27 research outputs found
Kolmogorov widths under holomorphic mappings
If is a bounded linear operator mapping the Banach space into the
Banach space and is a compact set in , then the Kolmogorov widths of
the image do not exceed those of multiplied by the norm of . We
extend this result from linear maps to holomorphic mappings from to
in the following sense: when the widths of are for some
r\textgreater{}1, then those of are for any s \textless{}
r-1, We then use these results to prove various theorems about Kolmogorov
widths of manifolds consisting of solutions to certain parametrized PDEs.
Results of this type are important in the numerical analysis of reduced bases
and other reduced modeling methods, since the best possible performance of such
methods is governed by the rate of decay of the Kolmogorov widths of the
solution manifold
Lipschitz dependence of the coefficients on the resolvent and greedy approximation for scalar elliptic problems
We analyze the inverse problem of identifying the diffusivity coefficient of
a scalar elliptic equation as a function of the resolvent operator. We prove
that, within the class of measurable coefficients, bounded above and below by
positive constants, the resolvent determines the diffusivity in an unique
manner. Furthermore we prove that the inverse mapping from resolvent to the
coefficient is Lipschitz in suitable topologies. This result plays a key role
when applying greedy algorithms to the approximation of parameter-dependent
elliptic problems in an uniform and robust manner, independent of the given
source terms. In one space dimension the results can be improved using the
explicit expression of solutions, which allows to link distances between one
resolvent and a linear combination of finitely many others and the
corresponding distances on coefficients. These results are also extended to
multi-dimensional elliptic equations with variable density coefficients. We
also point out towards some possible extensions and open problems
Recursive POD expansion for the advection-diffusion-reaction equation
This paper deals with the approximation of advection-diffusion-reaction
equation solution by reduced order methods. We use the Recursive POD approximation for multivariate functions introduced in [M. AZAÏEZ, F. BEN BELGACEM, T. CHACÓN REBOLLO, Recursive POD expansion for reactiondiffusion equation, Adv.Model. and Simul. in Eng. Sci. (2016) 3:3. DOI 10.1186/s40323-016-0060-1] and applied to the low tensor representation
of the solution of the reaction-diffusion partial differential equation. In this
contribution we extend the Recursive POD approximation for multivariate functions with an arbitrary number of parameters, for which we prove general error estimates. The method is used to approximate the solutions of the advection-diffusion-reaction equation. We prove spectral error estimates, in which the spectral convergence rate depends only on the diffusion interval, while the error estimates are affected by a factor that grows exponentially with the advection velocity, and are independent of the reaction rate if this lives in a bounded set. These error estimates are based upon the analyticity of the solution of these equations as a function of the parameters (advection
velocity, diffusion, reaction rate). We present several numerical tests, strongly consistent with the theoretical error estimates.Ministerio de Economía y CompetitividadAgence nationale de la rechercheGruppo Nazionale per il Calcolo ScientificoUE ERA-PLANE
Recursive POD expansion for the advection-diffusion-reaction equation
This paper deals with the approximation of advection-diffusion-reaction
equation solution by reduced order methods. We use the Recursive POD approximation for multivariate functions introduced in [M. AZAÏEZ, F. BEN BELGACEM, T. CHACÓN REBOLLO, Recursive POD expansion for reactiondiffusion equation, Adv.Model. and Simul. in Eng. Sci. (2016) 3:3. DOI 10.1186/s40323-016-0060-1] and applied to the low tensor representation
of the solution of the reaction-diffusion partial differential equation. In this
contribution we extend the Recursive POD approximation for multivariate functions with an arbitrary number of parameters, for which we prove general error estimates. The method is used to approximate the solutions of the advection-diffusion-reaction equation. We prove spectral error estimates, in which the spectral convergence rate depends only on the diffusion interval, while the error estimates are affected by a factor that grows exponentially with the advection velocity, and are independent of the reaction rate if this lives in a bounded set. These error estimates are based upon the analyticity of the solution of these equations as a function of the parameters (advection
velocity, diffusion, reaction rate). We present several numerical tests, strongly consistent with the theoretical error estimates.Ministerio de Economía y CompetitividadAgence nationale de la rechercheGruppo Nazionale per il Calcolo ScientificoUE ERA-PLANE
Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
Kolmogorov -widths and low-rank approximations are studied for families of
elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay
of the -widths can be controlled by that of the error achieved by best
-term approximations using polynomials in the parametric variable. However,
we prove that in certain relevant instances where the diffusion coefficients
are piecewise constant over a partition of the physical domain, the -widths
exhibit significantly faster decay. This, in turn, yields a theoretical
justification of the fast convergence of reduced basis or POD methods when
treating such parametric PDEs. Our results are confirmed by numerical
experiments, which also reveal the influence of the partition geometry on the
decay of the -widths.Comment: 27 pages, 6 figure
Stochastic optimization methods for the simultaneous control of parameter-dependent systems
We address the application of stochastic optimization methods for the
simultaneous control of parameter-dependent systems. In particular, we focus on
the classical Stochastic Gradient Descent (SGD) approach of Robbins and Monro,
and on the recently developed Continuous Stochastic Gradient (CSG) algorithm.
We consider the problem of computing simultaneous controls through the
minimization of a cost functional defined as the superposition of individual
costs for each realization of the system. We compare the performances of these
stochastic approaches, in terms of their computational complexity, with those
of the more classical Gradient Descent (GD) and Conjugate Gradient (CG)
algorithms, and we discuss the advantages and disadvantages of each
methodology. In agreement with well-established results in the machine learning
context, we show how the SGD and CSG algorithms can significantly reduce the
computational burden when treating control problems depending on a large amount
of parameters. This is corroborated by numerical experiments
Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes
Training nonlinear parametrizations such as deep neural networks to
numerically approximate solutions of partial differential equations is often
based on minimizing a loss that includes the residual, which is analytically
available in limited settings only. At the same time, empirically estimating
the training loss is challenging because residuals and related quantities can
have high variance, especially for transport-dominated and high-dimensional
problems that exhibit local features such as waves and coherent structures.
Thus, estimators based on data samples from un-informed, uniform distributions
are inefficient. This work introduces Neural Galerkin schemes that estimate the
training loss with data from adaptive distributions, which are empirically
represented via ensembles of particles. The ensembles are actively adapted by
evolving the particles with dynamics coupled to the nonlinear parametrizations
of the solution fields so that the ensembles remain informative for estimating
the training loss. Numerical experiments indicate that few dynamic particles
are sufficient for obtaining accurate empirical estimates of the training loss,
even for problems with local features and with high-dimensional spatial
domains