739 research outputs found
Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems
We present a method for approximating the solution of a parameterized,
nonlinear dynamical system using an affine combination of solutions computed at
other points in the input parameter space. The coefficients of the affine
combination are computed with a nonlinear least squares procedure that
minimizes the residual of the governing equations. The approximation properties
of this residual minimizing scheme are comparable to existing reduced basis and
POD-Galerkin model reduction methods, but its implementation requires only
independent evaluations of the nonlinear forcing function. It is particularly
appropriate when one wishes to approximate the states at a few points in time
without time marching from the initial conditions. We prove some interesting
characteristics of the scheme including an interpolatory property, and we
present heuristics for mitigating the effects of the ill-conditioning and
reducing the overall cost of the method. We apply the method to representative
numerical examples from kinetics - a three state system with one parameter
controlling the stiffness - and conductive heat transfer - a nonlinear
parabolic PDE with a random field model for the thermal conductivity.Comment: 28 pages, 8 figures, 2 table
Discovering an active subspace in a single-diode solar cell model
Predictions from science and engineering models depend on the values of the
model's input parameters. As the number of parameters increases, algorithmic
parameter studies like optimization or uncertainty quantification require many
more model evaluations. One way to combat this curse of dimensionality is to
seek an alternative parameterization with fewer variables that produces
comparable predictions. The active subspace is a low-dimensional linear
subspace defined by important directions in the model's input space; input
perturbations along these directions change the model's prediction more, on
average, than perturbations orthogonal to the important directions. We describe
a method for checking if a model admits an exploitable active subspace, and we
apply this method to a single-diode solar cell model with five input
parameters. We find that the maximum power of the solar cell has a dominant
one-dimensional active subspace, which enables us to perform thorough parameter
studies in one dimension instead of five
A co-locating fast file system for UNIX
Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1998.Includes bibliographical references (p. 53-54).by Constantine Sapuntzakis.M.Eng
Factorizing the Stochastic Galerkin System
Recent work has explored solver strategies for the linear system of equations
arising from a spectral Galerkin approximation of the solution of PDEs with
parameterized (or stochastic) inputs. We consider the related problem of a
matrix equation whose matrix and right hand side depend on a set of parameters
(e.g. a PDE with stochastic inputs semidiscretized in space) and examine the
linear system arising from a similar Galerkin approximation of the solution. We
derive a useful factorization of this system of equations, which yields bounds
on the eigenvalues, clues to preconditioning, and a flexible implementation
method for a wide array of problems. We complement this analysis with (i) a
numerical study of preconditioners on a standard elliptic PDE test problem and
(ii) a fluids application using existing CFD codes; the MATLAB codes used in
the numerical studies are available online.Comment: 13 pages, 4 figures, 2 table
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