93,282 research outputs found
Higher order Quasi-Monte Carlo integration for Bayesian Estimation
We analyze combined Quasi-Monte Carlo quadrature and Finite Element
approximations in Bayesian estimation of solutions to countably-parametric
operator equations with holomorphic dependence on the parameters as considered
in [Cl.~Schillings and Ch.~Schwab: Sparsity in Bayesian Inversion of Parametric
Operator Equations. Inverse Problems, {\bf 30}, (2014)]. Such problems arise in
numerical uncertainty quantification and in Bayesian inversion of operator
equations with distributed uncertain inputs, such as uncertain coefficients,
uncertain domains or uncertain source terms and boundary data. We show that the
parametric Bayesian posterior densities belong to a class of weighted Bochner
spaces of functions of countably many variables, with a particular structure of
the QMC quadrature weights: up to a (problem-dependent, and possibly large)
finite dimension product weights can be used, and beyond this dimension,
weighted spaces with so-called SPOD weights are used to describe the solution
regularity. We establish error bounds for higher order Quasi-Monte Carlo
quadrature for the Bayesian estimation based on [J.~Dick, Q.T.~LeGia and
Ch.~Schwab, Higher order Quasi-Monte Carlo integration for holomorphic,
parametric operator equations, Report 2014-23, SAM, ETH Z\"urich]. It implies,
in particular, regularity of the parametric solution and of the
countably-parametric Bayesian posterior density in SPOD weighted spaces. This,
in turn, implies that the Quasi-Monte Carlo quadrature methods in [J. Dick,
F.Y.~Kuo, Q.T.~Le Gia, D.~Nuyens, Ch.~Schwab, Higher order QMC Galerkin
discretization for parametric operator equations, SINUM (2014)] are applicable
to these problem classes, with dimension-independent convergence rates
\calO(N^{-1/p}) of -point HoQMC approximated Bayesian estimates, where
depends only on the sparsity class of the uncertain input in the
Bayesian estimation.Comment: arXiv admin note: text overlap with arXiv:1409.218
Primal-Dual Reduced Basis Methods for Convex Minimization Variational Problems: Robust True Solution A Posteriori Error Certification and Adaptive Greedy Algorithms
In this paper, with the parametric symmetric coercive elliptic boundary value
problem as an example of the primal-dual variational problems satisfying the
strong duality, we develop primal-dual reduced basis methods (PD-RBM) with
robust true error certifications and discuss three versions of greedy
algorithms to balance the finite element error, the exact reduced basis error,
and the adaptive mesh refinements.
For a class of convex minimization variational problems which has
corresponding dual problems satisfying the strong duality, the primal-dual gap
between the primal and dual functionals can be used as a posteriori error
estimator. This primal-dual gap error estimator is robust with respect to the
parameters of the problem, and it can be used for both mesh refinements of
finite element methods and the true RB error certification.
With the help of integrations by parts formula, the primal-dual variational
theory is developed for the symmetric coercive elliptic boundary value problems
with non-homogeneous boundary conditions by both the conjugate function and
Lagrangian theories. A generalized Prager-Synge identity, which is the
primal-dual gap error representation for this specific problem, is developed.
RBMs for both the primal and dual problems with robust error estimates are
developed. The dual variational problem often can be viewed as a constraint
optimization problem. In the paper, different from the standard saddle-point
finite element approximation, the dual RBM is treated as a Galerkin projection
by constructing RB spaces satisfying the homogeneous constraint.
Inspired by the greedy algorithm with spatio-parameter adaptivity of
\cite{Yano:18}, adaptive balanced greedy algorithms with primal-dual finite
element and reduced basis error estimators are discussed. Numerical tests are
presented to test the PD-RBM with adaptive balanced greedy algorithms.Comment: 40 pages. The paper is accepted by SIAM Journal on Scientific
Computin
Robust a posteriori error estimation for stochastic Galerkin formulations of parameter-dependent linear elasticity equations
The focus of this work is a posteriori error estimation for stochastic
Galerkin approximations of parameter-dependent linear elasticity equations. The
starting point is a three-field PDE model in which the Young's modulus is an
affine function of a countable set of parameters. We analyse the weak
formulation, its stability with respect to a weighted norm and discuss
approximation using stochastic Galerkin mixed finite element methods
(SG-MFEMs). We introduce a novel a posteriori error estimation scheme and
establish upper and lower bounds for the SG-MFEM error. The constants in the
bounds are independent of the Poisson ratio as well as the SG-MFEM
discretisation parameters. In addition, we discuss proxies for the error
reduction associated with certain enrichments of the SG-MFEM spaces and we use
these to develop an adaptive algorithm that terminates when the estimated error
falls below a user-prescribed tolerance. We prove that both the a posteriori
error estimate and the error reduction proxies are reliable and efficient in
the incompressible limit case. Numerical results are presented to validate the
theory. All experiments were performed using open source (IFISS) software that
is available online.Comment: 23 pages, 5 figure
Optimal Experimental Design for Infinite-dimensional Bayesian Inverse Problems Governed by PDEs: A Review
We present a review of methods for optimal experimental design (OED) for
Bayesian inverse problems governed by partial differential equations with
infinite-dimensional parameters. The focus is on problems where one seeks to
optimize the placement of measurement points, at which data are collected, such
that the uncertainty in the estimated parameters is minimized. We present the
mathematical foundations of OED in this context and survey the computational
methods for the class of OED problems under study. We also outline some
directions for future research in this area.Comment: 37 pages; minor revisions; added more references; article accepted
for publication in Inverse Problem
Nitsche-XFEM for optimal control problems governed by elliptic PDEs with interfaces
For the optimal control problem governed by elliptic equations with
interfaces, we present a numerical method based on the Hansbo's Nitsche-XFEM.
We followed the Hinze's variational discretization concept to discretize the
continuous problem on a uniform mesh. We derive optimal error estimates of the
state, co-state and control both in mesh dependent norm and L2 norm. In
addition, our method is suitable for the model with non-homogeneous interface
condition. Numerical results confirmed our theoretical results, with the
implementation details discussed
A posteriori error estimation in a finite element method for reconstruction of dielectric permittivity
We present a posteriori error estimates for finite element approximations in
a minimization approach to a coefficient inverse problem. The problem is that
of reconstructing the dielectric permittivity , , from
boundary measurements of the electric field. The electric field is related to
the permittivity via Maxwell's equations. The reconstruction procedure is based
on minimization of a Tikhonov functional where the permittivity, the electric
field and a Lagrangian multiplier function are approximated by peicewise
polynomials. Our main result is an estimate for the difference between the
computed coefficient and the true minimizer , in
terms of the computed functions.Comment: 17 pages, 1 figur
Adaptive Estimation for Nonlinear Systems using Reproducing Kernel Hilbert Spaces
This paper extends a conventional, general framework for online adaptive
estimation problems for systems governed by unknown nonlinear ordinary
differential equations. The central feature of the theory introduced in this
paper represents the unknown function as a member of a reproducing kernel
Hilbert space (RKHS) and defines a distributed parameter system (DPS) that
governs state estimates and estimates of the unknown function. This paper 1)
derives sufficient conditions for the existence and stability of the infinite
dimensional online estimation problem, 2) derives existence and stability of
finite dimensional approximations of the infinite dimensional approximations,
and 3) determines sufficient conditions for the convergence of finite
dimensional approximations to the infinite dimensional online estimates. A new
condition for persistency of excitation in a RKHS in terms of its evaluation
functionals is introduced in the paper that enables proof of convergence of the
finite dimensional approximations of the unknown function in the RKHS. This
paper studies two particular choices of the RKHS, those that are generated by
exponential functions and those that are generated by multiscale kernels
defined from a multiresolution analysis.Comment: 24 pages, Submitted to CMAM
Spatial Mat\'ern fields driven by non-Gaussian noise
The article studies non-Gaussian extensions of a recently discovered link
between certain Gaussian random fields, expressed as solutions to stochastic
partial differential equations (SPDEs), and Gaussian Markov random fields. The
focus is on non-Gaussian random fields with Mat\'ern covariance functions, and
in particular we show how the SPDE formulation of a Laplace moving average
model can be used to obtain an efficient simulation method as well as an
accurate parameter estimation technique for the model. This should be seen as a
demonstration of how these techniques can be used, and generalizations to more
general SPDEs are readily available
Sparse approximate inverses of Gramians and impulse response matrices of large-scale interconnected systems
In this paper we show that inverses of well-conditioned, finite-time Gramians
and impulse response matrices of large-scale interconnected systems described
by sparse state-space models, can be approximated by sparse matrices. The
approximation methodology established in this paper opens the door to the
development of novel methods for distributed estimation, identification and
control of large-scale interconnected systems. The novel estimators
(controllers) compute local estimates (control actions) simply as linear
combinations of inputs and outputs (states) of local subsystems. The size of
these local data sets essentially depends on the condition number of the
finite-time observability (controllability) Gramian. Furthermore, the developed
theory shows that the sparsity patterns of the system matrices of the
distributed estimators (controllers) are primarily determined by the sparsity
patterns of state-space matrices of large-scale systems. The computational and
memory complexity of the approximation algorithms are , where is the
number of local subsystems of the interconnected system. Consequently, the
proposed approximation methodology is computationally feasible for
interconnected systems with an extremely large number of local subsystems.Comment: 11 pages, 4 figures. Submitted to IEEE Transactions on Automatic
Contro
Goal-oriented a posteriori error control for nonstationary convection-dominated transport problems
The numerical approximation of convection-dominated problems continues to
remain subject of strong interest. Families of stabilization techniques for
finite element methods were developed in the past. Adaptive techniques based on
a posteriori error estimates offer potential for further improvements. However,
there is still a lack in robust a posteriori error estimates in natural norms
of the discretizations. Here we combine the dual weighted residual method for
goal-oriented error control with stabilized finite element approximations. By a
duality argument an error representation is derived on that a space-time
adaptive approach is built. It differs from former works on the dual weighted
residual method. Numerical experiments illustrate that our schemes are capable
to resolve layers and sharp fronts with high accuracy and to further reduce
spurious oscillations of approximations
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