21 research outputs found
On the Compressive Spectral Method
The authors of [Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 6634--6639] proposed sparse Fourier domain approximation of solutions to multiscale PDE problems by soft thresholding. We show here that the method enjoys a number of desirable numerical and analytic properties, including convergence for linear PDEs and a modified equation resulting from the sparse approximation. We also extend the method to solve elliptic equations and introduce sparse approximation of differential operators in the Fourier domain. The effectiveness of the method is demonstrated on homogenization examples, where its complexity is dependent only on the sparsity of the problem and constant in many cases
Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control
In this work, we explore finite-dimensional linear representations of
nonlinear dynamical systems by restricting the Koopman operator to an invariant
subspace. The Koopman operator is an infinite-dimensional linear operator that
evolves observable functions of the state-space of a dynamical system [Koopman
1931, PNAS]. Dominant terms in the Koopman expansion are typically computed
using dynamic mode decomposition (DMD). DMD uses linear measurements of the
state variables, and it has recently been shown that this may be too
restrictive for nonlinear systems [Williams et al. 2015, JNLS]. Choosing
nonlinear observable functions to form an invariant subspace where it is
possible to obtain linear models, especially those that are useful for control,
is an open challenge.
Here, we investigate the choice of observable functions for Koopman analysis
that enable the use of optimal linear control techniques on nonlinear problems.
First, to include a cost on the state of the system, as in linear quadratic
regulator (LQR) control, it is helpful to include these states in the
observable subspace, as in DMD. However, we find that this is only possible
when there is a single isolated fixed point, as systems with multiple fixed
points or more complicated attractors are not globally topologically conjugate
to a finite-dimensional linear system, and cannot be represented by a
finite-dimensional linear Koopman subspace that includes the state. We then
present a data-driven strategy to identify relevant observable functions for
Koopman analysis using a new algorithm to determine terms in a dynamical system
by sparse regression of the data in a nonlinear function space [Brunton et al.
2015, arxiv]; we show how this algorithm is related to DMD. Finally, we
demonstrate how to design optimal control laws for nonlinear systems using
techniques from linear optimal control on Koopman invariant subspaces.Comment: 20 pages, 5 figures, 2 code
Sparse Generalized Multiscale Finite Element Methods and their applications
In a number of previous papers, local (coarse grid) multiscale model
reduction techniques are developed using a Generalized Multiscale Finite
Element Method. In these approaches, multiscale basis functions are constructed
using local snapshot spaces, where a snapshot space is a large space that
represents the solution behavior in a coarse block. In a number of applications
(e.g., those discussed in the paper), one may have a sparsity in the snapshot
space for an appropriate choice of a snapshot space. More precisely, the
solution may only involve a portion of the snapshot space. In this case, one
can use sparsity techniques to identify multiscale basis functions. In this
paper, we consider two such sparse local multiscale model reduction approaches.
In the first approach (which is used for parameter-dependent multiscale
PDEs), we use local minimization techniques, such as sparse POD, to identify
multiscale basis functions, which are sparse in the snapshot space. These
minimization techniques use minimization to find local multiscale basis
functions, which are further used for finding the solution. In the second
approach (which is used for the Helmholtz equation), we directly apply
minimization techniques to solve the underlying PDEs. This approach is more
expensive as it involves a large snapshot space; however, in this example, we
can not identify a local minimization principle, such as local generalized SVD