Infinite-dimensional Operator Research Articles

A Rigorous Integrator and Global Existence for Higher-Dimensional Semilinear Parabolic PDEs via Semigroup Theory

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift–Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta–Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.

Dynamical systems and complex networks: a Koopman operator perspective

Abstract The Koopman operator has entered and transformed many research areas over the last years. Although the underlying concept—representing highly nonlinear dynamical systems by infinite-dimensional linear operators—has been known for a long time, the availability of large data sets and efficient machine learning algorithms for estimating the Koopman operator from data make this framework extremely powerful and popular. Koopman operator theory allows us to gain insights into the characteristic global properties of a system without requiring detailed mathematical models. We will show how these methods can also be used to analyze complex networks and highlight relationships between Koopman operators and graph Laplacians.

Convergence analysis of novel discontinuous Galerkin methods for a convection dominated problem

In this paper, we propose and analyze a numerically stable and convergent scheme for a convection-diffusion-reaction equation in the convection-dominated regime. Discontinuous Galerkin (DG) methods are considered since standard finite element methods for the convection-dominated equation cause spurious oscillations. We choose to follow a DG finite element differential calculus framework introduced in Feng et al. (2016) and approximate the infinite-dimensional operators in the equation with the finite-dimensional DG differential operators. Specifically, we construct the numerical method by using the dual-wind discontinuous Galerkin (DWDG) formulation for the diffusive term and the average discrete gradient operator for the convective term along with standard DG stabilization. We prove that the method converges optimally in the convection-dominated regime. Numerical results are provided to support the theoretical findings.

Meta Koopman decomposition for time series forecasting under temporal distribution shifts

Time series forecasting facilitates various real-world applications and has attracted great research interests. In real-world scenarios, time series forecasting models confront a fundamental issue of temporal distribution shifts (TDS), i.e., the statistical properties of time series are evolving over time. In this paper, we utilize Koopman theory to address temporal distribution shifts. Koopman theory states any time series can be mapped into a Koopman space by proper measurement functions with their dynamics modeled by the infinite-dimensional linear Koopman operator. Therefore, time series under different distributions can be modeled by different Koopman operators. Considering the linearity of Koopman operators, the Koopman operators for representing time series under different distributions can be decomposed as a linear combination of a set of Koopman operators, which we termed as meta Koopman operators. We further theoretically show the infinite-dimensional Koopman operators can be approximated by finite matrix multiplications and the meta Koopman operators are equivalent to a set of matrices. Based on the analysis, we propose an auto-encoder framework for implementing the meta Koopman decomposition of time series, which is theoretically capable of handling TDS. Extensive experiments conducted on four real-world time series datasets demonstrate the superiority of the proposed model in tackling temporal distribution shifts.

Data-Driven Fault Detection and Isolation for Multirotor System Using Koopman Operator

This paper presents a data-driven fault detection and isolation (FDI) for a multirotor system using Koopman operator and Luenberger observer. Koopman operator is an infinite-dimensional linear operator that can transform nonlinear dynamical systems into linear ones. Using this transformation, our aim is to apply the linear fault detection method to the nonlinear system. Initially, a Koopman operator-based linear model is derived to represent the multirotor system, considering factors like non-diagonal inertial tensor, center of gravity variations, aerodynamic effects, and actuator dynamics. Various candidate lifting functions are evaluated for prediction performance and compared using the root mean square error to identify the most suitable one. Subsequently, a Koopman operator-based Luenberger observer is proposed using the lifted linear model to generate residuals for identifying faulty actuators. Simulation and experimental results demonstrate the effectiveness of the proposed observer in detecting actuator faults such as bias and loss of effectiveness, without the need for an explicitly defined fault dataset.

On embedding the Koopmanization into controlled nonlinear systems, its comparison with the Carleman linearisation and concerning results: beyond the feedback linearisation

The feedback linearisation has found its striking applications in many practical situations for control problems that exploit the ideas from the linear system theory. Towards linearisation techniques for controlled nonlinear systems, operator-theoretic frameworks from functional analysis have received attentions as well. The Koopman operator is a linear, but an infinite dimensional operator. Most notably, the novelties of the paper are the following. (i) Sketching a rigorous and comprehensive proof of the Koopmanization of the controlled nonlinear system using the notion of the operator-invariant subspace and unifying the minimiser inequality formula with the Koopmanized system. The proof unfolds that the Koopmanization meets the bilinearisation in control context with the augmented state space. (ii) A formal construction of an observer encompassing the generalised Riccati equation in the Koopman setting of the original controlled nonlinear system. The Carleman version of the generalised Riccati equation is rephrased as well. (iii) Unfolding the usefulness of the Koopman observer. To achieve extensive illustrations of the theory of the paper, we construct two Koopman observers for two practical control problems, i.e. a controlled Duffing system with a cubic nonlinearity and a controlled system with a square nonlinearity. Then, we compare the Koopman observers with the Carleman observers.

Approximation of translation invariant Koopman operators for coupled non-linear systems.

Many physical systems exhibit translational invariance, meaning that the underlying physical laws are independent of the position in space. Data driven approximations of the infinite dimensional but linear Koopman operator of non-linear dynamical systems need to be physically informed in order to respect such physical symmetries. In the current work, we introduce a translation invariant extended dynamic mode decomposition (tieDMD) for coupled non-linear systems on periodic domains. This is achieved by exploiting a block-diagonal structure of the Koopman operator in Fourier space. Variants of tieDMD are applied to data obtained on one-dimensional periodic domains from the non-linear phase-diffusion equation, the Burgers equation, the Korteweg-de Vries equation, and a coupled FitzHugh-Nagumo system of partial differential equations. The reconstruction capability of tieDMD is compared to existing linear and non-linear variants of the dynamic mode decomposition applied to the same data. For the regarded data, tieDMD consistently shows superior capabilities in data reconstruction.

Fredholm theory on Krein spaces and its application to pseudospectrum

In this note, we introduce a notion of the J -kernel of a bounded linear operator on a Krein space and study the J -Fredholm theory for Krein space operators. Using J -Fredholm theory, we discuss and study the J -essential pseudospectra of bounded linear operators on Krein spaces, and some of their properties are studied. Furthermore, we discuss the structure of the J -essential pseudospectra of infinite-dimensional Hamiltonian operators.

An order relation between eigenvalues and symplectic eigenvalues of a class of infinite-dimensional operators

An order relation between eigenvalues and symplectic eigenvalues of a class of infinite-dimensional operators

Koopman neural operator as a mesh-free solver of non-linear partial differential equations

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-Bénard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modeling, and precipitation forecasting).

Galerkin approximation for [formula omitted]-control of the stable parabolic system under Dirichlet boundary control

In this paper, we explore state feedback control for the H∞ disturbance-attenuation problem in stable parabolic systems with in-domain distributed disturbances under Dirichlet boundary control. Calculating the state feedback control involves solving an operator algebraic Riccati equation, which poses challenges in finding an analytic solution. A practical approach is to seek an approximate solution via finite-dimensional approximation. Specifically, we employ the Galerkin approximation, which generates a sequence of finite-dimensional systems that approximate the original infinite-dimensional system. All corresponding finite-dimensional disturbance-attenuation problems are solvable, and it is demonstrated that the sequence of solutions to the associated finite-dimensional algebraic Riccati equations converges in norm to the solution of the infinite-dimensional operator algebraic Riccati equation. Furthermore, the state feedback controls derived from the finite-dimensional algebraic Riccati equations are proven to be γ-admissible controls for the original system.

The Approximation Characteristics of Weighted Band-Limited Function Space

This article primarily investigates the width problem within weighted band-limited function space in a uniform setting. Through an analysis of the properties of s-numbers, we establish a connection between the widths of weighted band-limited function spaces and the s-numbers of infinite-dimensional diagonal operators. Furthermore, employing the discretization method, we estimate the exact asymptotic orders of Kolmogorov n-width and linear n-width in the weighted band-limited function space, which is characterized by the weight ω=ωk=|k|rk∈Z0.

On approximating the symplectic spectrum of infinite-dimensional operators

The symplectic eigenvalues play a significant role in the finite-mode quantum information theory, and Williamson’s normal form proves to be a valuable tool in this area. Understanding the symplectic spectrum of a Gaussian Covariance Operator is a crucial task. Recently, in 2018, an infinite-dimensional analogue of Williamson’s Normal form was discovered, which has been instrumental in studying the infinite-mode Gaussian quantum states. However, most existing results pertain to finite-dimensional operators, leaving a dearth of literature in the infinite-dimensional context. The focus of this article is on employing approximation techniques to estimate the symplectic spectrum of certain infinite-dimensional operators. These techniques are well-suited for a particular class of operators, including specific types of infinite-mode Gaussian Covariance Operators. Our approach involves computing the Williamson’s Normal form and deriving bounds for the symplectic spectrum of these operators. As a practical application, we explicitly compute the symplectic spectrum of Gaussian Covariance Operators. Through this research, we aim to contribute to the understanding of symplectic eigenvalues in the context of infinite-dimensional operators, opening new avenues for exploration in quantum information theory and related fields.

Dynamical Systems of an Infinite-Dimensional Nonlinear Operator on the Banach Space $l_{1}$

We investigate discrete-time dynamical systems generated by an infinite-dimensional nonlinear operator that maps the Banach space $l_{1}$ to itself. It is demonstrated that this operator possesses up to seven fixed points. By leveraging the specific form of our operator, we illustrate that analyzing the operator can be simplified to a two-dimensional approach. Subsequently, we provide a detailed description of all fixed points, invariant sets for the two-dimensional operator and determine the set of limit points for its trajectories. These results are then applied to find the set of limit points for trajectories generated by the infinite-dimensional operator.

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

Computation of prolate spheroidal wavefunctions (PSWFs) is notoriously difficult and time consuming. This paper applies operator theory to the discrete Fourier transform (DFT) to address the problem of computing PSWFs. The problem is turned into an infinite dimensional matrix operator eigenvalue problem, which we recognize as being the definition of the DPSSs. Truncation of the infinite matrix leads to a finite dimensional matrix eigenvalue problem which in turn yields what is known as the Slepian basis. These discrete-valued Slepian basis vectors can then be used as (approximately) discrete time evaluations of the PSWFs. Taking an inverse Fourier transform further demonstrates that continuous PSWFs can be reconstructed from the Slepian basis. The feasibility of this approach is shown via theoretical derivations followed by simulations to consider practical aspects. Simulations demonstrate that the level of errors between the reconstructed Slepian basis approach and true PSWFs are low when the orders of the eigenvectors are low but can become large when the orders of the eigenvectors are high. Accuracy can be increased by increasing the number of points used to generate the Slepian basis. Users need to balance accuracy with computational cost. For large time-bandwidth product PSWFs, the number of Slepian basis points required increases for a reconstruction to reach the same error as for low time-bandwidth products. However, when the time-bandwidth products increase and reach maximum concentration, the required number of points to achieve a given error level achieves steady state values. Furthermore, this method of reconstructing the PSWF from the Slepian basis can be more accurate when compared to the Shannon sampling approach and traditional quadrature approach for large time-bandwidth products. Finally, since the Slepian basis represents the (approximate) sampled values of PSWFs, when the number of points is sufficiently large, the reconstruction process can be omitted entirely so that the Slepian vectors can be used directly, without a reconstruction step.

Lyapunov functions and dynamics of infinite dimensional Volterra operators

The majority of research on quadratic stochastic operators (QSOs) was done on a finite-dimensional set of all probability distributions (also known as a simplex), thus it is intriguing to extend them for an infinite case. In particular, the infinite dimensional Volterra operators are discussed in the current paper. Due to the fact non-compactness of infinite dimensional simplex (unlike a finite case) makes the general study challenging, therefore, a sub-class of infinite dimensional Volterra operators is introduced. Furthermore, we construct Lyapunov functions which allow us to explore the dynamics of the introduced operators. Several examples are given with a full description of the limiting set.

Integral Quadratic Constraints with Infinite-Dimensional Channels.

Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such input-output analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinite-dimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinite-dimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity.

A Structure Preserving Shift-Invert Infinite Arnoldi Algorithm for a Class of Delay Eigenvalue Problems with Hamiltonian Symmetry

This work considers a class of delay eigenvalue problems that admit a spectrum similar to that of a Hamiltonian matrix, in the sense that the spectrum is symmetric with respect to both the real and imaginary axes. We propose a method to iteratively approximate the eigenvalues closest to a given purely real or imaginary shift, while preserving the symmetries of the spectrum. To this end, our method exploits the equivalence between the considered delay eigenvalue problem and the eigenvalue problem associated with a linear but infinite-dimensional operator. To compute the eigenvalues closest to the given shift, we apply a specifically chosen shift-invert transformation to this linear operator and compute the eigenvalues with the largest modulus of the new shifted and inverted operator using an (infinite) Arnoldi procedure. The advantage of the chosen shift-invert transformation is that the spectrum of the transformed operator has a “real skew-Hamiltonian”-like structure. Furthermore, it is proven that the Krylov subspace constructed by applying this operator satisfies an orthogonality property in terms of a specifically chosen bilinear form. By taking this property into account during the orthogonalization process, it is ensured that, even in the presence of rounding errors, the obtained approximation for, e.g., a simple, purely imaginary eigenvalue is simple and purely imaginary. The presented work can thus be seen as an extension of [V. Mehrmann and D. Watkins, SIAM J. Sci. Comput., 22 (2001), pp. 1905–1925] to the considered class of delay eigenvalue problems. Although the presented method is initially defined on function spaces, it can be implemented using finite-dimensional linear algebra operations. The performance of the resulting numerical algorithm is verified for two example problems: the first example illustrates the advantage of the proposed approach in preserving purely imaginary eigenvalues when working in finite precision, while the second one demonstrates its applicability to a large scale problem.

A Lyapunov-Based ISS Small-Gain Theorem for Infinite Networks of Nonlinear Systems

In this article, we show that an infinite network of input-to-state stable (ISS) subsystems, admitting ISS Lyapunov functions, itself admits an ISS Lyapunov function, provided that the couplings between the subsystems are sufficiently weak. The strength of the couplings is described in terms of the properties of an infinite-dimensional nonlinear positive operator, built from the interconnection gains. If this operator induces a uniformly globally asymptotically stable (UGAS) system, a Lyapunov function for the infinite network can be constructed. We analyze necessary and sufficient conditions for UGAS and relate them to small-gain conditions used in the stability analysis of finite networks.

Fixed points of an infinite-dimensional operator related to Gibbs measures

We describe fixed points of an infinite dimensional non-linear operator related to a hard core (HC) model with a countable set $\mathbb{N}$ of spin values on the Cayley tree. This operator is defined by a countable set of parameters $\lambda_{i}>0$, $a_{ij}\in \{0,1\}$, $ i,j \in \mathbb{N}$. We find a sufficient condition on these parameters under which the operator has unique fixed point. When this condition is not satisfied then we show that the operator may have up to five fixed points. Also, we prove that every fixed point generates a normalisable boundary law and therefore defines a Gibbs measure for the given HC-model.