1,210 research outputs found
Nonlinear Methods for Model Reduction
The usual approach to model reduction for parametric partial differential
equations (PDEs) is to construct a linear space which approximates well
the solution manifold consisting of all solutions with
the vector of parameters. This linear reduced model is then used for
various tasks such as building an online forward solver for the PDE or
estimating parameters from data observations. It is well understood in other
problems of numerical computation that nonlinear methods such as adaptive
approximation, -term approximation, and certain tree-based methods may
provide improved numerical efficiency. For model reduction, a nonlinear method
would replace the linear space by a nonlinear space . This idea
has already been suggested in recent papers on model reduction where the
parameter domain is decomposed into a finite number of cells and a linear space
of low dimension is assigned to each cell.
Up to this point, little is known in terms of performance guarantees for such
a nonlinear strategy. Moreover, most numerical experiments for nonlinear model
reduction use a parameter dimension of only one or two. In this work, a step is
made towards a more cohesive theory for nonlinear model reduction. Framing
these methods in the general setting of library approximation allows us to give
a first comparison of their performance with those of standard linear
approximation for any general compact set. We then turn to the study these
methods for solution manifolds of parametrized elliptic PDEs. We study a very
specific example of library approximation where the parameter domain is split
into a finite number of rectangular cells and where different reduced
affine spaces of dimension are assigned to each cell. The performance of
this nonlinear procedure is analyzed from the viewpoint of accuracy of
approximation versus and
Model predictive control techniques for hybrid systems
This paper describes the main issues encountered when applying model predictive control to hybrid processes. Hybrid model predictive control (HMPC) is a research field non-fully developed with many open challenges. The paper describes some of the techniques proposed by the research community to overcome the main problems encountered. Issues related to the stability and the solution of the optimization problem are also discussed. The paper ends by describing the results of a benchmark exercise in which several HMPC schemes were applied to a solar air conditioning plant.Ministerio de EduaciĂłn y Ciencia DPI2007-66718-C04-01Ministerio de EduaciĂłn y Ciencia DPI2008-0581
Low Complexity Regularization of Linear Inverse Problems
Inverse problems and regularization theory is a central theme in contemporary
signal processing, where the goal is to reconstruct an unknown signal from
partial indirect, and possibly noisy, measurements of it. A now standard method
for recovering the unknown signal is to solve a convex optimization problem
that enforces some prior knowledge about its structure. This has proved
efficient in many problems routinely encountered in imaging sciences,
statistics and machine learning. This chapter delivers a review of recent
advances in the field where the regularization prior promotes solutions
conforming to some notion of simplicity/low-complexity. These priors encompass
as popular examples sparsity and group sparsity (to capture the compressibility
of natural signals and images), total variation and analysis sparsity (to
promote piecewise regularity), and low-rank (as natural extension of sparsity
to matrix-valued data). Our aim is to provide a unified treatment of all these
regularizations under a single umbrella, namely the theory of partial
smoothness. This framework is very general and accommodates all low-complexity
regularizers just mentioned, as well as many others. Partial smoothness turns
out to be the canonical way to encode low-dimensional models that can be linear
spaces or more general smooth manifolds. This review is intended to serve as a
one stop shop toward the understanding of the theoretical properties of the
so-regularized solutions. It covers a large spectrum including: (i) recovery
guarantees and stability to noise, both in terms of -stability and
model (manifold) identification; (ii) sensitivity analysis to perturbations of
the parameters involved (in particular the observations), with applications to
unbiased risk estimation ; (iii) convergence properties of the forward-backward
proximal splitting scheme, that is particularly well suited to solve the
corresponding large-scale regularized optimization problem
Global optimization for low-dimensional switching linear regression and bounded-error estimation
The paper provides global optimization algorithms for two particularly
difficult nonconvex problems raised by hybrid system identification: switching
linear regression and bounded-error estimation. While most works focus on local
optimization heuristics without global optimality guarantees or with guarantees
valid only under restrictive conditions, the proposed approach always yields a
solution with a certificate of global optimality. This approach relies on a
branch-and-bound strategy for which we devise lower bounds that can be
efficiently computed. In order to obtain scalable algorithms with respect to
the number of data, we directly optimize the model parameters in a continuous
optimization setting without involving integer variables. Numerical experiments
show that the proposed algorithms offer a higher accuracy than convex
relaxations with a reasonable computational burden for hybrid system
identification. In addition, we discuss how bounded-error estimation is related
to robust estimation in the presence of outliers and exact recovery under
sparse noise, for which we also obtain promising numerical results
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