October 17, 2018
Dimitris_Giannakis
▦ Data-driven approaches for spectral analysis of ergodic dynamical systems ▦

     It is a remarkable fact, realized in the work of Koopman in the 1930s, that a general deterministic dynamical system can be characterized through its action on spaces of observables (functions of the state) through intrinsically linear evolution operators, acting by composition with the flow map. In the setting of measure-preserving, ergodic dynamics, these operators form a unitary group, whose spectral properties are useful for coherent pattern extraction and prediction of observables, among many applications. In this talk, we will discuss methods for data-driven approximation of the spectra of such unitary groups, focusing, in particular, on systems with mixing (chaotic) dynamics and continuous spectra. These methods utilize techniques from reproducing kernel Hilbert space (RKHS) theory to approximate the generator of the unitary Koopman group (an unbounded operator with complicated spectral behavior), through compact, skew-adjoint operators acting on a suitable RKHS of observables. These "compactified" generators have well-defined, purely atomic spectral measures, which are shown to converge to the spectrum of the generator in a limit of vanishing regularization parameter. In addition, the spectral measures of the compactified generators identify coherent observables under the dynamics through corresponding eigenfunctions, and have an associated functional calculus, allowing one to approximate functions of the generator. In particular, exponentiating the generator leads to an approximation of the unitary Koopman operator, which can be used to perform prediction of observables. The RKHS structure also allows stable, data-driven formulations of this framework that converge under fairly general assumptions on the system and observation modality. We illustrate this approach with applications to toy dynamical systems and examples drawn from climate dynamics.

October 24, 2018
Max Budninskiy
▦ Operator-adapted wavelets for finite-element differential forms ▦

     We introduce an operator-adapted multiresolution analysis for finite-element differential forms. From a given continuous, linear, bijective, and self-adjoint positive-definite operator L, a hierarchy of basis functions and associated wavelets for discrete differential forms is constructed in a fine-to-coarse fashion and in quasilinear time. The resulting wavelets are L-orthogonal across all scales, and can be used to derive a Galerkin discretization of the operator such that the resulting stiffness matrix becomes block-diagonal, with uniformly well-conditioned and sparse blocks. Because our approach applies to arbitrary differential p-forms, we can derive both scalar-valued and vector-valued wavelets block-diagonalizing a prescribed operator. We also discuss the generality of the construction by pointing out that it applies to various types of computational grids, offers arbitrary smoothness orders of basis functions and wavelets, and can accommodate linear differential constraints such as divergence-freeness. We demonstrate the benefits of the operator-adapted multiresolution decomposition for coarse-graining and model reduction of linear and nonlinear partial differential equations.

November 7, 2018
Matt Thomson
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November 28, 2018
Mathieu Desbrun
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January 16, 2109
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January 23, 2019
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