Finite-dimensional Approximation Research Articles

Koopman Modeling for Optimal Control of the Perimeter of Multi-region Urban Traffic Networks

The purpose of perimeter control is to regulate the transfer flow between the perimeters of the urban traffic network, so that the vehicle aggregation in each region is maintained at a desired level. It is regarded as one of the most important methods to solve traffic congestion in urban road networks. Accurate mathematical modeling of perimeter controlled traffic dynamics remains a challenge due to its high nonlinearity and uncertainty. Machine learning methods have been used to learn traffic dynamic models for perimeter control. However, these existing techniques lack generalization and interpretability. In this paper, we propose a Koopman modeling approach (i.e., a two-stage method) that uses a new eigenfunction of the Koopman operator based on a novel L0 norm approximation to construct an interpretable finite-dimensional approximation of the Koopman operator, which is a linear operator that describes how eigenfunctions evolve along the trajectory of a specific nonlinear dynamical system. The main advantage of utilizing the Koopman operator is that it can characterize the nonlinear system in a global linear lifted feature space. Furthermore, an algorithm based on the Koopman modeling method and model predictive control is proposed for real-time perimeter control of urban road networks. Some experiments are carried out to demonstrate the effectiveness of the proposed algorithm.

Mathematical Model of Drilling Mud Movement along a Shale Shaker Screen

Introduction. When drilling oil or gas wells, rock from the bottomhole is brought to the surface by drilling mud, which is cleaned of sludge by shale shakers at the first stage. Shale shakers are equipped with such a screen and create such trajectories of vibration of the frame in order to solve a dual problem: on the one hand, to provide the cleaning of the drilling mud coming from the well mouth, and on the other hand — to maintain the proper degree of cleaning. To correctly select the nomenclature of screens, it is necessary to reliably determine the throughput capacity of a shale shaker in the real-time setting. This will allow you to pre-order the required number of screens with the required cell size. Previously, studies were conducted by sieving a mixture of calibrated granules on a standard shaker or by straining mineral oil through a fixed screen. However, this does not fully correspond to the actual conditions of drilling mud screening. The objective of this article was to simulate the throughput of a drilling shale shaker under real conditions by calculating the movement of drilling mud along a vibrating screen using its previously obtained specific throughput capacity.Materials and Methods. When creating a mathematical model of the full throughput capacity of a shale shaker, a previously known mathematical model of fluid flow in an open channel and a finite-dimensional approximation in the form of a sequence of sections of concentrated containers connected by hydraulic conductivities (by analogy with the approximation of long electrical lines and extended gas pipelines by the finite element method) were used. The fluid flow rate over the screen was determined according to Chézy law. In this case, the cross-sectional area of the flow above the screen for a specific section was specified as the arithmetic mean between the initial and final values of the drilling mud height in the corresponding section.Results. A chain calculation scheme for sieving drilling mud was compiled. Based on the material balance, mathematical models were developed for determining the throughput and height of drilling mud above the screen: for the first section of the screen and the first nodal point of the screen; for subsequent screen sections and subsequent screen nodal points; for the last section of the screen and the last nodal point of the screen.A mathematical model of the drilling mud movement along the shale shaker screen has been obtained, which, due to the finite-dimensional representation of this flow along n sections of the screen length, is a system of n integral equations describing concentrated tanks, and n – 1 algebraic equations describing the flow of drilling mud between tanks.Discussion and Conclusion. The movement of drilling mud along the length of the shale shaker screen corresponds to the movement of a steady uneven flow in an open channel. However, in the first case, the volume flow along the length is variable (decreasing to zero), and in the second case, it is constant along the length. Therefore, the use of only one differential equation for steady uneven flow in an open channel is not sufficient in this case. The resulting mathematical model of the drilling mud movement along the shale shaker screen is a mathematical basis for the development of a modeling program to determine the throughput of the shale shaker for real operating conditions, i.e., for a specific drilling mud, a given screen or a given drive system of the shale shaker.

The classical and quantum particle on a flag manifold

Abstract In the present paper we consider two related problems, i.e. the description of geodesics and the calculation of the spectrum of the Laplace–Beltrami operator on a flag manifold. We show that there exists a family of invariant metrics such that both problems can be solved simply and explicitly. In order to determine the spectrum of the Laplace–Beltrami operator, we construct natural, finite-dimensional approximations (of spin chain type) to the Hilbert space of functions on a flag manifold.

Robustness of Hilbert space-valued stochastic volatility models

In this paper, we show that Hilbert space-valued stochastic models are robust with respect to perturbations, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic-volatility-modulated Ornstein–Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite-dimensional approximations of the driving compound Poisson process and semigroup generator, respectively, when considering operator-valued Barndorff-Nielsen and Shephard stochastic volatility models. We also give results on square root approximations. In all cases, we provide explicit bounds for the induced error in terms of the approximation of the underlying parameter. We discuss some applications to robustness of prices of options on forwards and volatility.

On a hemivariational inequality with non-monotone operator

We investigate the existence of generalized variational solutions to abstract hemivariational inequality driven by a coercive and locally Lipschitz operator with no further monotonicity conditions imposed on it. A finite dimensional Ritz type approximation scheme is derived. Applications to competitive systems driven by the generalized ( p , q ) - Laplacian with convection are given.

A finite-dimensional approximation for partial differential equations on Wasserstein space

This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the convergence of the viscosity solutions of the finite-dimensional PDE to the viscosity solutions of the PDE on Wasserstein space, provided that uniqueness holds for the latter, and heavily relies on an adaptation of the Barles & Souganidis monotone scheme (Barles and Souganidis, 1991) to our context, as well as on a key precompactness result for semimartingale measures. We illustrate our convergence result with the example of the Hamilton–Jacobi–Bellman and Bellman–Isaacs equations arising in stochastic control and differential games, and propose an extension to the case of path-dependent PDEs.

On the functional regression model and its finite-dimensional approximations

AbstractThe problem of linearly predicting a scalar response Y from a functional (random) explanatory variable $$X=X(t),\ t\in I$$ X = X ( t ) , t ∈ I is considered. It is argued that the term “linearly” can be interpreted in several meaningful ways. Thus, one could interpret that (up to a random noise) Y could be expressed as a linear combination of a finite family of marginals $$X(t_i)$$ X ( t i ) of the process X, or a limit of a sequence of such linear combinations. This simple point of view (which has some precedents in the literature) leads to a formulation of the linear model in terms of the RKHS space generated by the covariance function of the process X(t). It turns out that such RKHS-based formulation includes the standard functional linear model, based on the inner product in the space $$L^2[0,1]$$ L 2 [ 0 , 1 ] , as a particular case. It includes as well all models in which Y is assumed to be (up to an additive noise) a linear combination of a finite number of linear projections of X. Some consistency results are proved which, in particular, lead to an asymptotic approximation of the predictions derived from the general (functional) linear model in terms of finite-dimensional models based on a finite family of marginals $$X(t_i)$$ X ( t i ) , for an increasing grid of points $$t_j$$ t j in I. We also include a discussion on the crucial notion of coefficient of determination (aimed at assessing the fit of the model) in this setting. A few experimental results are given.

The Discrete Nonlinear Schrödinger Equation with Linear Gain and Nonlinear Loss: The Infinite Lattice with Nonzero Boundary Conditions and Its Finite-Dimensional Approximations

The Discrete Nonlinear Schrödinger Equation with Linear Gain and Nonlinear Loss: The Infinite Lattice with Nonzero Boundary Conditions and Its Finite-Dimensional Approximations

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.

A geometric framework for distributed frequency models

Geometric control theory, developed by Basile and Marro, and independently, by Wonham and Morse in the 1970s revolves around characterizing the properties of finite dimensional, linear and time-invariant systems using geometry. Some examples of these properties are invariance, controllability and observability. The task addressed in this paper is to develop the geometric tools for fractional systems using the diffusive representation (also known as the distributed frequency) model. The mathematics involved in this approach is different from the classical case as fractional systems are inherently infinite dimensional. Unlike the integer order derivative, the fractional derivative is a non-local operator, and so the evolution of the so-called pseudo-state depends on not just its current value but also its past history. Thus, the notion of an initial condition for a fractional system can come in different forms. This leads to different kinds of fractional derivative operators. With some of these operators, the semigroup property is lost. With the distributed frequency model, the initial condition comes in the form of an initialization function. This takes care of the infinite dimensional nature of fractional systems. Furthermore, the distributed frequency model retains the semigroup property. This is useful in developing invariance and controlled invariance for fractional systems. Moreover, these properties of fractional systems are verified numerically using a higher order finite-dimensional approximation, which retains all the geometric properties of the distributed frequency model.

Closed-loop Koopman operator approximation

This paper proposes a method to identify a Koopman model of a feedback-controlled system given a known controller. The Koopman operator allows a nonlinear system to be rewritten as an infinite-dimensional linear system by viewing it in terms of an infinite set of lifting functions. A finite-dimensional approximation of the Koopman operator can be identified from data by choosing a finite subset of lifting functions and solving a regression problem in the lifted space. Existing methods are designed to identify open-loop systems. However, it is impractical or impossible to run experiments on some systems, such as unstable systems, in an open-loop fashion. The proposed method leverages the linearity of the Koopman operator, along with knowledge of the controller and the structure of the closed-loop (CL) system, to simultaneously identify the CL and plant systems. The advantages of the proposed CL Koopman operator approximation method are demonstrated in simulation using a Duffing oscillator and experimentally using a rotary inverted pendulum system. An open-source software implementation of the proposed method is publicly available, along with the experimental dataset generated for this paper.

Kernel-algorithms in frame-approximations

With view to applications, we present here general classes of non-orthogonal expansions in Hilbert space. The main purpose of our paper is a new approach to design of algorithms of Kaczmarz type in the framework of operators in Hilbert space. Our work includes a diverse list of optimization problems, new Karhunen–Loève transforms, and Principal Component Analysis (PCA) for digital images. A key feature of our algorithms is our use of recursive systems of projection operators. For this we also make use of specific reproducing kernel Hilbert spaces, kernel factorizations, and finite-dimensional approximations. Our projection algorithms are designed with view to maximum likelihood solutions, minimization of “cost” problems, identification of principal components, and data-dimension reduction.

Singular Hamiltonian elliptic systems involving double exponential growth in dimension two

In this research, we are interested to investigate the existence of nontrivial weak solutions to the following Hamiltonian elliptic system −div(ω(x)∇u)=g(v)|x|a,x∈B1(0),−div(ω(x)∇v)=f(u)|x|b,x∈B1(0),with Dirichlet boundary conditions, where a,b∈[0,2), the weight ω(x) is of logarithmic type and the nonlinearities f and g possess double exponential growth. To establish the existence of solutions, our approach involves utilizing the linking theorem and a finite-dimensional approximation.

Entropic transfer operators

We propose a new concept for the regularization and discretization of transfer and Koopman operators in dynamical systems. Our approach is based on the entropically regularized optimal transport between two probability measures. In particular, we use optimal transport plans in order to construct a finite-dimensional approximation of some transfer or Koopman operator which can be analyzed computationally. We prove that the spectrum of the discretized operator converges to the one of the regularized original operator, give a detailed analysis of the relation between the discretized and the original peripheral spectrum for a rotation map on the n-torus and provide code for three numerical experiments, including one based on the raw trajectory data of a small biomolecule from which its dominant conformations are recovered.

Data-Driven Stochastic Optimal Control With Safety Constraints Using Linear Transfer Operators

We provide a data-driven framework for optimal control of a continuous-time control-affine stochastic dynamical system. The proposed framework relies on the linear operator theory involving Perron-Frobenius (P-F) and Koopman operators. Our first result involving the P-F operator provides a convex formulation to the optimal control problem in the dual space of densities. This convex formulation of the stochastic optimal control problem leads to an infinite-dimensional convex program. The finite-dimensional approximation of the convex program is obtained using a data-driven approximation of the linear operators. We provide rigorous error bounds and convergence rate for the data-driven approximation of the linear operators and comment on the convergence rate of the optimal control w.r.t. data size. Our second results demonstrate using the Koopman operator, dual to the P-F operator, for the stochastic optimal control design. We show that the Hamilton Jacobi Bellman (HJB) equation can be expressed using the Koopman operator. We provide an iterative procedure along the lines of a popular policy iteration algorithm based on the data-driven approximation of the Koopman operator for solving the HJB equation. The two formulations, namely the convex formulation involving the P-F operator and the Koopman-based formulation using the HJB equation, can be viewed as dual to each other where the duality follows due to the dual nature of the P-F and Koopman operators. Finally, we present examples to demonstrate the efficacy of the developed framework and verify the convergence rates for the operator and optimal control numerically.

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

This manuscript proposes a class of fractional stochastic integro-differential equations (FSIDEs) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of approximate solutions, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators, and fixed point approach. Furthermore, we examine the convergence of the Faedo–Galerkin (F–G) approximate solutions to the mild solution of our given problem. Finally, a concrete example involving a partial differential equation is provided to validate the main abstract results.

Optimal control for a nonlinear Schrödinger problem perturbed by multiplicative fractional noise

The aim of this paper is to investigate the existence of optimal solutions for control problems involving nonlinear stochastic Schrödinger equations perturbed by a multiplicative fractional Brownian motion. Finite-dimensional approximations and a linearization method are employed and lead to ε-optimal solutions for the considered problems.

Plane crack problems within strain gradient elasticity and mixed finite element implementation

An alternative approach is proposed and applied to solve boundary value problems within the strain gradient elasticity theory. A mixed variation formulation of the finite element method (FEM) based on the concept of the Galerkin method is used. To construct finite-dimensional subspaces separate approximations of displacements, deformations, stresses, and their gradients are implemented by choosing the different sets of piecewise polynomial basis functions, interrelated by the stability condition of the mixed FEM approximation. This significantly simplifies the pre-requirement for approximating functions to belong to class C1 and allows one to use the simplest triangular finite elements with a linear approximation of displacements under uniform or near-uniform triangulation conditions. Global unknowns in a discrete problem are nodal displacements, while the strains and stresses and their gradients are treated as local unknowns. The conditions of existence, uniqueness, and continuous dependence of the solution on the problem’s initial data are formulated for discrete equations of mixed FEM. These are solved by a modified iteration procedure, where the global stiffness matrix for classical elasticity problems is treated as a preconditioning matrix with fictitious elastic moduli. This avoids the need to form a global stiffness matrix for the problem of strain gradient elasticity since it is enough to calculate only the residual vector in the current approximation. A set of modeling plane crack problems is solved. The obtained solutions agree with the results available in the relevant literature. Good convergence is achieved by refining the mesh for all scale parameters. All three problems under study exhibit specific qualitative features characterizing strain gradient solutions namely crack stiffness increase with length scale parameter and cusp-like closure effect.

Koopman-Based MPC With Learned Dynamics: Hierarchical Neural Network Approach.

This article presents a data-driven control strategy for nonlinear dynamical systems, enabling the construction of a Koopman-based linear system associated with nonlinear dynamics. The primary idea is to apply the deep learning technique to the Koopman framework for globally linearizing nonlinear dynamics and impose a Koopman-based model predictive control (MPC) approach to stabilize the nonlinear dynamical systems. In this work, we first generalize the Koopman framework to nonlinear control systems, enabling comprehensive linear analysis and control methods to be effective for nonlinear systems. We next present a hierarchical neural network (HNN) approach to deal with the crucial challenge of the finite-dimensional Koopman representation approximation. In particular, a scale-invariant constrained network in the HNN includes four modules, in which a predictor module and a linear module can accurately approximate the finite Koopman eigenfunctions and Koopman operator, respectively, thus forming the lifted linear system. Then, we design the Koopman-based MPC scheme for controlling nonlinear systems with constraints by adopting the modified MPC with a saturation-like function on the lifted linear system. Importantly, the Koopman-based MPC enjoys higher computational efficiency compared to the classical linear MPC and nonlinear MPC methods. Finally, a physical experiment on an overhead crane system is provided to demonstrate the effectiveness of the proposed data-driven control framework.

Controlled transport of fluid particles by microrotors in a Stokes flow using linear transfer operators

The manipulation of a collection of fluid particles in a low Reynolds number environment has several important applications. As we demonstrate in this paper, this manipulation problem is related to the scientific question of how fluid flow structures direct Lagrangian transport. We investigate this problem of directing the transport by manipulating the flow, specifically in the Stokes flow context, by controlling the strengths of two rotors fixed in space. We demonstrate a novel dynamical systems approach for this problem and apply this method to several scenarios of Stokes flow in unbounded and bounded domains. Furthermore, we show that the time-varying flow field produced by the optimal control can be understood in terms of dynamical structures such as coherent sets that define Lagrangian transport. We model the time evolution of the fluid particle density using finite-dimensional approximations of the Liouville operators for the microrotor flow fields. Using these operators, the particle transport problem is framed as an optimal control problem, which we solve numerically. This framework is then applied to the problem of transporting a blob of fluid particles in domains with different boundary conditions: free space, near to a plane wall, in a circular confinement, and the transport of two distributions of particles to a common target. These examples demonstrate the effectiveness of the proposed framework and also shed light on the effects of boundaries on the ability to achieve a desired fluid transport using a rotor-driven flow.