Solving Operator Equations Research Articles

SOLVING OPERATOR EQUATIONS Ax = Y AND Ax = y IN ALGL

In this paper the following is proved: Let L be a subspace lattice on a Hilbert space H and X and Y be operators acting on a Hilbert space H. If XE

Nonlinear boundary integral equations method for contact problems of the elasticity theory

When implementing variational methods for solving complex contact problems, there are difficulties, associated with non-convexity of minimized energy function of the system of interacting bodies and non-differentiability of this function at the desired point of its minimum. These difficulties do not allow to use gradient methods and convex analysis methods to minimize the energy function, therefore, numerical procedures for finding minimum points of such functions are cumbersome in program implementation and sometimes make it impossible to obtain the contact problem solution with sufficient accuracy. Non-variational method, based on using nonlinear operator equations with no difficulties during its implementation is proposed in the paper. Applying these equations allows to use modern achievements of nonlinear functional analysis, fixed-point theory of continuous mappings, theory of iterative methods for solving operator equations for both proving theorems of existence of solutions to contact problems, and developing effective iterative procedures for approximate solutions. Nonlinear boundary integral equations, used in this paper to simulate the contact interaction of elastic bodies, allow (unlike other similar equations) to take into account both the linkage and partial slip on the contact surface of bodies, and loading history of these bodies. Based on these equations, simple and efficient iterative procedures for approximate solutions to the contact problems are developed. A numerical solution of the contact problem on indenting the elastic sphere in the elastic half-space is obtained, and comparison of results with the known problem solution is made.

Approximate multiplication in adaptive wavelet methods

Abstract Cohen, Dahmen and DeVore designed in [Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comp., 2001, 70(233), 27–75] and [Adaptive wavelet methods II¶beyond the elliptic case, Found. Comput. Math., 2002, 2(3), 203–245] a general concept for solving operator equations. Its essential steps are: transformation of the variational formulation into the well-conditioned infinite-dimensional l 2-problem, finding the convergent iteration process for the l 2-problem and finally using its finite dimensional approximation which works with an inexact right-hand side and approximate matrix-vector multiplication. In our contribution, we pay attention to approximate matrix-vector multiplication which is enabled by an off-diagonal decay of entries of the wavelet stiffness matrices. We propose a more efficient technique which better utilizes actual decay of matrix and vector entries and we also prove that this multiplication algorithm is asymptotically optimal in the sense that storage and number of floating point operations, needed to resolve the problem with desired accuracy, remain proportional to the problem size when the resolution of the discretization is refined.

Using canonical dual frames in adaptive Richardson iterative method for solving operator equations

Using canonical dual frames in adaptive Richardson iterative method for solving operator equations

Some Newton-like methods with sharper error estimates for solving operator equations in Banach spaces

It is well known that the rate of convergence of S-iteration process introduced by Agarwal et al. (pp. 61-79) is faster than Picard iteration process for contraction operators. Following the ideas of S-iteration process, we introduce some Newton-like algorithms to solve the non-linear operator equation in Banach space setting. We study the semi-local as well as local convergence analysis of our algorithms. The rate of convergence of our algorithms are faster than the modified Newton method.Mathematics Subject Classification 2010: 49M15; 65K10; 47H10.

Dilations of Γ-contractions by solving operator equations

For a contraction P and a bounded commutant S of P, we seek a solution X of the operator equationS−S⁎P=(I−P⁎P)12X(I−P⁎P)12, where X is a bounded operator on Ran¯(I−P⁎P)12 with numerical radius of X being not greater than 1. A pair of bounded operators (S,P) which has the domainΓ={(z1+z2,z1z2):|z1|⩽1,|z2|⩽1}⊆C2 as a spectral set, is called a Γ-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a Γ-contraction (S,P). This allows us to construct an explicit Γ-isometric dilation of a Γ-contraction (S,P). We prove the other way too, i.e., for a commuting pair (S,P) with ‖P‖⩽1 and the spectral radius of S being not greater than 2, the existence of a solution to the above equation implies that (S,P) is a Γ-contraction. We show that for a pure Γ-contraction (S,P), there is a bounded operator C with numerical radius not greater than 1, such that S=C+C⁎P. Any Γ-isometry can be written in this form where P now is an isometry commuting with C and C⁎. Any Γ-unitary is of this form as well with P and C being commuting unitaries. Examples of Γ-contractions on reproducing kernel Hilbert spaces and their Γ-isometric dilations are discussed.

Two projection methods for Skew-Hermitian operator equations

In this paper we present a projection method for solving operator equations with bounded operator in Hilbert spaces. We prove the existence of the solution for the approximate equation, and we perform the error analysis. We apply the method for solving the Cauchy integral equations in L 2 ( [ 0 , 1 ] , C ) for two cases: Galerkin projections and Kulkarni projections respectively, using a sequence of orthogonal finite rank projections. Numerical examples illustrate the theoretical results.

Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse

The method of approximate inverse is a mollification method for stably solving inverse problems. In its original form it has been developed to solve operator equations in L2-spaces and general Hilbert spaces. We show that the method of approximate inverse can be extended to solve linear, ill-posed problems in Banach spaces. This paper is restricted to function spaces. The method itself consists of evaluations of dual pairings of the given data with reconstruction kernels that are associated with mollifiers and the dual of the operator. We first define what we mean by a mollifier in general Banach spaces and then investigate two settings more exactly: the case of Lp-spaces and the case of the Banach space of continuous functions on a compact set. For both settings we present the criteria turning the method of approximate inverse into a regularization method and prove convergence with rates. As an application we refer to x-ray diffractometry which is a technique of non-destructive testing that is concerned with computing the stress tensor of a specimen. Since one knows that the stress tensor is smooth, x-ray diffractometry can appropriately be modelled by a Banach space setting using continuous functions.

Projection-Iterative Methods for a Class of Difference Equations

We develop a theoretical framework for projection-iterative methods to solve operator equations of the form Au + Bu = f, where A is a Toeplitz operator in a Banach space $$l_{p}({\mathbb{N}}) \quad (1 \leq p < \infty)$$ , B is considered as a perturbation (of general form) of A, and f is a given element in this space. The methods are adopted for application to general situations, in particular, to the equations in which A need not be a Fredholm operator. The idea to involve iteration procedures and the technique which we apply allow to obtain conditions on perturbations for convergence and effective error estimates in terms of some weighted spaces (without any restrictions on the norms for perturbations). Based on established evaluations we derive further information about decaying properties of the solutions. The obtained results are illustrated by considering concrete classes of equations as, for instance, equations corresponding to Jacobi type operators.

Mathematical models of some nonlinear systems and their computer realization

A model of an interphase layer in composite media is investigated. Functional spaces are introduced and general operator equations are constructed that include operators of interface conditions in addition to boundary conditions. Projective-iterative methods for solving operator equations in the stationary case are considered. The result of computer realization of an inhomogeneous nonlinear boundary-value problem is illustrated.

On reduction of informational expenses in solving ill-posed problems with not exactly given input data

A class of approximate methods to solve operator equations of first kind with not exactly given input data is constructed. For involved methods their optimality by the order on sets of sourcewise represented solutions is proved and the bound of informational expenses is obtained. These algorithms are numerically implemented in an efficient way. An example of application of two such algorithms is given.

Computation of differential operators in aggregated wavelet frame coordinates

Adaptive wavelet algorithms for solving operator equations have been shown to converge with the best possible rates in linear complexity. For the latter statement, all costs are taken into account, i.e. also the cost of approximating entries from the infinite stiffness matrix with respect to the wavelet basis using suitable quadrature. A difficulty is the construction of a suitable wavelet basis on the generally non-trivially shaped domain on which the equation is posed. In view of this, recently corresponding algorithms have been proposed that require only a wavelet frame instead of a basis. By employing an overlapping decomposition of the domain, where each subdomain is the smooth parametric image of the unit cube, and by lifting a wavelet basis on this cube to each of the subdomains, the union of these collections defines such a frame. A potential bottleneck within this approach is the efficient approximation of entries corresponding to pairs of wavelets from different collections. Indeed, such wavelets are piecewise smooth with respect to mutually non-nested partitions. In this paper, considering partial differential operators and spline wavelets on the subdomains, we propose an easy implementable quadrature scheme to approximate the required entries, which allows the fully discrete adaptive frame algorithm to converge with the optimal rate in linear complexity.

Inexact-Newton method for solving operator equations in infinite-dimensional spaces

In this paper, we develop an inexact-Newton method for solving nonsmooth operator equations in infinite-dimensional spaces. The linear convergence and superlinear convergence of inexact-Newton method under some conditions are shown. Then, we characterize the order of convergence in terms of the rate of convergence of the relative residuals. The present inexact-Newton method could be viewed as the extensions of previous ones with same convergent results in finite-dimensional spaces.

Wavelets with patchwise cancellation properties

We construct wavelets on general n n -dimensional domains or manifolds via a domain decomposition technique, resulting in so-called composite wavelets. With this construction, wavelets with supports that extend to more than one patch are only continuous over the patch interfaces. Normally, this limited smoothness restricts the possibility for matrix compression, and with that the application of these wavelets in (adaptive) methods for solving operator equations. By modifying the scaling functions on the interval, and with that on the n n -cube that serves as parameter domain, we obtain composite wavelets that have patchwise cancellation properties of any required order, meaning that the restriction of any wavelet to each patch is again a wavelet. This is also true when the wavelets are required to satisfy zeroth order homogeneous Dirichlet boundary conditions on (part of) the boundary. As a result, compression estimates now depend only on the patchwise smoothness of the wavelets that one may choose. Also taking stability into account, our composite wavelets have all the properties for the application to the (adaptive) solution of well-posed operator equations of orders 2 t 2 t for t ∈ ( − 1 2 , 3 2 ) t \in (-\frac {1}{2},\frac {3}{2}) .

Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces

The aim of this paper is twofold. First, several basic mathematical concepts involved in the construction and study of Bregman type iterative algorithms are presented from a unified analytic perspective. Also, some gaps in the current knowledge about those concepts are filled in. Second, we employ existing results on total convexity, sequential consistency, uniform convexity and relative projections in order to define and study the convergence of a new Bregman type iterative method of solving operator equations.

On a quasi-optimal regularized projection method for solving operator equations of the first kind

A new projection-iterative method was worked out for the numerical solution of equations of the first kind. This method consists of combining the Showalter regularization method with an adaptive projection discretization scheme. Quasi-optimality of the proposed method was proved on wide classes of ill-posed problems and its efficiency was established in the sense of volume of the discrete information used.

On Convergence Domains of Newton's and Modified Newton Methods

ABSTRACT We analyze convergence domains of Newton's and the modified Newton methods for solving operator equations in Banach spaces assuming first that the operator in question is ω-smooth in a ball centered at the starting point. It is shown that the gap between convergence domains of these two methods cannot be closed under ω-smoothness. Its exact size for Hölder smooth operators is computed. Then we proceed to investigate their convergence domains under regular smoothness. As our analysis reveals, both domains are the same and wider than their counterparts in the previous case.

Lipschitz Operators and the Solvability of Non-linear Operator Equations

Let U and V be Banach spaces, L and N be non-linear operators from U into V. L is said to be Lipschitz if L 1(L) := sup{∥Lx - Ly∥ · ∥x - y∥-1 : x ≠ y} is finite. In this paper, we give some basic properties of Lipschitz operators and then discuss the unique solvability, exact solvability, approximate solvability of the operator equations Lx = y and Lx + Nx = y. Under some conditions we prove the equivalence of these solvabilities. We also give an estimation for the relative-errors of the solutions of these two systems and an application of our method to a non-linear control system.

On the Compressibility of Operators in Wavelet Coordinates

In [Found. Comput. Math., 2 (2002), pp. 203--245], Cohen, Dahmen, and DeVore proposed an adaptive wavelet algorithm for solving operator equations. Assuming that the operator defines a boundedly invertible mapping between a Hilbert space and its dual, and that a Riesz basis of wavelet type for this Hilbert space is available, the operator equation can be transformed into an equivalent well-posed infinite matrix-vector system. This system is solved by an iterative method, where each application of the infinite stiffness matrix is replaced by an adaptive approximation. For a certain range of s>0, determined by the compressibility of the stiffness matrix, i.e., by how well it can be approximated by sparse matrices, it was proven that if the errors of best linear combinations from the wavelet bases with N terms are $\mathcal{O}(N^{-s})$, then approximations yielded by the adaptive method with N terms also have errors of $\mathcal{O}(N^{-s})$, where their computation takes only $\mathcal{O}(N)$ operations. With the available estimates for both differential and singular integral operators, the compressibility of the stiffness matrix appears to limit the rate of convergence of the adaptive method, in the sense that for solutions that have a sufficiently high (Besov) regularity, these best N-term approximations converge with a better rate than can be shown for the approximations produced by the adaptive method. In this paper, considering piecewise smooth wavelets as spline or finite element wavelets, and using modified sparse matrix approximations, we derive improved results concerning compressibility. From these results it will follow that for the full range of s for which, under appropriate smoothness conditions, convergence of the best N-term approximations of $\mathcal{O}(N^{-s})$ can be shown, the adaptive method converges with that rate.

Kantorovich's Majorization and Functional Equations

We consider systems of nonlinear difference equations arising when convergence analysis of an iterative method for solving operator equations in Banach spaces is carried out via Kantorovich's technique of majorization. The main challenge in this context is to determine the convergence domain of the corresponding majorant generator. As it turns out, dealing with this task leads to solution of functional equations of a certain kind. After considering several examples, we formulate two generic models and develop an approach to their solution.