120 research outputs found
Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations
We consider stochastic programs where the distribution of the uncertain
parameters is only observable through a finite training dataset. Using the
Wasserstein metric, we construct a ball in the space of (multivariate and
non-discrete) probability distributions centered at the uniform distribution on
the training samples, and we seek decisions that perform best in view of the
worst-case distribution within this Wasserstein ball. The state-of-the-art
methods for solving the resulting distributionally robust optimization problems
rely on global optimization techniques, which quickly become computationally
excruciating. In this paper we demonstrate that, under mild assumptions, the
distributionally robust optimization problems over Wasserstein balls can in
fact be reformulated as finite convex programs---in many interesting cases even
as tractable linear programs. Leveraging recent measure concentration results,
we also show that their solutions enjoy powerful finite-sample performance
guarantees. Our theoretical results are exemplified in mean-risk portfolio
optimization as well as uncertainty quantification.Comment: 42 pages, 10 figure
Multilevel Accelerated Quadrature for PDEs with Log-Normally Distributed Diffusion Coefficient
This article is dedicated to multilevel quadrature methods for the rapid solution of stochastic partial differential equations with a log-normally distributed diffusion coefficient. The key idea of such approaches is a sparse-grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's moments with focus on the mean and the variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi-Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments. This article is dedicated to multilevel quadrature methods forthe rapid solution of stochastic partial differential equationswith a log-normally distributed diffusion coefficient. The key ideaof such approaches is a sparse-grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's moments with focus on the mean and the variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi-Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments
Parameter Synthesis for Markov Models
Markov chain analysis is a key technique in reliability engineering. A
practical obstacle is that all probabilities in Markov models need to be known.
However, system quantities such as failure rates or packet loss ratios, etc.
are often not---or only partially---known. This motivates considering
parametric models with transitions labeled with functions over parameters.
Whereas traditional Markov chain analysis evaluates a reliability metric for a
single, fixed set of probabilities, analysing parametric Markov models focuses
on synthesising parameter values that establish a given reliability or
performance specification . Examples are: what component failure rates
ensure the probability of a system breakdown to be below 0.00000001?, or which
failure rates maximise reliability? This paper presents various analysis
algorithms for parametric Markov chains and Markov decision processes. We focus
on three problems: (a) do all parameter values within a given region satisfy
?, (b) which regions satisfy and which ones do not?, and (c)
an approximate version of (b) focusing on covering a large fraction of all
possible parameter values. We give a detailed account of the various
algorithms, present a software tool realising these techniques, and report on
an extensive experimental evaluation on benchmarks that span a wide range of
applications.Comment: 38 page
Multilevel accelerated quadrature for PDEs with log-normal distributed random coefficient
This article is dedicated to multilevel quadrature methods for the rapid solution of stochastic partial differential equations with a log-normal distributed diffusion coefficient. The key idea of these approaches is a sparse grid approximation of the occurring product space between the stochastic and the spatial variable. We develop the mathematical theory and present error estimates for the computation of the solution's statistical moments with focus on the mean and variance. Especially, the present framework covers the multilevel Monte Carlo method and the multilevel quasi Monte Carlo method as special cases. The theoretical findings are supplemented by numerical experiments
Quadrature methods for elliptic PDEs with random diffusion
In this thesis, we consider elliptic boundary value problems with
random diffusion coefficients. Such equations arise in many
engineering applications, for example, in the modelling of
subsurface flows in porous media, such as rocks.
To describe the subsurface flow, it is convenient to use
Darcy's law. The key ingredient in this approach is the hydraulic
conductivity. In most cases, this hydraulic conductivity is approximated
from a discrete number of measurements and, hence, it is common to
endow it with uncertainty, i.e. model it as a random field.
This random field is usually characterized
by its mean field and its covariance function.
Naturally, this randomness propagates through the model which
yields that the solution is a random field as well.
The thesis on hand is concerned with the effective computation
of statistical quantities of this random solution, like the expectation,
the variance, and higher order moments.
In order to compute these quantities, a suitable representation of the
random field which describes the hydraulic conductivity needs to be
computed from the mean field and the covariance function.
This is realized by the Karhunen-Loeve expansion which
separates the spatial variable and the stochastic variable. In general, the
number of random variables and spatial functions used in this expansion
is infinite and needs to be truncated appropriately.
The number of random variables which are required depends on the
smoothness of the covariance function and grows with the desired accuracy.
Since the solution also depends on these random variables, each moment
of the solution appears as a high-dimensional Bochner integral over the
image space of the collection of random variables. This integral has to be
approximated by quadrature methods where each function evaluation
corresponds to a PDE solve.
In this thesis, the Monte Carlo, quasi-Monte Carlo, Gaussian tensor product, and
Gaussian sparse grid quadrature is analyzed to deal with this high-dimensional
integration problem.
In the first part, the necessary regularity requirements of the integrand and
its powers are provided in order to guarantee convergence of the different
methods.
It turns out that all the powers of the solution depend, like the solution itself,
anisotropic on the different random variables which means in this case that
there is a decaying dependence on the different random variables.
This dependence can be used to overcome, at least up to a certain extent, the
curse of dimensionality of the quadrature problem.
This is reflected in the proofs of the convergence rates of the different
quadrature methods which can be found in the second part of this thesis.
The last part is concerned with multilevel quadrature approaches to keep
the computational cost low. As mentioned earlier, we need to solve a partial
differential equation for each quadrature point.
The common approach is to apply a finite element approximation scheme on
a refinement level which corresponds to the desired accuracy.
Hence, the total computational cost is given by the product of the number
of quadrature points times the cost to compute one finite element solution
on a relatively high refinement level.
The multilevel idea is to use a telescoping sum decomposition of the quantity
of interest with respect to different spatial refinement levels and use
quadrature methods with different accuracies for each summand.
Roughly speaking, the multilevel approach spends a lot of quadrature points
on a low spatial refinement and only a few on the higher refinement levels.
This reduces the computational complexity but requires further regularity
on the integrand which is proven for the considered problems in this thesis
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