Infinite-dimensional Space Research Articles

Singular limit of BSDES and optimal control of two scale systems with jumps in infinite dimensional spaces

The paper is devoted to a stochastic optimal control problem for a two scale, infinite dimensional, stochastic system. The state of the system consists of “slow” and “fast” component and its evolution is driven by both continuousWiener noises and discontinuous Poisson-type noises. The presence of discontinuous noises is the main feature of the present work. We use the theory of backward stochastic differential equations (BSDEs) to prove that, as the speed of the fast component diverges, the value function of the control problem converges to the solution of a reduced forward backward system that, in turn, is related to a reduced stochastic optimal control problem. The results of this paper generalize to the case of discontinuous noise the ones in [18] and [24].

Dynamic mean-field theory for continuous random networks

Abstract This article studies the dynamics of the mean-field approximation of continuous random networks. These networks are stochastic integrodifferential equations driven by Gaussian noise. The kernels in the integral operators are realizations of generalized Gaussian random variables. The equation controls the time evolution of a macroscopic state interpreted as neural activity, which depends on position and time. The position is an element of a measurable space. Such a network corresponds to a statistical field theory (STF) given by a momenta-generating functional. Discrete versions of the mentioned networks appeared in spin glasses and as models of artificial neural networks (NNs). Each of these discrete networks corresponds to a lattice SFT, where the action contains a finite number of neurons and two scalar fields for each neuron. Recently, it has been proposed that these networks can be used as models for deep learning. In this application, the number of neurons is astronomical; consequently, continuous models are required. In this article, we develop mathematically rigorous, continuous versions of the mean-field theory approximation and the double-copy system that allow us to derive a condition for the criticality of continuous stochastic networks via the largest Lyapunov exponent. It is essential to mention that the classical methods for mean-field theory approximation and the double-copy based on the stationary phase approximation cannot be used here because we are dealing with oscillatory integrals on infinite dimensional spaces. To our knowledge, the approach presented here is completely new. We use two basic architectures; in the first one, the space of neurons is the real line, and then the neurons are organized in one layer; in the second one, the space of neurons is the p-adic line, and then the neurons are organized in an infinite, fractal, tree-like structure. We also studied a toy model of a continuous Gaussian network with a continuous phase transition. This behavior matches the critical brain hypothesis, which states that certain biological neuronal networks work near phase transitions.

Simulating Electronic Structure on Bosonic Quantum Computers.

Quantum harmonic oscillators, or qumodes, provide a promising and versatile framework for quantum computing. Unlike qubits, which are limited to two discrete levels, qumodes have an infinite-dimensional Hilbert space, making them well-suited for a wide range of quantum simulations. In this work, we focus on the molecular electronic structure problem. We propose an approach to map the electronic Hamiltonian into a qumode bosonic problem that can be solved on bosonic quantum devices using the variational quantum eigensolver (VQE). Our approach is demonstrated through the computation of ground potential energy surfaces for benchmark model systems, including H2 and the linear H4 molecule. The preparation of trial qumode states and the computation of expectation values leverage universal ansatzes based on the echoed conditional displacement (ECD), or the selective number-dependent arbitrary phase (SNAP) operations. These techniques are compatible with circuit quantum electrodynamics (cQED) platforms, where microwave resonators coupled to superconducting transmon qubits can offer an efficient hardware realization. This work establishes a new pathway for simulating many-fermion systems, highlighting the potential of hybrid qubit-qumode quantum devices in advancing quantum computational chemistry.

Modes of Convergence of Sequences of Holomorphic Functions: A Linear Point of View

In this paper, pointwise convergence, uniform convergence and compact convergence of sequences of holomorphic functions on an open subset of the complex plane are compared from a linear point of view. In fact, it is proved the existence of large linear algebras consisting, except for zero, of sequences of holomorphic functions tending to zero compactly but not uniformly on the open set or of sequences of holomorphic functions tending pointwisely to zero but not compactly. Also dense linear subspaces in an appropriate Fréchet space as well as infinite dimensional Banach spaces of sequences converging to zero in the mentioned modes are shown to exist.

Precise quantum control of molecular rotation toward a desired orientation

The lack of a direct map between control fields and desired control objectives poses a significant challenge in applying quantum control theory to quantum technologies. Here, we propose an analytical framework to precisely control a limited set of quantum states and construct desired coherent superpositions using a well-designed laser pulse sequence with optimal amplitudes, phases, and delays. This theoretical framework that corresponds to a multilevel pulse-area theorem establishes a straightforward mapping between the control parameters of the pulse sequence and the amplitudes and phases of rotational states within a specific subspace. As an example, we utilize this approach to generate 15 distinct and desired rotational superpositions of ultracold polar molecules, leading to 15 desired field-free molecular orientations. By optimizing the superposition of the lowest 16 rotational states, we demonstrate that this approach can achieve a maximum orientation value of |〈cosθ〉|max above 0.99, which is very close to the global optimal value of 1 that could be achieved in an infinite-dimensional state space. This work marks a significant advancement in achieving precise control over multilevel subsystems within molecules. It holds potential applications in molecular alignment and orientation, as well as in various interdisciplinary fields related to the precise quantum control of ultracold polar molecules, opening up considerable opportunities in molecular-based quantum techniques. Published by the American Physical Society 2025

Uniqueness Results of Semilinear Parabolic Equations in Infinite-Dimensional Hilbert Spaces

This paper is devoted to the uniqueness of solutions for a class of nonhomogeneous stationary partial differential equations related to Hamilton–Jacobi-type equations in infinite-dimensional Hilbert spaces. Specifically, the uniqueness of the viscosity solution is established by employing the inf/sup-convolution approach in a separable infinite-dimensional Hilbert space. The proof is based on the Faedo–Galerkin approximate method by assuming the existence of a Hilbert–Schmidt operator and by employing modulus continuity and Lipschitz arguments. The results are of interest regarding the stochastic optimal control problem.

Trajectory controllability of higher-order fractional neutral stochastic system with non-instantaneous impulses via state-dependent delay with numerical simulation followed by hearth wall degradation process

Nowadays, engineers and biochemical industries have benefited greatly from controllability analysis and its computational methods. In this paper, the strongest notion of controllability called trajectory controllability (TC) of higher-order fractional neutral stochastic integrodifferential systems (FNSIDEs) with non-instantaneous impulsive (NI) via state-dependent delay is studied. The existence and uniqueness of solutions are proved in the infinite-dimensional space by using Mönch-type fixed-point theorem with the Hausdroff measure of noncompactness without compactness assumption on the semigroup. Further, the control problem is considered to establish TC results for FNSIDEs with NII. Then after, a fractional neutral stochastic model is discussed in the example section which is extended by the numerical simulation and the optimization technique supported by the Nelder-Mead method is used to identify the control that makes the state equations track a certain control. Finally, the remaining usable life, which can be described by either the probability density function or the point estimate of the mathematical expectation under certain specific stochastic assumptions, is typically defined and a case study on certain hearth wall degradation processes to validate the proposed method in practice.

On the solution of the Yang–Baxter–like operator equation on Hilbert spaces

This paper deals with solutions of the Yang–Baxter–like operator equation AXA = XAX on the infinite dimensional Hilbert space H , where A is a rank-two bounded linear operator and X is the unknown bounded linear operator. For an operator X, necessary and sufficient conditions such that X is a solution of the equation AXA = XAX are given. With these criteria, it is in fact shown that all solutions of the nonlinear equation AXA = XAX under the corresponding conditions can be derived merely through solving some systems of linear equations.

Nonlinear maps preserving asymptotic equivalence on B(X)

Let X be an infinite-dimensional complex Banach space. By B(X) we denote the algebra of all bounded linear operators on X. It is shown that if Φ : B(X) → B(X) is a surjective map satisfying A + B is asymptotically equivalent to C if and only if Φ(A) + Φ(B) is asymptotically equivalent to Φ(C), then either there exist bijective continuous linear or conjugate-linear maps T, S : X → X such that Φ(A) = TAS for every A ∈ B(X), or there exist bijective continuous linear or conjugate-linear maps T : X* → X and S : X → X * such that Φ(A) = TA* S for every A ∈ B(X).

Deconvolution Density Estimation with Penalized MLE

Deconvolution is the important problem of estimating the distribution of a quantity of interest from a sample with additive measurement error. Nearly all infinite-dimensional deconvolution methods in the literature use Fourier transformations. These methods are mathematically neat, but unstable, and produce bad estimates when signal-noise ratio or sample size are low. A popular alternative is to maximize penalized likelihood for a finite-dimensional basis expansion of the unknown density. We develop a new method to optimize penalized likelihood over the infinite-dimensional space of all functions. This gives the stability of regularized likelihood methods without restricting the space of solutions. Our method compares favorably with state-of-the-art methods on simulated and real data, particularly for small sample size or low signal-noise ratio. We also provide the first results on the consistency and rate of convergence of penalized maximum likelihood estimates for density deconvolution. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Spectral submanifolds in time delay systems

Abstract The concept of spectral submanifolds, a powerful method of model order reduction of nonlinear systems, is extended to time delay systems that have infinite dimensional phase space representation. The proposed sun-star calculus based algorithm results in system reduction to manifolds which are constructed corresponding to either a real eigenvalue or to a pair of complex conjugate eigenvalues of the linearized system. Furthermore, it allows an improved approximation of self-excited oscillations exactly at the parameter point of interest, which could be further away from the corresponding Hopf bifurcation point. The paper includes case studies that demonstrate the capabilities of the algorithm.

Improving student understanding of quantum measurement in infinite-dimensional Hilbert space using a research-based multiple-choice question sequence

Research-based multiple-choice questions implemented in class with peer instruction have been shown to be an effective tool for improving students’ engagement and learning outcomes. Moreover, multiple-choice questions that are carefully sequenced to build on each other can be particularly helpful for students to develop a systematic understanding of concepts pertaining to a theme. Here, we discuss the development, validation, and implementation of a multiple-choice question sequence (MQS) on the topic of quantum measurement in the context of wave functions in the infinite-dimensional Hilbert space. This MQS was developed using students’ common difficulties with quantum measurements as a guide and was implemented in a junior-/senior-level quantum mechanics course at a large research university in the U.S. We compare student performance on assessment tasks focusing on quantum measurement before and after the implementation of the MQS and discuss how different difficulties were reduced and how to further improve students’ conceptual understanding of quantum measurement in infinite-dimensional Hilbert space. Published by the American Physical Society 2025

Stochastic data-driven Bouligand–Landweber method for solving non-smooth inverse problems

Abstract In this study, we present and analyze a novel variant of the stochastic gradient descent method, referred as Stochastic data-driven Bouligand–Landweber iteration tailored for addressing the system of non-smooth ill-posed inverse problems. Our method incorporates the utilization of training data, using a bounded linear operator, which guides the iterative procedure. At each iteration step, the method randomly chooses one equation from the nonlinear system with data-driven term. When dealing with the precise or exact data, it has been established that mean square iteration error converges to zero. However, when confronted with the noisy data, we employ our approach in conjunction with a predefined stopping criterion, which we refer to as an a priori stopping rule. We provide a comprehensive theoretical foundation, establishing convergence and stability for this scheme within the realm of infinite-dimensional Hilbert spaces. These theoretical underpinnings are further bolstered by a numerical experiment on a system of linearly ill-posed problems and by discussing an example that fulfills the assumptions of the paper.

Simultaneous orthogonalization of inner products in infinite-dimensional vector spaces

For an arbitrary field K and a family of inner products in a K -vector space V of arbitrary dimension, we study necessary and sufficient conditions in order to have a basis which is orthogonal relative to all the inner products. If the family contains a nondegenerate element plus a compatibility condition, then under mild hypotheses the simultaneous orthogonalization can be achieved. So we investigate several constructions whose purpose is to add a nondegenerate element to a degenerate family and we study under what conditions the enlarged family is nondegenerate.

Equilibria in a large economy with externalities, non-convex preferences and an infinite-dimensional commodity space via relaxation

Equilibria in a large economy with externalities, non-convex preferences and an infinite-dimensional commodity space via relaxation

The topology of a chaotic attractor in the Kuramoto-Sivashinsky equation.

The Birman-Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within a chaotic attractor and the topology of that attractor, for three-dimensional systems. In certain cases, the fractal dimension of a chaotic attractor in a partial differential equation (PDE) is less than three, even though that attractor is embedded within an infinite-dimensional space. Here, we study the Kuramoto-Sivashinsky PDE at the onset of chaos. We use two different dimensionality-reduction techniques-proper orthogonal decomposition and an autoencoder neural network-to find two different mappings of the chaotic attractor into three dimensions. By finding the image of the attractor's UPOs in these reduced spaces and examining their linking numbers, we construct templates for the branched manifold, which encodes the topological properties of the attractor. The templates obtained using two different dimensionality reduction methods are equivalent. The organization of the periodic orbits is identical and consistent symbolic sequences for low-period UPOs are derived. While this is not a formal mathematical proof, this agreement is strong evidence that the dimensional reduction is robust, in this case, and that an accurate topological characterization of the chaotic attractor of the chaotic PDE has been achieved.

Koopman operator theory and dynamic mode decomposition in data-driven science and engineering: A comprehensive review

Poincaré's geometric representation, while historically fundamental in dynamical system analysis, faces challenges with high-dimensional and uncertain systems in modern engineering and data analysis. This article extensively explores Koopman Operator Theory (KOT) and Dynamic Mode Decomposition (DMD) within data-driven science and engineering and advocates for a conceptual shift toward observable dynamics, emphasizing KOT's capacity to capture nonlinear dynamics in infinite-dimensional space. The potential practical applications of Koopman-based methods are highlighted. Leveraging Poincaré's framework, the limitations of traditional methods are discussed. The review also addresses the growing significance of data-driven methodologies for modelling, predicting, and controlling complex systems.

Finite and Infinite Dimensional Reproducing Kernel Hilbert Space Approach for Bagley–Torvik Equation

Finite and Infinite Dimensional Reproducing Kernel Hilbert Space Approach for Bagley–Torvik Equation

Kinetic Theory with Casimir Invariants-Toward Understanding of Self-Organization by Topological Constraints.

A topological constraint, characterized by the Casimir invariant, imparts non-trivial structures in a complex system. We construct a kinetic theory in a constrained phase space (infinite-dimensional function space of macroscopic fields), and characterize a self-organized structure as a thermal equilibrium on a leaf of foliated phase space. By introducing a model of a grand canonical ensemble, the Casimir invariant is interpreted as the number of topological particles.

Solving a nonlinear matrix equation by Newton's method

The study of nonlinear boundary value problems, generally speaking, is not solved with respect to the derivative for ordinary differential equations. This area of research is a traditional focus of the Kyiv school of nonlinear oscillations, initiated in the 1930s by M.M. Krylov and M.M. Bogolyubov. Their work was continued by S. Campbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, and O.A. Boychuk, who focused on differential-algebraic boundary value problems. The linearization of nonlinear systems of differential equations that are not solved with respect to the derivative leads to linear differential-algebraic equations. Solving these linear differential-algebraic equations can be considered an ill-posed problem, as small perturbations in the equations can lead to significant changes in their solutions. The study of linear differential-algebraic equations, using the central canonical form and perfect pairs and triples of matrices, is the subject of monographs by A.M. Samoilenko, M.O. Perestiuk, V.P. Yakovets, O.A. Boychuk, as well as numerous works by foreign authors such as S. Campbell, J.R. Magnus, and V.F. Chistyakov. In the article by O.A. Boychuk and O.A. Pokutnyi, conditions for the solvability of a nonlinear differential-algebraic system are proposed, utilizing a generalized central canonical form. A key assumption in this work is that the linear part of the differential-algebraic system can be reduced to a canonical form introduced by V.F. Chistyakov. Additionally, studies on linear differential equations with an irreversible operator in the derivative in infinite-dimensional spaces were initiated by S.G. Crane, A.G. Rutkas, L.A. Vlasenko, G.V. Demidenko, M.V. Falaleev, and S.P. Zubova, as well as numerous works by foreign authors A. Favini, A. Yagi, and R.E. Showalter. We have studied the problem of determining constructive conditions for existence and methods for constructing solutions to nonlinear differential-algebraic boundary value problems. In particular, we focus on the problem of finding equilibrium positions for these problems. A distinctive feature of such problems is the inability to apply classical Newton’s method. To find the equilibrium positions, the article uses the generalized Newton-Kantorovich method.