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Multilevel Representations of Isotropic Gaussian Random Fields on the Sphere
No full textSeries expansions of isotropic Gaussian random fields on S2 with independent Gaussian coefficients and localized basis functions are constructed. Such representations provide an alternative to the standard Karhunen-Loève expansions of isotropic random fields in terms of spherical harmonics. Their multilevel localized structure of basis functions is especially useful in adaptive algorithms. The basis functions are obtained by applying the square root of the covariance operator to spherical needlets. Localization of the resulting covariance-dependent multilevel basis is shown under decay conditions on the angular power spectrum of the random field. In addition, numerical illustrations are given and an application to random elliptic PDEs on the sphere is analyzed
Deep learning Markov and Koopman models with physical constraints
No full textAbstract
The long-timescale behavior of complex dynamical systems can be described by linear
Markov or Koopman models in a suitable latent space. Recent variational approaches
allow the latent space representation and the linear dynamical model to be optimized via
unsupervised machine learning methods. Incorporation of physical constraints such as time-
reversibility or stochasticity into the dynamical model has been established for a linear, but
not for arbitrarily nonlinear (deep learning) representations of the latent space. Here we
develop theory and methods for deep learning Markov and Koopman models that can
bear such physical constraints. We prove that the model is an universal approximator for
reversible Markov processes and that it can be optimized with either maximum likelihood
or the variational approach of Markov processes (VAMP). We demonstrate that the model
performs equally well for equilibrium and systematically better for biased data compared
to existing approaches, thus providing a tool to study the long-timescale processes of
dynamical systems
Data-driven analysis of complex dynamical systems
No full textThe main focus of this thesis is the data-driven analysis of complex dynamical systems. Although we will consider mainly molecular dynamics and fluid dynamics problems, the presented methods can be applied to arbitrary dynamical systems. In fact, in order to apply these methods, no a priori knowledge about the system is required, only simulation or measurement data. Such data-driven methods got a lot of attention recently due to the availability of large data sets. Gaining insight into the characteristic properties of a system by analyzing such data sets is akin to the metaphorical search for a needle in a haystack. The goal of data-driven methods is to extract relevant information about global properties of the underlying system, whose governing equations might be unknown. Global information can be obtained by analyzing the eigenvalues and eigenfunctions of transfer operators associated with the system
On strongly nonlinear gravity waves in a vertically sheared atmosphere
No full textWe investigate strongly nonlinear stationary gravity waves which experience refraction due to a thinvertical shear layer of horizontal background wind. The velocity amplitude of the waves is of the same orderof magnitude as the background flow and hence the self-induced mean flow alters the modulation propertiesto leading order. In this theoretical study, we show that the stability of such a refracted wave depends on theclassical modulation stability criterion for each individual layer, above and below the shearing. Additionally,the stability is conditioned by novel instability criteria providing bounds on the mean-flow horizontal windand the amplitude of the wave. A necessary condition for instability is that the mean-flow horizontal wind inthe upper layer is stronger than the wind in the lower layer.Keywords:Internal gravity waves; Modulation equations; She
VarGrad: A Low-Variance Gradient Estimator for Variational Inference
Get PDFWe analyse the properties of an unbiased gradient estimator of the evidence lowerbound (ELBO) for variational inference, based on the score function method withleave-one-out control variates. We show that this gradient estimator can be obtainedusing a new loss, defined as the variance of the log-ratio between the exact posteriorand the variational approximation, which we call thelog-variance loss. Undercertain conditions, the gradient of the log-variance loss equals the gradient of the(negative)ELBO. We show theoretically that this gradient estimator, which we callVarGraddue to its connection to the log-variance loss, exhibits lower variance thanthe score function method in certain settings, and that the leave-one-out controlvariate coefficients are close to the optimal ones. We empirically demonstrate thatVarGrad offers a favourable variance versus computation trade-off compared toother state-of-the-art estimators on a discrete variational autoencoder (VAE
Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
No full textWe derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems
Fractal homogenization of multiscale interface problems
Get PDFInspired by continuum mechanical contact problems with geological fault networks, we consider elliptic second order differential equations with jump conditions on a sequence of multiscale networks of interfaces with a finite number of nonseparating scales. Our aim is to derive and analyze a description of the asymptotic limit of infinitely many scales in order to quantify the effect of resolving the network only up to some finite number of interfaces, and to consider all further effects as homogeneous. As classical homogenization techniques are not suited for this kind of geometrical setting, we suggest a new concept, called fractal homogenization, to derive and analyze an asymptotic limit problem from a corresponding sequence of finite-scale interface problems. We provide an intuitive characterization of the corresponding fractal solution space in terms of generalized jumps and gradients together with continuous embeddings into and , . We show existence and uniqueness of the solution of the asymptotic limit problem and exponential convergence of the approximating finite-scale solutions. Computational experiments involving a related numerical homogenization technique illustrate our theoretical findings
Liouville-type equations for the n-particle distribution functions of an open system
Get PDFIn this work we derive a mathematical model for an open system that exchanges particles and momentum with a reservoir from their joint Hamiltonian dynamics. The complexity of this many-particle problem is addressed by introducing a countable set of n-particle phase space distribution functions just for the open subsystem, while accounting for the reservoir only in terms of statistical expectations. From the Liouville equation for the full system we derive a set of coupled Liouville-type equations for the n-particle distributions by marginalization with respect to reservoir states. The resulting equation hierarchy describes the external momentum forcing of the open system by the reservoir across its boundaries, and it covers the effects of particle exchanges, which induce probability transfers between the n- and (n+1)-particle distributions. Similarities and differences with the Bergmann-Lebowitz model of open systems (P.G.Bergmann, J.L. Lebowitz, Phys.Rev., 99:578--587 (1955)) are discussed in the context of the implementation of these guiding principles in a computational scheme for molecular simulations
Analysis of a compressible Stokes-flow with degenerating and singular viscosity
Get PDFIn this paper we show the existence of a weak solution for a compressible single-phase Stokes flow with mass transport
accounting for the degeneracy and the singular behavior of a density-dependent viscosity. The analysis is based
on an implicit time-discrete scheme and a Galerkin-approximation in space. Convergence of the discrete solutions is obtained
thanks to a diffusive regularization of p-Laplacian type in the transport equation that allows for refined compactness
arguments on subdomains
Coarse-graining of non-reversible stochasticdifferential equations: quantitative results and connections to averaging
No full textThis work is concerned with model reduction of stochastic differential equations and builds on the idea of replacing drift and noise coefficients of preselected relevant, e.g., slow variables by their conditional expectations. We extend recent results by Legoll and Leli\` evre [Nonlinearity 23 (2010), pp. 2131--2165] and Duong et al. [Nonlinearity, 31 (2018), pp. 4517--4567] on effective reversible dynamics by conditional expectations to the setting of general nonreversible processes with nonconstant diffusion coefficient. We prove relative entropy and Wasserstein error estimates for the difference between the time marginals of the effective and original dynamics as well as an entropy error bound for the corresponding path space measures. A comparison with the averaging principle for systems with time-scale separation reveals that, unlike in the reversible setting, the effective dynamics for a nonreversible system need not agree with the averaged equations. We present a thorough comparison for the Ornstein--Uhlenbeck process and make a conjecture about necessary and sufficient conditions for when averaged and effective dynamics agree for nonlinear nonreversible processes. The theoretical results are illustrated with suitable numerical examples