2,021 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
A posteriori modeling error estimates in the optimization of two-scale elastic composite materials
The a posteriori analysis of the discretization error and the modeling error
is studied for a compliance cost functional in the context of the optimization
of composite elastic materials and a two-scale linearized elasticity model. A
mechanically simple, parametrized microscopic supporting structure is chosen
and the parameters describing the structure are determined minimizing the
compliance objective. An a posteriori error estimate is derived which includes
the modeling error caused by the replacement of a nested laminate
microstructure by this considerably simpler microstructure. Indeed, nested
laminates are known to realize the minimal compliance and provide a benchmark
for the quality of the microstructures. To estimate the local difference in the
compliance functional the dual weighted residual approach is used. Different
numerical experiments show that the resulting adaptive scheme leads to simple
parametrized microscopic supporting structures that can compete with the
optimal nested laminate construction. The derived a posteriori error indicators
allow to verify that the suggested simplified microstructures achieve the
optimal value of the compliance up to a few percent. Furthermore, it is shown
how discretization error and modeling error can be balanced by choosing an
optimal level of grid refinement. Our two scale results with a single scale
microstructure can provide guidance towards the design of a producible
macroscopic fine scale pattern
Reduced Basis Method for Parametrized Elliptic Optimal Control Problems
We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique. Copyright \ua9 by SIAM. Unauthorized reproduction of this article is prohibited
Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
This paper introduces tensorial calculus techniques in the framework of
Proper Orthogonal Decomposition (POD) to reduce the computational complexity of
the reduced nonlinear terms. The resulting method, named tensorial POD, can be
applied to polynomial nonlinearities of any degree . Such nonlinear terms
have an on-line complexity of , where is the
dimension of POD basis, and therefore is independent of full space dimension.
However it is efficient only for quadratic nonlinear terms since for higher
nonlinearities standard POD proves to be less time consuming once the POD basis
dimension is increased. Numerical experiments are carried out with a two
dimensional shallow water equation (SWE) test problem to compare the
performance of tensorial POD, standard POD, and POD/Discrete Empirical
Interpolation Method (DEIM). Numerical results show that tensorial POD
decreases by times the computational cost of the on-line stage of
standard POD for configurations using more than model variables. The
tensorial POD SWE model was only slower than the POD/DEIM SWE model
but the implementation effort is considerably increased. Tensorial calculus was
again employed to construct a new algorithm allowing POD/DEIM shallow water
equation model to compute its off-line stage faster than the standard and
tensorial POD approaches.Comment: 23 pages, 8 figures, 5 table
An error indicator-based adaptive reduced order model for nonlinear structural mechanics -- application to high-pressure turbine blades
The industrial application motivating this work is the fatigue computation of
aircraft engines' high-pressure turbine blades. The material model involves
nonlinear elastoviscoplastic behavior laws, for which the parameters depend on
the temperature. For this application, the temperature loading is not
accurately known and can reach values relatively close to the creep
temperature: important nonlinear effects occur and the solution strongly
depends on the used thermal loading. We consider a nonlinear reduced order
model able to compute, in the exploitation phase, the behavior of the blade for
a new temperature field loading. The sensitivity of the solution to the
temperature makes {the classical unenriched proper orthogonal decomposition
method} fail. In this work, we propose a new error indicator, quantifying the
error made by the reduced order model in computational complexity independent
of the size of the high-fidelity reference model. In our framework, when the
{error indicator} becomes larger than a given tolerance, the reduced order
model is updated using one time step solution of the high-fidelity reference
model. The approach is illustrated on a series of academic test cases and
applied on a setting of industrial complexity involving 5 million degrees of
freedom, where the whole procedure is computed in parallel with distributed
memory
A certified RB method for PDE-constrained parametric optimization problems
Abstract
We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an "optimize-then-reduce" approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows
- …