72,892 research outputs found
A Stochastic Gradient Method with Mesh Refinement for PDE Constrained Optimization under Uncertainty
Models incorporating uncertain inputs, such as random forces or material
parameters, have been of increasing interest in PDE-constrained optimization.
In this paper, we focus on the efficient numerical minimization of a convex and
smooth tracking-type functional subject to a linear partial differential
equation with random coefficients and box constraints. The approach we take is
based on stochastic approximation where, in place of a true gradient, a
stochastic gradient is chosen using one sample from a known probability
distribution. Feasibility is maintained by performing a projection at each
iteration. In the application of this method to PDE-constrained optimization
under uncertainty, new challenges arise. We observe the discretization error
made by approximating the stochastic gradient using finite elements. Analyzing
the interplay between PDE discretization and stochastic error, we develop a
mesh refinement strategy coupled with decreasing step sizes. Additionally, we
develop a mesh refinement strategy for the modified algorithm using iterate
averaging and larger step sizes. The effectiveness of the approach is
demonstrated numerically for different random field choices
Adaptive finite element method assisted by stochastic simulation of chemical systems
Stochastic models of chemical systems are often analysed by solving the corresponding\ud
Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud
distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density
A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws
In this article we consider one-dimensional random systems of hyperbolic
conservation laws. We first establish existence and uniqueness of random
entropy admissible solutions for initial value problems of conservation laws
which involve random initial data and random flux functions. Based on these
results we present an a posteriori error analysis for a numerical approximation
of the random entropy admissible solution. For the stochastic discretization,
we consider a non-intrusive approach, the Stochastic Collocation method. The
spatio-temporal discretization relies on the Runge--Kutta Discontinuous
Galerkin method. We derive the a posteriori estimator using continuous
reconstructions of the discrete solution. Combined with the relative entropy
stability framework this yields computable error bounds for the entire
space-stochastic discretization error. The estimator admits a splitting into a
stochastic and a deterministic (space-time) part, allowing for a novel
residual-based space-stochastic adaptive mesh refinement algorithm. We conclude
with various numerical examples investigating the scaling properties of the
residuals and illustrating the efficiency of the proposed adaptive algorithm
Stochastic calculus over symmetric Markov processes without time reversal
We refine stochastic calculus for symmetric Markov processes without using
time reverse operators. Under some conditions on the jump functions of locally
square integrable martingale additive functionals, we extend Nakao's
divergence-like continuous additive functional of zero energy and the
stochastic integral with respect to it under the law for quasi-everywhere
starting points, which are refinements of the previous results under the law
for almost everywhere starting points. This refinement of stochastic calculus
enables us to establish a generalized Fukushima decomposition for a certain
class of functions locally in the domain of Dirichlet form and a generalized
It\^{o} formula. (With Errata.)Comment: Published in at http://dx.doi.org/10.1214/09-AOP516 and Errata at
http://dx.doi.org/10.1214/11-AOP700 the Annals of Probability
(http://www.imstat.org/aop/) by the Institute of Mathematical Statistics
(http://www.imstat.org
Logic-based machine learning using a bounded hypothesis space: the lattice structure, refinement operators and a genetic algorithm approach
Rich representation inherited from computational logic makes logic-based machine learning a competent method for application domains involving relational background knowledge and structured data. There is however a trade-off between the expressive power of the representation and the computational costs. Inductive Logic Programming (ILP) systems employ different kind of biases and heuristics to cope with the complexity of the search, which otherwise is intractable. Searching the hypothesis space bounded below by a bottom clause is the basis of several state-of-the-art ILP systems (e.g. Progol and Aleph). However, the structure of the search space and the properties of the refinement operators for theses systems have not been previously characterised. The contributions of this thesis can be summarised as follows: (i) characterising the properties, structure and morphisms of bounded subsumption lattice (ii) analysis of bounded refinement operators and stochastic refinement and (iii) implementation and empirical evaluation of stochastic search algorithms and in particular a Genetic Algorithm (GA) approach for bounded subsumption. In this thesis we introduce the concept of bounded subsumption and study the lattice and cover structure of bounded subsumption. We show the morphisms between the lattice of bounded subsumption, an atomic lattice and the lattice of partitions. We also show that ideal refinement operators exist for bounded subsumption and that, by contrast with general subsumption, efficient least and minimal generalisation operators can be designed for bounded subsumption. In this thesis we also show how refinement operators can be adapted for a stochastic search and give an analysis of refinement operators within the framework of stochastic refinement search. We also discuss genetic search for learning first-order clauses and describe a framework for genetic and stochastic refinement search for bounded subsumption. on. Finally, ILP algorithms and implementations which are based on this framework are described and evaluated.Open Acces
Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems
Stochastic physical problems governed by nonlinear conservation laws are
challenging due to solution discontinuities in stochastic and physical space.
In this paper, we present a level set method to track discontinuities in
stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed
function that vanishes at discontinuities, the iso-zero of the level set
problem coincide with the discontinuities of the conservation law. The level
set problem is solved on a sequence of successively finer grids in stochastic
space. The method is adaptive in the sense that costly evaluations of the
conservation law of interest are only performed in the vicinity of the
discontinuities during the refinement stage. In regions of stochastic space
where the solution is smooth, a surrogate method replaces expensive evaluations
of the conservation law. The proposed method is tested in conjunction with
different sets of localized orthogonal basis functions on simplex elements, as
well as frames based on piecewise polynomials conforming to the level set
function. The performance of the proposed method is compared to existing
adaptive multi-element generalized polynomial chaos methods
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