2,654 research outputs found
A fast Monte-Carlo method with a Reduced Basis of Control Variates applied to Uncertainty Propagation and Bayesian Estimation
The Reduced-Basis Control-Variate Monte-Carlo method was introduced recently
in [S. Boyaval and T. Leli\`evre, CMS, 8 2010] as an improved Monte-Carlo
method, for the fast estimation of many parametrized expected values at many
parameter values. We provide here a more complete analysis of the method
including precise error estimates and convergence results. We also numerically
demonstrate that it can be useful to some parametrized frameworks in
Uncertainty Quantification, in particular (i) the case where the parametrized
expectation is a scalar output of the solution to a Partial Differential
Equation (PDE) with stochastic coefficients (an Uncertainty Propagation
problem), and (ii) the case where the parametrized expectation is the Bayesian
estimator of a scalar output in a similar PDE context. Moreover, in each case,
a PDE has to be solved many times for many values of its coefficients. This is
costly and we also use a reduced basis of PDE solutions like in [S. Boyaval, C.
Le Bris, Nguyen C., Y. Maday and T. Patera, CMAME, 198 2009]. This is the first
combination of various Reduced-Basis ideas to our knowledge, here with a view
to reducing as much as possible the computational cost of a simple approach to
Uncertainty Quantification
Enhanced goal-oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems
This work focuses on providing accurate low-cost approximations of stochastic Âżnite elements simulations in the framework of linear elasticity. In a previous work, an adaptive strategy was introduced as an improved Monte-Carlo method for multi-dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal-oriented error estimator and estimates of CPU (computational processing unit) cost gain. Technical insights of these two topics are presented in details, and numerical examples show the interest of these new developments.Postprint (author's final draft
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
Bayesian Estimation for Continuous-Time Sparse Stochastic Processes
We consider continuous-time sparse stochastic processes from which we have
only a finite number of noisy/noiseless samples. Our goal is to estimate the
noiseless samples (denoising) and the signal in-between (interpolation
problem).
By relying on tools from the theory of splines, we derive the joint a priori
distribution of the samples and show how this probability density function can
be factorized. The factorization enables us to tractably implement the maximum
a posteriori and minimum mean-square error (MMSE) criteria as two statistical
approaches for estimating the unknowns. We compare the derived statistical
methods with well-known techniques for the recovery of sparse signals, such as
the norm and Log (- relaxation) regularization
methods. The simulation results show that, under certain conditions, the
performance of the regularization techniques can be very close to that of the
MMSE estimator.Comment: To appear in IEEE TS
Statistical Physics Analysis of Maximum a Posteriori Estimation for Multi-channel Hidden Markov Models
The performance of Maximum a posteriori (MAP) estimation is studied
analytically for binary symmetric multi-channel Hidden Markov processes. We
reduce the estimation problem to a 1D Ising spin model and define order
parameters that correspond to different characteristics of the MAP-estimated
sequence. The solution to the MAP estimation problem has different operational
regimes separated by first order phase transitions. The transition points for
-channel system with identical noise levels, are uniquely determined by
being odd or even, irrespective of the actual number of channels. We
demonstrate that for lower noise intensities, the number of solutions is
uniquely determined for odd , whereas for even there are exponentially
many solutions. We also develop a semi analytical approach to calculate the
estimation error without resorting to brute force simulations. Finally, we
examine the tradeoff between a system with single low-noise channel and one
with multiple noisy channels.Comment: The paper has been submitted to Journal of Statistical Physics with
submission number JOSS-S-12-0039
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