arXiv subjects

Dimitrios Giannakis

Publications and source records attributed to Dimitrios Giannakis.

At least 37 records · Page 2Linked to original sources

arXiv2023

On harmonic Hilbert spaces on compact abelian groups

Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted $L^2$ spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach $^*$-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian group $G$ to have the same spectrum as the $C^*$-algebra of continuous functions on $G$. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on $L^2(G)$, and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier--Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation.

math.FA↗

arXiv2022

Quantum Mechanics for Closure of Dynamical Systems

We propose a scheme for data-driven parameterization of unresolved dimensions of dynamical systems based on the mathematical framework of quantum mechanics and Koopman operator theory. Given a system in which some components of the state are unknown, this method involves defining a surrogate system in a time-dependent quantum state which determines the fluxes from the unresolved degrees of freedom at each timestep. The quantum state is a density operator on a finite-dimensional Hilbert space of classical observables and evolves over time under an action induced by the Koopman operator. The quantum state also updates with new values of the resolved variables according to a quantum Bayes' law, implemented via an operator-valued feature map. Kernel methods are utilized to learn data-driven basis functions and represent quantum states, observables, and evolution operators as matrices. The resulting computational schemes are automatically positivity-preserving, aiding in the physical consistency of the parameterized system. We analyze the results of two different modalities of this methodology applied to the Lorenz 63 and Lorenz 96 multiscale systems, and show how this approach preserves important statistical and qualitative properties of the underlying chaotic systems.

math.DS↗

arXiv2022

Data Assimilation in Operator Algebras

We develop an algebraic framework for sequential data assimilation of partially observed dynamical systems. In this framework, Bayesian data assimilation is embedded in a non-abelian operator algebra, which provides a representation of observables by multiplication operators and probability densities by density operators (quantum states). In the algebraic approach, the forecast step of data assimilation is represented by a quantum operation induced by the Koopman operator of the dynamical system. Moreover, the analysis step is described by a quantum effect, which generalizes the Bayesian observational update rule. Projecting this formulation to finite-dimensional matrix algebras leads to new computational data assimilation schemes that are (i) automatically positivity-preserving; and (ii) amenable to consistent data-driven approximation using kernel methods for machine learning. Moreover, these methods are natural candidates for implementation on quantum computers. Applications to data assimilation of the Lorenz 96 multiscale system and the El Nino Southern Oscillation in a climate model show promising results in terms of forecast skill and uncertainty quantification.

math.ST↗

arXiv2022

Graph-theoretic algorithms for Kolmogorov operators: Approximating solutions and their gradients in elliptic and parabolic problems on manifolds

We employ kernel-based approaches that use samples from a probability distribution to approximate a Kolmogorov operator on a manifold. The self-tuning variable-bandwidth kernel method [Berry & Harlim, Appl. Comput. Harmon. Anal., 40(1):68--96, 2016] computes a large, sparse matrix that approximates the differential operator. Here, we use the eigendecomposition of the discretization to (i) invert the operator, solving a differential equation, and (ii) represent gradient vector fields on the manifold. These methods only require samples from the underlying distribution and, therefore, can be applied in high dimensions or on geometrically complex manifolds when spatial discretizations are not available. We also employ an efficient $k$-$d$ tree algorithm to compute the sparse kernel matrix, which is a computational bottleneck.

math.NA↗

arXiv2021

Embedding classical dynamics in a quantum computer

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides a new operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map for representing classical states by density operators on a reproducing kernel Hilbert space, $\mathcal H $, and an embedding of classical observables into self-adjoint operators on $\mathcal H$. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of $\mathcal H$, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve an exponential quantum computational advantage, we project the state of the quantum system to a density matrix on a $2^n$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through a quantum circuit of size $O(n)$. Furthermore, the circuit features a state preparation stage, also of size $O(n)$, and a quantum Fourier transform stage of size $O(n^2)$, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions as $n\to\infty$. We present simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.

quant-ph↗

arXiv2021

Spectral analysis of climate dynamics with operator-theoretic approaches

The Earth's climate system is a classical example of a multiscale, multiphysics dynamical system with an extremely large number of active degrees of freedom, exhibiting variability on scales ranging from micrometers and seconds in cloud microphysics, to thousands of kilometers and centuries in ocean dynamics. Yet, despite this dynamical complexity, climate dynamics is known to exhibit coherent modes of variability. A primary example is the El Niño Southern Oscillation (ENSO), the dominant mode of interannual (3-5 yr) variability in the climate system. The objective and robust characterization of this and other important phenomena presents a long-standing challenge in Earth system science, the resolution of which would lead to improved scientific understanding and prediction of climate dynamics, as well as assessment of their impacts on human and natural systems. Here, we show that the spectral theory of dynamical systems, combined with techniques from data science, provides an effective means for extracting coherent modes of climate variability from high-dimensional model and observational data, requiring no frequency prefiltering, but recovering multiple timescales and their interactions. Lifecycle composites of ENSO are shown to improve upon results from conventional indices in terms of dynamical consistency and physical interpretability. In addition, the role of combination modes between ENSO and the annual cycle in ENSO diversity is elucidated.

physics.ao-ph↗

arXiv2021

Reproducing kernel Hilbert space compactification of unitary evolution groups

A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. It is based on an approximation of the generator of the system by a compact operator $W_τ$ on a reproducing kernel Hilbert space (RKHS). A key tool is that $W_τ$ is skew-adjoint , and thus can be characterized by a unique projection-valued measure, discrete by compactness, and an associated orthonormal basis of eigenfunctions. These eigenfunctions can be ordered in terms of a Dirichlet energy on the RKHS, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, the regularized generator has a well-defined Borel functional calculus allowing the construction of a unitary evolution group $\{ e^{t W_τ} \}_{t\in\mathbb{R}}$ on the RKHS, which approximates the unitary Koopman evolution group of the original system. We establish convergence results for the spectrum and Borel functional calculus of the regularized generator to those of the original system in the limit $τ\to 0^+$. Convergence results are also established for a data-driven formulation, where these operators are approximated using finite-rank operators obtained from observed time series. An advantage of working with an RKHS structure is that one can perform pointwise evaluation and interpolation through bounded linear operators, which is not possible in $L^p$ spaces. This enables the out of sample evaluation of data-approximated eigenfunctions, as well as data-driven forecasts initialized with pointwise initial data (as opposed to probability densities in $L^p$). The pattern extraction and prediction framework is numerically applied to three instances of ergodic dynamical systems with atomic and continuous spectra.

math.DS↗

arXiv2021

Kernel-based Prediction of Non-Markovian Time Series

A nonparametric method to predict non-Markovian time series of partially observed dynamics is developed. The prediction problem we consider is a supervised learning task of finding a regression function that takes a delay embedded observable to the observable at a future time. When delay embedding theory is applicable, the proposed regression function is a consistent estimator of the flow map induced by the delay embedding. Furthermore, the corresponding Mori-Zwanzig equation governing the evolution of the observable simplifies to only a Markovian term, represented by the regression function. We realize this supervised learning task with a class of kernel-based linear estimators, the kernel analog forecast (KAF), which are consistent in the limit of large data. In a scenario with a high-dimensional covariate space, we employ a Markovian kernel smoothing method which is computationally cheaper than the Nyström projection method for realizing KAF. In addition to the guaranteed theoretical convergence, we numerically demonstrate the effectiveness of this approach on higher-dimensional problems where the relevant kernel features are difficult to capture with the Nyström method. Given noisy training data, we propose a nonparametric smoother as a de-noising method. Numerically, we show that the proposed smoother is more accurate than EnKF and 4Dvar in de-noising signals corrupted by independent (but not necessarily identically distributed) noise, even if the smoother is constructed using a data set corrupted by white noise. We show skillful prediction using the KAF constructed from the denoised data.

stat.ME↗

arXiv2020

Kernel Analog Forecasting: Multiscale Test Problems

Data-driven prediction is becoming increasingly widespread as the volume of data available grows and as algorithmic development matches this growth. The nature of the predictions made, and the manner in which they should be interpreted, depends crucially on the extent to which the variables chosen for prediction are Markovian, or approximately Markovian. Multiscale systems provide a framework in which this issue can be analyzed. In this work kernel analog forecasting methods are studied from the perspective of data generated by multiscale dynamical systems. The problems chosen exhibit a variety of different Markovian closures, using both averaging and homogenization; furthermore, settings where scale-separation is not present and the predicted variables are non-Markovian, are also considered. The studies provide guidance for the interpretation of data-driven prediction methods when used in practice.

math.ST↗

arXiv2020

Delay-coordinate maps, coherence, and approximate spectra of evolution operators

The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are $ε$-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound $ε$ controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, $ ε$ can be made arbitrarily small as the embedding window increases so long as the corresponding eigenvalues remain sufficiently isolated in the spectrum of the integral operator. It is also shown that the time-autocorrelation functions of such observables are $ε$-approximate Koopman eigenvalue, exhibiting a well-defined characteristic oscillatory frequency (estimated using the Koopman generator) and a slowly-decaying modulating envelope. The results hold for measure-preserving, ergodic dynamical systems of arbitrary spectral character, including mixing systems with continuous spectrum and no non-constant Koopman eigenfunctions in $L^2$. Numerical examples reveal a coherent observable of the Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus over approximately 10 Lyapunov timescales.

math.DS↗

arXiv2020

Bridging data science and dynamical systems theory

This short review describes mathematical techniques for statistical analysis and prediction in dynamical systems. Two problems are discussed, namely (i) the supervised learning problem of forecasting the time evolution of an observable under potentially incomplete observations at forecast initialization; and (ii) the unsupervised learning problem of identification of observables of the system with a coherent dynamical evolution. We discuss how ideas from from operator-theoretic ergodic theory combined with statistical learning theory provide an effective route to address these problems, leading to methods well-adapted to handle nonlinear dynamics, with convergence guarantees as the amount of training data increases.

math.ST↗

arXiv2020

An information-geometric approach to feature extraction and moment reconstruction in dynamical systems

We propose a dimension reduction framework for feature extraction and moment reconstruction in dynamical systems that operates on spaces of probability measures induced by observables of the system rather than directly in the original data space of the observables themselves as in more conventional methods. Our approach is based on the fact that orbits of a dynamical system induce probability measures over the measurable space defined by (partial) observations of the system. We equip the space of these probability measures with a divergence, i.e., a distance between probability distributions, and use this divergence to define a kernel integral operator. The eigenfunctions of this operator create an orthonormal basis of functions that capture different timescales of the dynamical system. One of our main results shows that the evolution of the moments of the dynamics-dependent probability measures can be related to a time-averaging operator on the original dynamical system. Using this result, we show that the moments can be expanded in the eigenfunction basis, thus opening up the avenue for nonparametric forecasting of the moments. If the collection of probability measures is itself a manifold, we can in addition equip the statistical manifold with the Riemannian metric and use techniques from information geometry. We present applications to ergodic dynamical systems on the 2-torus and the Lorenz 63 system, and show on a real-world example that a small number of eigenvectors is sufficient to reconstruct the moments (here the first four moments) of an atmospheric time series, i.e., the realtime multivariate Madden-Julian oscillation index.

stat.ML↗

arXiv2020

Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques

Kernel analog forecasting (KAF), alternatively known as kernel principal component regression, is a kernel method used for nonparametric statistical forecasting of dynamically generated time series data. This paper synthesizes descriptions of kernel methods and Koopman operator theory in order to provide a single consistent account of KAF. The framework presented here illuminates the property of the KAF method that, under measure-preserving and ergodic dynamics, it consistently approximates the conditional expectation of observables that are acted upon by the Koopman operator of the dynamical system and are conditioned on the observed data at forecast initialization. More precisely, KAF yields optimal predictions, in the sense of minimal root mean square error with respect to the invariant measure, in the asymptotic limit of large data. The presented framework facilitates, moreover, the analysis of generalization error and quantification of uncertainty. Extensions of KAF to the construction of conditional variance and conditional probability functions, as well as to non-symmetric kernels, are also shown. Illustrations of various aspects of KAF are provided with applications to simple examples, namely a periodic flow on the circle and the chaotic Lorenz 63 system.

math.ST↗

arXiv2020

Spectral exterior calculus

A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space dimension. To achieve this, the exterior calculus is reformulated entirely in terms of the eigenvalues and eigenfunctions of the Laplacian operator on functions. The exterior derivatives of these eigenfunctions (and their wedge products) are shown to form a frame (a type of spanning set) for appropriate $L^2$ spaces of $k$-forms, as well as higher-order Sobolev spaces. Formulas are derived to express the Laplace-de Rham operators on forms in terms of the eigenfunctions and eigenvalues of the Laplacian on functions. By representing the Laplace-de Rham operators in this frame, spectral convergence results are obtained via Galerkin approximation techniques. Numerical examples demonstrate accurate recovery of eigenvalues and eigenforms of the Laplace-de Rham operator on 1-forms. The correct Betti numbers are obtained from the kernel of this operator approximated from data sampled on several orientable and non-orientable manifolds, and the eigenforms are visualized via their corresponding vector fields. These vector fields form a natural orthonormal basis for the space of square-integrable vector fields, and are ordered by a Dirichlet energy functional which measures oscillatory behavior. The spectral framework also shows promising results on a non-smooth example (the Lorenz 63 attractor), suggesting that a spectral formulation of exterior calculus may be feasible in spaces with no differentiable structure.

math.DG↗

arXiv2020

Quantum dynamics of the classical harmonic oscillator

A correspondence is established between measure-preserving, ergodic dynamics of a classical harmonic oscillator and a quantum mechanical gauge theory on two-dimensional Minkowski space. This correspondence is realized through an isometric embedding of the $L^2(μ)$ space on the circle associated with the oscillator's invariant measure, $ μ$, into a Hilbert space $\mathcal{H}$ of sections of a $\mathbb{C}$-line bundle over Minkowski space. This bundle is equipped with a covariant derivative induced from an SO$^+$(1,1) gauge field (connection 1-form) on the corresponding inertial frame bundle, satisfying the Yang-Mills equations. Under this embedding, the Hamiltonian operator of a Lorentz-invariant quantum system, constructed as a natural Laplace-type operator on bundle sections, pulls back to the generator of the unitary group of Koopman operators governing the evolution of classical observables of the harmonic oscillator, with Koopman eigenfunctions of zero, positive, and negative eigenfrequency corresponding to quantum eigenstates of zero ("vacuum"), positive ("matter"), and negative ("antimatter") energy. The embedding also induces a pair of operators acting on classical observables of the harmonic oscillator, exhibiting canonical position-momentum commutation relationships. These operators have the structure of order-$1/2$ fractional derivatives, and therefore display a form of non-locality. In a second part of this work, we study a quantum mechanical representation of the classical harmonic oscillator using a one-parameter family of reproducing kernel Hilbert algebras associated with the transition kernel of a fractional diffusion on the circle, where a stronger notion of classical-quantum consistency is established than in the $L^2(μ)$ case.

math.DS↗

arXiv2019

Extended-range statistical ENSO prediction through operator-theoretic techniques for nonlinear dynamics

Forecasting the El Nino-Southern Oscillation (ENSO) has been a subject of vigorous research due to the important role of the phenomenon in climate dynamics and its worldwide socioeconomic impacts. Over the past decades, numerous models for ENSO prediction have been developed, among which statistical models approximating ENSO evolution by linear dynamics have received significant attention owing to their simplicity and comparable forecast skill to first-principles models at short lead times. Yet, due to highly nonlinear and chaotic dynamics (particularly during ENSO initiation), such models have limited skill for longer-term forecasts beyond half a year. To resolve this limitation, here we employ a new nonparametric statistical approach based on analog forecasting, called kernel analog forecasting (KAF), which avoids assumptions on the underlying dynamics through the use of for nonlinear kernel methods for machine learning and dimension reduction of high-dimensional datasets. Through a rigorous connection with Koopman operator theory for dynamical systems, KAF yields statistically optimal predictions of future ENSO states as conditional expectations, given noisy and potentially incomplete data at forecast initialization. Here, using industrial-era Indo-Pacific sea surface temperature (SST) as training data, the method is shown to successfully predict the Nino 3.4 index in a 1998-2017 verification period out to a 10-month lead, which corresponds to an increase of 3-7 months over a benchmark linear inverse model (LIM), while significantly improving upon the ENSO predictability "spring barrier". An analysis of a 1300-yr control integration of a comprehensive climate model (CCSM4) further demonstrates that the enhanced predictability afforded by KAF holds over potentially much longer leads, extending to 24 months versus 18 months in the benchmark LIM.

physics.ao-ph↗

arXiv2019

Koopman spectra in reproducing kernel Hilbert spaces

Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the $L^2$ space of observables associated with the invariant measure. Koopman eigenfunctions represent the quasiperiodic, or non-mixing, component of the dynamics. The extraction of these eigenfunctions and their associated eigenfrequencies from a given time series is a non-trivial problem when the underlying system has a dense point spectrum, or a continuous spectrum behaving similarly to noise. This paper describes methods for identifying Koopman eigenfrequencies and eigenfunctions from a discretely sampled time series generated by such a system with unknown dynamics. Our main result gives necessary and sufficient conditions for a Fourier function, defined on $N$ states sampled along an orbit of the dynamics, to be extensible to a Koopman eigenfunction on the whole state space, lying in a reproducing kernel Hilbert space (RKHS). In particular, we show that such an extension exists if and only if the RKHS norm of the Fourier function does not diverge as $ N \to \infty $, in which case the corresponding Fourier frequency is also a Koopman eigenfrequency. If such an RKHS extension does not exist, we can still construct an $L^2$ approximation of the eigenfunction. Numerical experiments on mixed-spectrum systems with weak periodic components demonstrate that this approach has significantly higher skill in identifying Koopman eigenfrequencies compared to conventional spectral estimation techniques based on the discrete Fourier transform.

math.DS↗

arXiv2019

Quantum mechanics and data assimilation

A framework for data assimilation combining aspects of operator-theoretic ergodic theory and quantum mechanics is developed. This framework adapts the Dirac--von Neumann formalism of quantum dynamics and measurement to perform sequential data assimilation (filtering) of a partially observed, measure-preserving dynamical system, using the Koopman operator on the $L^2$ space associated with the invariant measure as an analog of the Heisenberg evolution operator in quantum mechanics. In addition, the state of the data assimilation system is represented by a trace-class operator analogous to the density operator in quantum mechanics, and the assimilated observables by self-adjoint multiplication operators. An averaging approach is also introduced, rendering the spectrum of the assimilated observables discrete, and thus amenable to numerical approximation. We present a data-driven formulation of the quantum mechanical data assimilation approach, utilizing kernel methods from machine learning and delay-coordinate maps of dynamical systems to represent the evolution and measurement operators via matrices in a data-driven basis. The data-driven formulation is structurally similar to its infinite-dimensional counterpart, and shown to converge in a limit of large data under mild assumptions. Applications to periodic oscillators and the Lorenz 63 system demonstrate that the framework is able to naturally handle highly non-Gaussian statistics, complex state space geometries, and chaotic dynamics.

math-ph↗