398,886 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
Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems
This work develops a nonlinear multigrid method for diffusion problems
discretized by cell-centered finite volume methods on general unstructured
grids. The multigrid hierarchy is constructed algebraically using aggregation
of degrees of freedom and spectral decomposition of reference linear operators
associated with the aggregates. For rapid convergence, it is important that the
resulting coarse spaces have good approximation properties. In our approach,
the approximation quality can be directly improved by including more spectral
degrees of freedom in the coarsening process. Further, by exploiting local
coarsening and a piecewise-constant approximation when evaluating the nonlinear
component, the coarse level problems are assembled and solved without ever
re-visiting the fine level, an essential element for multigrid algorithms to
achieve optimal scalability. Numerical examples comparing relative performance
of the proposed nonlinear multigrid solvers with standard single-level
approaches -- Picard's and Newton's methods -- are presented. Results show that
the proposed solver consistently outperforms the single-level methods, both in
efficiency and robustness
Linear/Quadratic Programming-Based Optimal Power Flow using Linear Power Flow and Absolute Loss Approximations
This paper presents novel methods to approximate the nonlinear AC optimal
power flow (OPF) into tractable linear/quadratic programming (LP/QP) based OPF
problems that can be used for power system planning and operation. We derive a
linear power flow approximation and consider a convex reformulation of the
power losses in the form of absolute value functions. We show four ways how to
incorporate this approximation into LP/QP based OPF problems. In a
comprehensive case study the usefulness of our OPF methods is analyzed and
compared with an existing OPF relaxation and approximation method. As a result,
the errors on voltage magnitudes and angles are reasonable, while obtaining
near-optimal results for typical scenarios. We find that our methods reduce
significantly the computational complexity compared to the nonlinear AC-OPF
making them a good choice for planning purposes
Solving DSGE Models with a Nonlinear Moving Average
We introduce a nonlinear infinite moving average as an alternative to the standard state-space policy function for solving nonlinear DSGE models. Perturbation of the nonlinear moving average policy function provides a direct mapping from a history of innovations to endogenous variables, decomposes the contributions from individual orders of uncertainty and nonlinearity, and enables familiar impulse response analysis in nonlinear settings. When the linear approximation is saddle stable and free of unit roots, higher order terms are likewise saddle stable and first order corrections for uncertainty are zero. We derive the third order approximation explicitly and examine the accuracy of the method using Euler equation tests.Perturbation, nonlinear impulse response, DSGE, solution methods
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