398,886 research outputs found

    Nonlinear Methods for Model Reduction

    Full text link
    The usual approach to model reduction for parametric partial differential equations (PDEs) is to construct a linear space VnV_n which approximates well the solution manifold M\mathcal{M} consisting of all solutions u(y)u(y) with yy the vector of parameters. This linear reduced model VnV_n 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, nn-term approximation, and certain tree-based methods may provide improved numerical efficiency. For model reduction, a nonlinear method would replace the linear space VnV_n by a nonlinear space Σn\Sigma_n. 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 NN of rectangular cells and where different reduced affine spaces of dimension mm are assigned to each cell. The performance of this nonlinear procedure is analyzed from the viewpoint of accuracy of approximation versus mm and NN

    Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems

    Full text link
    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

    Full text link
    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

    Get PDF
    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
    • …
    corecore