions allow for a bespoke adaptation procedure that lends
itself to a Markov chain with efficient problem dependent state space
exploration and improved burn-in properties. Simulation studies show that our
novel MCMC algorithm outperforms competitors when sampling from distributions
with a complex shape, highly correlated components or multiple modes.Comment: 24 pages, 12 figure
MATHICSE Technical Report : Multilevel Monte Carlo approximation of functions
Many applications across sciences and technologies require a careful quantification of non-deterministic effects to a system output, for example when evaluating the system's reliability or when gearing it towards more robust operation conditions. At the heart of these considerations lies an accurate characterization of uncertain system outputs. In this work we introduce and analyze novel multilevel Monte Carlo techniques for an efficient characterization of an uncertain system output's distribution. These techniques rely on accurately approximating general parametric expectations, i.e. expectations that depend on a parameter, uniformly on an interval. Applications of interest include, for example, the approximation of the characteristic function and of the cumulative distribution function of an uncertain system output. A further important consequence of the introduced approximation techniques for parametric expectations (i.e. for functions) is that they allow to construct multilevel Monte Carlo estimators for various robustness indicators, such as for a quantile (also known as value-at-risk) and for the conditional value-at-risk. These robustness indicators cannot be expressed as moments and are thus not easily accessible usually. In fact, here we provide a framework that allows to simultaneously estimate a cumulative distribution function, a quantile, and the associated conditional value-at-risk of an uncertain system output at the cost of a single multilevel Monte Carlo simulation, while each estimated quantity satisfies a prescribed tolerance goal