805 research outputs found
The stochastic counterpart of conservation laws with heterogeneous conductivity fields: application to deterministic problems and uncertainty quantification
Conservation laws in the form of elliptic and parabolic partial differential
equations (PDEs) are fundamental to the modeling of many problems such as heat
transfer and flow in porous media. Many of such PDEs are stochastic due to the
presence of uncertainty in the conductivity field. Based on the relation
between stochastic diffusion processes and PDEs, Monte Carlo (MC) methods are
available to solve these PDEs. These methods are especially relevant for cases
where we are interested in the solution in a small subset of the domain. The
existing MC methods based on the stochastic formulation require restrictively
small time steps for high variance conductivity fields. Moreover, in many
applications the conductivity is piecewise constant and the existing methods
are not readily applicable in these cases. Here we provide an algorithm to
solve one-dimensional elliptic problems that bypasses these two limitations.
The methodology is demonstrated using problems governed by deterministic and
stochastic PDEs. It is shown that the method provides an efficient alternative
to compute the statistical moments of the solution to a stochastic PDE at any
point in the domain. A variance reduction scheme is proposed for applying the
method for efficient mean calculations
Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs
This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on with , and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters . We
establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence of the random inputs, and prove convergence rates of best -term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best -term truncations can practically be computed, by greedy-type algorithms
as in [SG, Gi1], or by multilevel Monte-Carlo methods as in
[KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]
Novel ensemble algorithms for random two-domain parabolic problems
In this paper, three efficient ensemble algorithms are proposed for
fast-solving the random fluid-fluid interaction model. Such a model can be
simplified as coupling two heat equations with random diffusion coefficients
and a friction parameter due to its complexity and uncertainty. We utilize the
Monte Carlo method for the coupled model with random inputs to derive some
deterministic fluid-fluid numerical models and use the ensemble idea to realize
the fast computation of multiple problems. Our remarkable feature of these
algorithms is employing the same coefficient matrix for multiple linear
systems, significantly reducing the computational cost. By data-passing
partitioned techniques, we can decouple the numerical models into two smaller
sub-domain problems and achieve parallel computation. Theoretically, we derive
that both algorithms are unconditionally stable and convergent. Finally,
numerical experiments are conducted not only to support the theoretical results
but also to validate the exclusive feature of the proposed algorithms
Higher-order Quasi-Monte Carlo Training of Deep Neural Networks
We present a novel algorithmic approach and an error analysis leveraging
Quasi-Monte Carlo points for training deep neural network (DNN) surrogates of
Data-to-Observable (DtO) maps in engineering design. Our analysis reveals
higher-order consistent, deterministic choices of training points in the input
data space for deep and shallow Neural Networks with holomorphic activation
functions such as tanh. These novel training points are proved to facilitate
higher-order decay (in terms of the number of training samples) of the
underlying generalization error, with consistency error bounds that are free
from the curse of dimensionality in the input data space, provided that DNN
weights in hidden layers satisfy certain summability conditions. We present
numerical experiments for DtO maps from elliptic and parabolic PDEs with
uncertain inputs that confirm the theoretical analysis
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