Finite-dimensional Approximation Research Articles

On the finite dimensional approximations of some mixed variational inequalities

We construct the finite-dimensional approximations for mixed variational inequalities with pseudomonotone operators and convex non-differentiable functionals in Banach spaces. Such variational inequalities arise in the mathematical description of the processes of an established filtration and in the problems of determining the equilibrium of soft shells. The convergence of these approximations are investigated.

Об одном численном алгоритме решения интегральных уравнений первого рода в пространствах L2, основанном на обобщенном принципе невязки

In this paper we consider a one-dimensional Fredholm integral equation of type I closed with a kernel having a solution in the class W12 [a,b] with homogeneous boundary conditions of the first kind at the point a. The problem is reduced to a new integral equation for the derivative of the desired solution. The resulting integral equations is subjected to finite-dimensional approximation of a special form, that allows to use the variational regularization Tikhonov’s method with the choice of regularization parameter according to the generalized discrepancy principle to reduce the problem to a special system of linear algebraic equations. A priori estimation of the accuracy of the resulting finite-stable approximate solution that takes into account the accuracy of the finite-dimensional approximation of the problem is also carried out. Using of this approach is made on the example of the problem of determining the phonon spectrum on its heat capacity, depending on the temperature, which is known to be reduced to integral equations of the first kind.

Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations

The paper is concerned with a semi-linear backward stochasticSchrödinger equation in $\mathbb{R}^d$ or in its bounded domain of a $C^2$ boundary. Galerkin's finite-dimensional approximation method is used and the harmonic role of the Laplacian is shown. The existence, uniqueness and regularity are given for the weak solution of the equation. A more general backward stochastic Hamiltonian partial differential equation is also discussed.

Bayesian Numerical Homogenization

Numerical homogenization, i.e., the finite-dimensional approximation of solution spaces of PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements. These basis elements are oftentimes found after a laborious process of scientific investigation and plain guesswork. Can this identification problem be facilitated? Is there a general recipe/decision framework for guiding the design of basis elements? We suggest that the answer to the above questions could be positive based on the reformulation of numerical homogenization as a Bayesian inference problem in which a given PDE with rough coefficients (or multiscale operator) is excited with noise (random right-hand side/source term) and one tries to estimate the value of the solution at a given point based on a finite number of observations. We apply this reformulation to the identification of bases for the numerical homogenization of arbitrary integro-differential equations and show that these bases have optimal recovery properties. In particular we show how rough polyharmonic splines can be rediscovered as the optimal solution of a Gaussian filtering problem.

On a Problem of Guarantee Optimization in Time-Delay Systems

A control problem under conditions of disturbances is considered for a linear time-delay dynamical system. The goal of the control is to minimize a non-terminal quality index that evaluates a motion history and realizations of control and disturbance actions. The control problem is posed within the game-theoretical approach. For calculating the optimal guaranteed result of the control and constructing a control scheme that ensures this result, two methods are proposed. The first one is based on an appropriate approximation of the quality index. The second one is based on a finite-dimensional approximation of the dynamical system. Both methods allow us to reduce the control problem to high-dimensional auxiliary differential games without delays and with terminal quality indices. An illustrative example is considered, simulations results are given.

Methods of infinite dimensional Morse theory for geodesics on Finsler manifolds

We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of H1-curves, and two splitting lemmas for the functionals on Banach manifolds of C1-curves. Two results on critical groups of iterated closed geodesics are also proved; their corresponding versions on Riemannian manifolds are based on the usual splitting lemma by Gromoll and Meyer (1969)). Our approach consists in deforming the square of the Finsler metric in a Lagrangian which is smooth also on the zero section and then in using the splitting lemma for nonsmooth functionals that the author recently developed in Lu (2011, 0000, 2013). The argument does not involve finite-dimensional approximations and any Palais’ result in Palais (1966). As an application, we extend to Finsler manifolds a result by V. Bangert and W. Klingenberg (1983) about the existence of infinitely many, geometrically distinct, closed geodesics on a compact Riemannian manifold.

Convergence rates in regularization for a system of nonlinear ill-posed equations with m-accretive operators

In this paper, we study an operator version of the modified Browder-Tikhonov regularization method for finding a common solution for a system of ill-posed operator equations involving m-accretive operators Ai, i =0 , ... , N, in a reflexive Banach space. The convergence rates of the regularized solutions are estimated not only in the infinite-dimensional space, but also in connection with its finite-dimensional approximations without the weakly sequential continuity of the dual mapping. MSC: 47H17; 47H20

Simulation of Fractional-Order Low-Pass Filters

The attenuation of standard analog low-pass filters corresponds to a multiple value of -6 decibels per octave. This quantified value is related to the order of the filter. The issue addressed here is concerned with the extension of integer orders to non integer orders, such that the attenuation of a low-pass filter can be continuously adjusted. Fractional differential systems are known to provide such asymptotic behaviors and many results about their simulation are available. But even for a fixed cutoff frequency, their combination does not generate an additive group with respect to the order and they involve stability problems. In this paper, a class of low-pass filters with orders between 0 (the filter is a unit gain) and 1 (standard one-pole filter) is defined to restore these properties. These infinite dimensional filters are not fractional differential but admit some well-posed representations into weighted integrals of standard one-pole filters. Based on this, finite dimensional approximations are proposed and recast into the framework of statespace representations-space representations. A special care is given to reduce the computational complexity, through the dimension of the state. In practice, this objective is reached for the complete family, without damaging the perceptive quality, with dimension 13. Then, an accurate low-cost digital version of this family is built in the time-domain. The accuracy of the digital filters is verified on the complete range of parameters (cutoff frequencies and fractional orders). Moreover, the stability is guaranteed, even for time-varying parameters. As an application, a plugin has been implemented which provides a new audio tool for tuning the cutoff frequency and the asymptotic slope in a continuous way. As a very special application, choosing a one-half order combined with a low cutoff frequency (20 Hz or less), the filter fed with a white noise provides a pink noise generator.

Finite-dimensional approximation of convex regularization via hexagonal pixel grids

This work extends the existing convergence analysis for discrete approximations of minimizers of convex regularization functionals. In particular, some solution concepts are generalized, namely the standard minimum norm solutions for squared-norm regularizers and the -minimizing solutions for general convex regularizers, respectively. A central part of the manuscript addresses finite-dimensional approximations of solutions of ill-posed operator equations with basis functions defined on hexagonal grids, which require the novel solution concept.

A Large Deviation Principle of Retarded Ornstein-Uhlenbeck Processes Driven by Lévy Noise

In this article, we develop a large deviation principle (LDP) for a class of retarded Ornstein-Uhlenbeck processes driven by Lévy processes. We first present a LDP result for time delay systems driven by cylindrical Wiener processes based on the large deviations of Gaussian processes. By using a contraction technique and passing on a finite-dimensional approximation, an LDP is obtained for stochastic time delay evolution equations driven by additive Lévy noise, whose solutions are generally not Lévy processes any more.

Existence of nontrivial solution for a 4-sublinear Schrödinger–Poisson system

We consider a Schrödinger–Poisson system in R 3 with potential indefinite in sign and a general 4-sublinear nonlinearity. Its variational functional does not satisfy the Palais–Smale condition in general. We use a finite-dimensional approximation method to obtain a sequence of approximate solutions. Then we prove that these approximate solutions converge weakly to a nontrivial solution of this system.

Second-Order Optimality Conditions for Singular Pontryagin Local Minimizers

In this article, we use the method of finite-dimensional approximations to derive second-order necessary optimality conditions for singular Pontryagin local minimizers for an unconstrained nonlinear Mayer problem.

Generalized method of Lie-algebraic discrete approximations for solving Cauchy problems with evolution equation

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

Finite-dimensional approximations of distributed RC networks

Abstract Spectral properties of ladder and spatial electrical networks are considered. Dynamic properties of the networks are characterised by eigenvalues of the Jacobi cyclic state matrix. The effective formulas for eigenvalues of appropriate uniform systems are given. Numerical calculations were made using MATLAB.

Finite-Rank Multivariate-Basis Expansions of the Resolvent Operator as a Means of Solving the Multivariable Lippmann–Schwinger Equation for Two-Particle Scattering

Finite-rank expansions of the two-body resolvent operator are explored as a tool for calculating the full three-dimensional two-body T-matrix without invoking the partial-wave decomposition. The separable expansions of the full resolvent that follow from finite-rank approximations of the free resolvent are employed in the Low equation to calculate the T-matrix elements. The finite-rank expansions of the free resolvent are generated via projections onto certain finite-dimensional approximation subspaces. Types of operator approximations considered include one-sided projections (right or left projections), tensor-product (or outer) projection and inner projection. Boolean combination of projections is explored as a means of going beyond tensor-product projection. Two types of multivariate basis functions are employed to construct the finite-dimensional approximation spaces and their projectors: (i) Tensor-product bases built from univariate local piecewise polynomials, and (ii) multivariate radial functions. Various combinations of approximation schemes and expansion bases are applied to the nucleon-nucleon scattering employing a model two-nucleon potential. The inner-projection approximation to the free resolvent is found to exhibit the best convergence with respect to the basis size. Our calculations indicate that radial function bases are very promising in the context of multivariable integral equations.

Optimal Stabilization Using Lyapunov Measures

Numerical solutions for the optimal feedback stabilization of discrete time dynamical systems is the focus of this technical note. Set-theoretic notion of almost everywhere stability introduced by the Lyapunov measure, weaker than conventional Lyapunov function-based stabilization methods, is used for optimal stabilization. The linear Perron-Frobenius transfer operator is used to pose the optimal stabilization problem as an infinite dimensional linear program. Set-oriented numerical methods are used to obtain the finite dimensional approximation of the linear program. We provide conditions for the existence of stabilizing feedback controls and show the optimal stabilizing feedback control can be obtained as a solution of a finite dimensional linear program. The approach is demonstrated on stabilization of period two orbit in a controlled standard map.

Coupled Controls-Computational Fluids Approach for the Estimation of the Concentration From a Moving Gaseous Source in a 2-D Domain With a Lyapunov-Guided Sensing Aerial Vehicle

The estimation of the gas concentration (process state) associated with an emitting stationary or moving source using a sensing aerial vehicle (SAV) is considered. The dispersion from such a gas source into the ambient atmosphere is representative of accidental or deliberate release of chemicals, or release of gases from the biological systems. Estimation of the concentration field provides a superior ability for source localization, assessment of possible adverse impacts, and eventual containment. The abstract and finite-dimensional approximation framework present couples theoretical estimation and control with computational fluid dynamics methods. The gas dispersion (process) model is based on the 2-D advection-diffusion equation with variable eddy diffusivities and ambient winds. The state estimator is a modified Luenberger observer with a collocated filter gain that is parameterized by the position of the SAV. The process-state (concentration) estimator is based on a 2-D adaptive, multigrid, multistep finite-volume method. The grid is adapted with local refinement and coarsening during the process-state estimation, to improve accuracy and efficiency. The 2-D motion dynamics of the SAV is incorporated into the spatial process and the SAV's guidance is directly linked to the performance of the state estimator. The computational model and the state estimator are coupled in the sense that grid refinement is affected by the SAV repositioning, and the guidance laws of the SAV are affected by grid refinement. Extensive numerical simulations serve to demonstrate the effectiveness of the coupled approach.

Numerical Solution For Random Forced Spde Via Galerkin Finite Element Method

In this paper, we present a deterministic finite element approach for solving a random forced Diffusion equation. Separation of random and deterministic variables is done by KarhunenLoeve expansion. Truncating the Karhunen-Loeve expansion of the permeability field leads to a finite dimensional approximation of the problem. The problem is discretized, in spatial part, using the finite-element method and the polynomial chaos expansion in stochastic part. Finally, using Kronecker product preconditioner and thus, preconditioned conjugate gradient method the governed system of equation is solved. Numerical experiments are presented for illustrating the theoretical results.

Distances and inference for covariance operators

A framework is developed for inference concerning the covariance operator of a functional random process, where the covariance operator itself is an object of interest for statistical analysis. Distances for comparing positive-definite covariance matrices are either extended or shown to be inapplicable to functional data. In particular, an infinite-dimensional analogue of the Procrustes size-and-shape distance is developed. Convergence of finite-dimensional approximations to the infinite-dimensional distance metrics is also shown. For inference, a Frechet estimator of both the covariance operator itself and the average covariance operator is introduced. A permutation procedure to test the equality of the covariance operators between two groups is also considered. Additionally, the use of such distances for extrapolation to make predictions is explored. As an example of the proposed methodology, the use of covariance operators has been suggested in a philological study of cross-linguistic dependence as a way to incorporate quantitative phonetic information. It is shown that distances between languages derived from phonetic covariance functions can provide insight into the relationships between the Romance languages.

Path toward fast calculations in quantum scattering theory

Finite-dimensional approximations for scattering theory operators and for the Lippmann-Schwinger equation are obtained by going over to a basis of stationary wave packets. A multidimensional wave-packet basis is used for the first time, which allows to avoid expanding wave functions in partial waves and thereby to reduce substantially, at intermediate energies, the dimensions of the basis space used. In this wave-packet subspace, a finite-dimensional analytic representation of the free-particle Green’s function is obtained, which simplifies essentially solving respective equations. The possibility of a massive-parallel treatment of the resulting equations is shown.