Finite Rank Approximations Research Articles

Transfer operator approach to ray-tracing in circular domains

The computation of wave-energy distributions in the mid-to-high frequency regime can be reduced to ray-tracing calculations. Solving the ray-tracing problem in terms of an operator equation for the energy density leads to an inhomogeneous equation which involves a Perron–Frobenius operator defined on a suitable Sobolev space. Even for fairly simple geometries, let alone realistic scenarios such as typical boundary value problems in room acoustics or for mechanical vibrations, numerical approximations are necessary. Here we study the convergence of approximation schemes by rigorous methods. For circular billiards we prove that convergence of finite-rank approximations using a Fourier basis follows a power law where the power depends on the smoothness of the source distribution driving the system. The relevance of our studies for more general geometries is illustrated by numerical examples.

Signal approximations by shifted Gaussians: a direct approach by finite linear systems

In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite–rank approximations by linear systems. The main result for this approach is to establish correctness of the finite–rank linear system under consideration. And the main result of the paper is to prove correctness of the finite–rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non–zero.

On the Correctness of Finite-Rank Approximations by Series of Shifted Gaussians

In this paper we consider interpolation problem connected with series by integer shifts of Gaussians. Known approaches for these problems met numerical difficulties. Due to it another method is considered based on finite-rank approximations by linear systems. The main result for this approach is to establish correctness of the finite-rank linear system under consideration. And the main result of the paper is to prove correctness of the finite-rank linear system approximation. For that an explicit formula for the main determinant of the linear system is derived to demonstrate that it is non-zero.

$V$-geometrical ergodicity of Markov kernels via finite-rank approximations

Under the standard drift/minorization and strong aperiodicity assumptions, this paper provides an original and quite direct approach of the $V$-geometrical ergodicity of a general Markov kernel $P$, which is by now a classical framework in Markov modelling. This is based on an explicit approximation of the iterates of $P$ by positive finite-rank operators, combined with the Krein-Rutman theorem in its version on topological dual spaces. Moreover this allows us to get a new bound on the spectral gap of the transition kernel. This new approach is expected to shed new light on the role and on the interest of the above mentioned drift/minorization and strong aperiodicity assumptions in $V$-geometrical ergodicity.

The S.V.D. of the Poisson Kernel

The solution of the Dirichlet problem for the Laplacian on bounded regions \(\Omega \) in \({\mathbb R}^{N}, \ N \ge 2\) is generally described via a boundary integral operator with the Poisson kernel. Under mild regularity conditions on the boundary, this operator is a compact linear transformation of \(L^2 (\partial \Omega ,d \sigma )\) to \(L^2_H(\Omega )\)—the Bergman space of \(L^2\)-harmonic functions on \(\Omega \). This paper describes the singular value decomposition of this operator and related results. The singular functions and singular values are constructed using Steklov eigenvalues and eigenfunctions of the biharmonic operator on \(\Omega \). These allow a spectral representation of the Bergman harmonic projection and the identification of an orthonormal basis of the real harmonic Bergman space \(L^2_H(\Omega )\). A reproducing kernel for \(L^2_H(\Omega )\) and an orthonormal basis of the space \(L^2 (\partial \Omega ,d \sigma )\) also are found. This enables the description of optimal finite rank approximations of the Poisson kernel with error estimates. Explicit spectral formulae for the normal derivatives of eigenfunctions for the Dirichlet Laplacian on \(\partial \Omega \) are obtained and used to identify a constant in an inequality of Hassell and Tao.

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.

A note on solving Cauchy integral equations of the second kind by projection

The purpose of this paper is to approximate the solution of a Cauchy integral equation of the second kind, using projection, finite rank approximations and a regularization procedure. We us the Kantorovich projection, the Sloan projection, the Galerkin projection, respectively. We give a general framework and we prove the existence of the solution for a projection schemes.

Ulam's method for some non-uniformly expanding maps

Certain dynamical systems on the interval with indifferent fixed points admit invariant probability measures which are absolutely continuous with respect to Lebesgue measure. These maps are often used as a model of intermittent dynamics, and they exhibit sub-exponential decay of correlations (due to the absence of a spectral gap in the underlying transfer operator). This paper concerns a class of these maps which are expanding (with convex branches), but admit an indifferent fixed point with tangency of $O(x^{1+\alpha})$ at $x=0$ ($0<\alpha<1$). The main results show that invariant probability measures can be rigorously approximated by a finite calculation. More precisely: Ulam's method (a sequence of computable finite rank approximations to the transfer operator) exhibits $L^1$ - convergence; and the $n$th approximate invariant density is accurate to at least $O(n^{-(1-\alpha)^2})$. Explicitly given non-uniform Ulam methods can improve this rate to $O(n^{-(1-\alpha)})$.

Finite rank approximations of expanding maps with neutral singularities

For a class of expanding maps with neutralsingularities we prove the validity of a finite rankapproximation scheme for the analysis of Sinai-Ruelle-Bowenmeasures. Earlier results of this sort were known onlyin the case of hyperbolic systems.

Optimal partition choice for invariant measure approximation for one-dimensional maps

The problem of finding absolutely continuous invariant measures (ACIMs) for a dynamical system can be formulated as a fixed point problem for a Markov operator (the Perron–Frobenius operator). This is an infinite-dimensional problem. Ulam's method replaces the Perron–Frobenius operator by a sequence of finite rank approximations whose fixed points are relatively easy to compute numerically. This paper concerns the optimal choice of Ulam approximations for one-dimensional maps; an adaptive partition selection is used to tailor the approximations to the structure of the invariant measure. The main idea is to select a partition which equally distributes the square root of the derivative of the invariant density amongst the bins of the partition. The results are illustrated for the logistic map where the ACIMs may have inverse square root singularities in their density functions. O(log n/n) convergence rates can be expected, whereas a non-adaptive algorithm yields O(n−1/2) at best. Studying the convergence of the adaptive algorithm allows an estimate to be made of the measure of the Jakobson parameter set (those logistic maps which admit an ACIM).

Markov finite approximation of Frobenius-Perron operator with doubly-stochastic tridiagonal action matrix

We extend the set of finite rank approximations of Frobenius-Perron operators, from projections, to general finite rank operators by indicating an arbitrary action matrix. We prove the convergence in the case for which the action matrix is double-stochastic and tridiagonal, and we find the rate of convergence.

A fast realization of preconditioned Conjugate Gradients for Wiener-Hopf integral equations

We present an efficient implementation of the Conjugate Gradients algorithm for Wiener-Hopf integral equations based on finite rank approximations of the integral operator and the corresponding preconditioner. The resulting algorithm is of linear complexity. Numerical experiments with this implementation of the preconditioned Conjugate Gradients algorithm show significant speed-up in the ill-conditioned case. This algorithm acts on ill-conditioned equations as a regularization algorithm.

Sampled-data control of continuous-time systems with an H∞ optimality criterion

The synthesis of sampled-data controllers for continuous-time systems is studied in the context of H ∞ control theory. The sampled-data H ∞ optimal control problem consists of finding a sampled-data controller which minimizes the L 2-induced norm from the continuous-time disturbance to the continuous-time output. The problem is here solved via a discrete system representation. This is achieved by describing the closed-loop system as a discrete system with finite state dimension, and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite-dimensional spaces. Via finite-rank approximations of a compact integral operator associated with the system, the sampled-data H ∞ control problem can be solved to any degree of accuracy. From the structure of the problem it follows that irrespective of the order of the approximation, a feasible sampled-data controller can be found which has the same order as the system.

Differentiable positive definite kernels and Lipschitz continuity

AbstractReade[11] has shown that positive definite kernels K(x, t) which satisfy a Lipschitz condition of order α on a bounded region have eigenvalues which are asymptotically O(1/n1+α). In this paper we extend this result to positive definite kernels whose symmetric derivative Krr(x, t) = ∂2rK(x, t)/∂xτ ∂tτ is in Lipα and establish λn(K) = O(1/n2r+1+α). If ∂Krr/∂t is in Lipα, the anticipated asymptotic estimate is also derived.The proofs use a well-known result of Chang [2], recently rederived by Ha [5], and estimates based upon finite rank approximations to the kernels in question. In these latter estimates we employ the familiar piecewise linear ‘hat’ basis functions of approximation theory.

Algebraic reduction and approximate intertwining relation in the strong approximation of Moller wave operators approach.

Nonrelativistic quantum scattering processes can be calculated from strong approximations of Moller wave operators, based on finite rank approximations of the full and asymptotic Hamiltonians. Firstly, similarly to Faddeev-type integral equations, we show that this approach also reduces a three-body system with separable two-body interactions to an effective two-body system. In the simplest case of an s-wave interaction without tensor forces, it reduces to a one-dimensional equation after partial wave decomposition. Secondly, we show that the intertwining relation is approximately valid in this approach and holds in the limit where the approximation parameters tend to infinity. We give a proof for the two-body system with short range interactions and for the long range Coulomb interaction. Implications for practical calculations are discussed.

Finite rank approximation of wave operators for short-range and Coulomb interactions

For the two-body system wave operators are shown to be strongly approximatable by finite rank operators which are obtained from finite rank approximations of the full and asymptotic Hamiltonian in the sense of strong resolvent convergence. This is demonstrated for short-range potentials plus the Coulomb potential with expansion functions chosen in momentum space as step functions. A generalization to the N-body system is indicated.

Finite rank approximations to integral operators which satisfy certain total positivity conditions

is totally positive [6]. Micchelli and Pinkus determined the Kolmogorov widths of sets determined by integral operators the kernels of which satisfy certain “total positivity” conditions. The integral operators which they considered in [ 12, 13, 141 act either from L”([O, 11) into L4([0, 11) (1 <qQ co) ([12, 141) or from Lp([O, 11) (1 QP< co) into L’([O, 1]) ([13, 141). In [14] (and in passing, in [ 121) they related the Kolmogorov widths to best approximations to the integral operators by operators of given finite rank.

Separable expansions to local potentials in a quasipotential approach

A recently proposed method which was successfully used to make separable expansions for the nonrelativistic two-body $t$ matrix is here used to construct separable expansions for the two-body $t$ matrix satisfying relativistic unitarity in a quasipotential approach. The present method, unlike other commonly used methods, does not require the calculations of eigenfunctions and eigenvalues of the kernel of the scattering equations and yields simple form factors. The method is illustrated for a simple nucleon-nucleon potential in the context of the Blankenbecler-Sugar equation.NUCLEAR REACTIONS Finite rank approximations to the $t$ matrix of a local potential, relativistic unitarity, Blankenbecler-Sugar equation solved, phase shift calculated.

An approximant for democratic representation of all Born terms

We propose an approximant which attempts to reconstruct a solution starting from the Born terms of a formal power series. The approximant follows closely the Fredholm solutions to Born–Neumann series of completely continuous operators and their finite rank approximations. We show that if the Fredholm solution is written as a combination of Born terms, it tends to become independent of the expansion parameter λ asymptotically. The proposed approximants then appear as solutions to differential equations approximately satisfied by the formal power series, a feature they share with kernel of finite rank approximations. All Born terms are put on equal footing and their respective weight is determined independently of λ by initial conditions; thus knowledge of the solution and (N−1) derivatives at one point is necessary. Analytic and crossing-symmetry properties are preserved by the proposed approximant, but unitarity is not insured and has to be examined specifically. Its error structure and its properties are studied and compared to Padé approximants.

Separable operator expansions for the t-matrix

Abstract A general scheme is developed for constructing finite-rank approximations to thet-matrix. The emphasis is on obtaining approximations that are good in a particular context (specifically, in the kernel of few-body problems), rather than on attaining good pointwise accuracy. The scheme in its most general form makes use of two arbitrary sets of expansion functions, and various choices for these functions are considered, and their virtues assessed. Other expansion methods (Weinberg series, unitary pole expansion, and the expansion of Ernst, Shakin and Thaler) can be obtained from the formalism through special choices of the expansion functions, but the methods proposed here, unlike those methods, do not require the solution of an eigenvalue problem. Numerical calculations, using simple choices for the expansion functions, are carried out for the Reid 1S0 soft-core N-N potential. They show that the convergence of the t-matrix expansion can be very good indeed, especially if it is used in the operator context for which it is designed.