233 research outputs found
Sequential Design with Mutual Information for Computer Experiments (MICE): Emulation of a Tsunami Model
Computer simulators can be computationally intensive to run over a large
number of input values, as required for optimization and various uncertainty
quantification tasks. The standard paradigm for the design and analysis of
computer experiments is to employ Gaussian random fields to model computer
simulators. Gaussian process models are trained on input-output data obtained
from simulation runs at various input values. Following this approach, we
propose a sequential design algorithm, MICE (Mutual Information for Computer
Experiments), that adaptively selects the input values at which to run the
computer simulator, in order to maximize the expected information gain (mutual
information) over the input space. The superior computational efficiency of the
MICE algorithm compared to other algorithms is demonstrated by test functions,
and a tsunami simulator with overall gains of up to 20% in that case
A sparse-grid isogeometric solver
Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS
as a basis for the approximation of the solution of PDEs. In this work, we
investigate to which extent IGA solvers can benefit from the so-called
sparse-grids construction in its combination technique form, which was first
introduced in the early 90s in the context of the approximation of
high-dimensional PDEs. The tests that we report show that, in accordance to the
literature, a sparse-grid construction can indeed be useful if the solution of
the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the
case of non-smooth solutions when some a-priori knowledge on the location of
the singularities of the solution can be exploited to devise suitable
non-equispaced meshes. Finally, we remark that sparse grids can be seen as a
simple way to parallelize pre-existing serial IGA solvers in a straightforward
fashion, which can be beneficial in many practical situations.Comment: updated version after revie
IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains
This paper proposes an extension of the Multi-Index Stochastic Collocation
(MISC) method for forward uncertainty quantification (UQ) problems in
computational domains of shape other than a square or cube, by exploiting
isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC
algorithm is very natural since they are tensor-based PDE solvers, which are
precisely what is required by the MISC machinery. Moreover, the
combination-technique formulation of MISC allows the straight-forward reuse of
existing implementations of IGA solvers. We present numerical results to
showcase the effectiveness of the proposed approach.Comment: version 3, version after revisio
Multilevel Double Loop Monte Carlo and Stochastic Collocation Methods with Importance Sampling for Bayesian Optimal Experimental Design
An optimal experimental set-up maximizes the value of data for statistical
inferences and predictions. The efficiency of strategies for finding optimal
experimental set-ups is particularly important for experiments that are
time-consuming or expensive to perform. For instance, in the situation when the
experiments are modeled by Partial Differential Equations (PDEs), multilevel
methods have been proven to dramatically reduce the computational complexity of
their single-level counterparts when estimating expected values. For a setting
where PDEs can model experiments, we propose two multilevel methods for
estimating a popular design criterion known as the expected information gain in
simulation-based Bayesian optimal experimental design. The expected information
gain criterion is of a nested expectation form, and only a handful of
multilevel methods have been proposed for problems of such form. We propose a
Multilevel Double Loop Monte Carlo (MLDLMC), which is a multilevel strategy
with Double Loop Monte Carlo (DLMC), and a Multilevel Double Loop Stochastic
Collocation (MLDLSC), which performs a high-dimensional integration by
deterministic quadrature on sparse grids. For both methods, the Laplace
approximation is used for importance sampling that significantly reduces the
computational work of estimating inner expectations. The optimal values of the
method parameters are determined by minimizing the average computational work,
subject to satisfying the desired error tolerance. The computational
efficiencies of the methods are demonstrated by estimating the expected
information gain for Bayesian inference of the fiber orientation in composite
laminate materials from an electrical impedance tomography experiment. MLDLSC
performs better than MLDLMC when the regularity of the quantity of interest,
with respect to the additive noise and the unknown parameters, can be
exploited
Fast Bayesian experimental design: Laplace-based importance sampling for the expected information gain
In calculating expected information gain in optimal Bayesian experimental
design, the computation of the inner loop in the classical double-loop Monte
Carlo requires a large number of samples and suffers from underflow if the
number of samples is small. These drawbacks can be avoided by using an
importance sampling approach. We present a computationally efficient method for
optimal Bayesian experimental design that introduces importance sampling based
on the Laplace method to the inner loop. We derive the optimal values for the
method parameters in which the average computational cost is minimized
according to the desired error tolerance. We use three numerical examples to
demonstrate the computational efficiency of our method compared with the
classical double-loop Monte Carlo, and a more recent single-loop Monte Carlo
method that uses the Laplace method as an approximation of the return value of
the inner loop. The first example is a scalar problem that is linear in the
uncertain parameter. The second example is a nonlinear scalar problem. The
third example deals with the optimal sensor placement for an electrical
impedance tomography experiment to recover the fiber orientation in laminate
composites.Comment: 42 pages, 35 figure
MATHICSE Technical Report : A quasi-optimal sparse grids procedure for groundwater flows
In this work we explore the extension of the quasi-optimal sparse grids method proposed in our previous work \On the optimal polynomial ap- proximation of stochastic PDEs by Galerkin and Collocation methods" to a Darcy problem where the permeability is modeled as a lognormal random field. We propose an explicit a-priori/a-posteriori procedure for the construc- tion of such quasi-optimal grid and show its effectivenenss on a numerical ex- ample. In this approach, the two main ingredients are an estimate of the decay of the Hermite coefficients of the solution an
Convergence of quasi-optimal Stochastic Galerkin methods for a class of PDES with random coefficients
In this work we consider quasi-optimal versions of the Stochastic Galerkin method for solving linear elliptic PDEs with stochastic coefficients. In particular, we consider the case of a finite number of random inputs and an analytic dependence of the solution of the PDE with respect to the parameters in a polydisc of the complex plane . We show that a quasi-optimal approximation is given by a Galerkin projection on a weighted (anisotropic) total degree space and prove a (sub)exponential convergence rate. As a specific application we consider a thermal conduction problem with non-overlapping inclusions of random conductivity. Numerical results show the sharpness of our estimates
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