Solving Linear Boundary Value Problems Via Non-commutative Gröbner Bases

A new approach for symbolically solving linear boundary value problems is presented. Rather than using general-purpose tools for obtaining parametrized solutions of the underlying ODE and fitting them against the specified boundary conditions (which may be quite expensive), the problem is interpreted as an operator inversion problem in a suitable Banach space setting. Using the concept of the oblique Moore—Penrose inverse, it is possible to transform the inversion problem into a system of operator equations that can be attacked by virtue of non-commutative Gröbner bases. The resulting operator solution can be represented as an integral operator having the classical Green’s function as its kernel. Although, at this stage of research, we cannot yet give an algorithmic formulation of the method and its domain of admissible inputs, we do believe that it has promising perspectives of automation and generalization; some of these perspectives are discussed.     

The Behaviour of Elastic Fields and Boundary Integral Mellin Techniques Near Conical Points

For the computation of the local singular behaviour of an homogeneous anisotropic clastic field near the three-dimensional vertex subjected to displacement boundary conditions, one can use a boundary integral equation of the first kind whose unkown is the boundary stress. Mellin transformation yields a one - dimensional integral equation on the intersection curve 7 of the cone with the unit sphere. The Mellin transformed operator defines the singular exponents and Jordan chains, which provide via inverse Mellin transformation a local expansion of the solution near the vertex. Based on Kondratiev's technique which yields a holomorphic operator pencil of elliptic boundary value problems on the cross - sectional interior and exterior intersection of the unit sphere with the conical interior and exterior original cones, respectively, and using results by Maz'ya and Kozlov, it can be shown how the Jordan chains of the one-dimensional boundary integral equation are related to the corresponding Jordan chains of the operator pencil and their jumps across γ. This allows a new and detailed analysis of the asymptotic behaviour of the boundary integral equation solutions near the vertex of the cone. In particular, the integral equation method developed by Schmitz, Volk and Wendland for the special case of the elastic Dirichlet problem in isotropic homogeneous materials could be completed and generalized to the anisotropic case.     

Computing numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions

In this article we compute numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions. A compactly perturbed half-plane Helmholtz problem is used to motivate this calculation, by treating it through integral equation techniques. These require the knowledge of the calculated Green’s function, and lead to a boundary element discretization. The Green’s function is computed using the inverse Fourier operator of its spectral transform, applying an inverse FFT for the regular part, and removing the singularities analytically. Finally, some numerical results for the Green’s function and for a benchmark resonance problem are shown.     

The construction of some efficient preconditioners in the boundary element method

The discretization of first kind boundary integral equations leads in general to a dense system of linear equations, whose spectral condition number depends on the discretization used. Here we describe a general preconditioning technique based on a boundary integral operator of opposite order. The corresponding spectral equivalence inequalities are independent of the special discretization used, i.e., independent of the triangulations and of the trial functions. Since the proposed preconditioning form involves a (pseudo)inverse operator, one needs for its discretization only a stability condition for obtaining a spectrally equivalent approximation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Inverse spectral problem for integro-differential operators

In this paper, we study the inverse spectral problem on a finite interval for the integro-differential operator ? which is the perturbation of the Sturm-Liouville operator by the Volterra integral operator. The potential q belongs to L ₂[0, π] and the kernel of the integral perturbation is integrable in its domain of definition. We obtain a local solution of the inverse reconstruction problem for the potential q, given the kernel of the integral perturbation, and prove the stability of this solution. For the spectral data we take the spectra of two operators given by the expression for ? and by two pairs of boundary conditions coinciding at one of the finite points.     

Application of the Factorization Method to Recover Cuts with Oblique Derivative Boundary Condition

Direct and inverse problems for the scattering of cracks with mixed oblique derivative boundary conditions from the incident plane wave are considered, which describe the scattering phenomenons such as the scattering of tidal waves by spits or reefs. The solvability of the direct scattering problem is proven by using the boundary integral equation method. In order to show the equivalent boundary integral system is Fredholm of index zero, some relationships concerning the tangential potential operator is used. Due to the mixed oblique derivative boundary conditions, we cannot employ the factorization method in a usual manner to reconstruct the cracks. An alternative technique is used in the theoretical analysis such that the far field operator can be factorized in an appropriate form and fulfills the range identity theorem. Finally, we present some numerical examples to demonstrate the feasibility and effectiveness of the factorization method.     

Linear boundary value problems for normally solvable operator equations in a Banach space

We consider linear boundary value problems for operator equations with generalized-invertible operator in a Banach or Hilbert space. We obtain solvability conditions for such problems and indicate the structure of their solutions. We construct a generalized Green operator and analyze its properties and the relationship with a generalized inverse operator of the linear boundary value problem. The suggested approach is illustrated in detail by an example.     

On some issues related to the moment problem for the band matrices with operator elements

The connection between the classical moment problem and the spectral theory of second order difference operators (or Jacobi matrices) is a thoroughly studied topic. Here we examine a similar connection in the case of the second order operator replaced by an operator generated by an infinite band matrix with operator elements. For such operators, we obtain an analog of the Stone theorem and consider the inverse spectral problem which amounts to restoring the operator from the moment sequence of its Weyl matrix. We establish the solvability criterion for such problems, find the conditions ensuring that the elements of the moment sequence admit an integral representation with respect to an operator valued measure and discuss an algorithm for the recovery of the operator. We also indicate a connection between the inverse problem method and the Hermite-Padé approximations.     

Reconstruction of the Dirac-Type Integro-Differential Operator From Nodal Data

Abstract

The inverse nodal problem for Dirac type integro-differential operator with the spectral parameter in the boundary conditions is studied. We prove that dense subset of the nodal points determines the coefficients of differential part of operator and gives partial information for integral part of it.     

Linear Noetherian boundary value problems for differential systems with an impulse action

A solvability criterion of linear nonhomogeneous boundary value problems is obtained for systems of ordinary differential equations with impulse action in the general case when the number of boundary conditions does not coincide with the order of the differential system (Noetherian problems). The generalized Green operator of such boundary value problems is constructed. Its connection with the generalized inverse operator of the operator of the original boundary value problem is shown.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 564–568, April, 1992.     

Hierarchical Interpolative Factorization for Elliptic Operators: Integral Equations

This paper introduces the hierarchical interpolative factorization for integral equations (HIF‐IE) associated with elliptic problems in two and three dimensions. This factorization takes the form of an approximate generalized LU decomposition that permits the efficient application of the discretized operator and its inverse. HIF‐IE is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features: (1) a matrix factorization framework for sparsifying structured dense matrices and (2) a recursive dimensional reduction strategy to decrease the cost. Thus, higher‐dimensional problems are effectively mapped to one dimension, and we conjecture that constructing, applying, and inverting the factorization all have linear or quasilinear complexity. Numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method, direct solver, and preconditioner. HIF‐IE is compatible with geometric adaptivity and can handle both boundary and volume problems.© 2016 Wiley Periodicals, Inc.     

Numerical methods for some inverse problems of heart electrophysiology

We consider numerical methods for solving inverse problems that arise in heart electrophysiology. The first inverse problem is the Cauchy problem for the Laplace equation. Its solution algorithm is based on the Tikhonov regularization method and the method of boundary integral equations. The second inverse problem is the problem of finding the discontinuity surface of the coefficient of conductivity of a medium on the basis of the potential and its normal derivative given on the exterior surface. For its numerical solution, we suggest a method based on the method of boundary integral equations and the assumption on a special representation of the unknown surface.     

On the convergence of a numerical method for a hypersingular integral equation on a closed surface

We consider a linear integral equation with a hypersingular integral treated in the sense of the Hadamard finite value. This equation arises in the solution of the Neumann boundary value problem for the Laplace equation with a representation of a solution in the form of a double-layer potential. We consider the case in which the interior or exterior boundary value problem is solved in a domain; whose boundary is a smooth closed surface, and an integral equation is written out on that surface. For the integral operator in that equation, we suggest quadrature formulas like the method of vortical frames with a regularization, which provides its approximation on the entire surface for the use of a nonstructured partition. We construct a numerical scheme for the integral equation on the basis of suggested quadrature formulas, prove an estimate for the norm of the inverse matrix of the related system of linear equations and the uniform convergence of numerical solutions to the exact solution of the hypersingular integral equation on the grid.     

Inverse Problem for a Fourth-Order Differential Operator with Nonseparated Boundary Conditions

Uniqueness theorems are proved for two inverse problems for a fourth-order differential operator with nonseparated boundary conditions. The first of the problems, which has technical applications, is the problem of identification of a differential equation and two boundary conditions, and the second problem is the problem of identification of a differential equation and four boundary conditions. One of two data sets is used as the spectral data of the problem. The first data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectral data of a system of three problems, and the second data set is the spectrum of the problem itself (or three of its eigenvalues) and the spectra of ten boundary value problems.     

Inverse spectral problem for the operators with non‐local potential

The main object under consideration in the paper is the second derivative operator on a finite interval with zero boundary conditions perturbed by a self‐adjoint integral operator with the degenerate kernel (non‐local potential). The inverse problem, i.e., the reconstruction of the perturbation from the spectral data, is solved by means of the step‐by‐step procedure based on the n‐interlacing property of the spectrum.     

Singular integral operators in a weighted L2-space

Cauchy singular integral operators are characterized as operators in a weighted L²-space. The integral operator arises from a singular integral equation with variable coefficients. An appropriate weight function associated with the singular integral operator is constructed, and the set of polynomials orthogonal with respect to this weight function is defined. The action of the operator on polynomial sets is studied, and the definition of the operator is extended to a weighted L²-space. In this space, the operator is shown to be bounded, and, in some cases, isometric. Formulas are developed for the composition of the singular integral operator and its one sided inverse.     

Nonstrictly hyperbolic systems of conservation laws of the conjugate type

We consider an inverse boundary problem for a general second order self-adjoint elliptic differential operator on a compact differential manifold with boundary. The inverse problem is that of the reconstruction of the manifold and operator via all but finite number of eigenvalues and traces on the boundary of the corresponding eigenfunctions of the operator. We prove that the data determine the manifold and the operator to within the group of the generalized gauge transformations. The proof is based upon a procedure of the reconstruction of a canonical object in the orbit of the group. This object, the canonical Schrödinger operator, is uniquely determined via its incomplete boundary spectral data.     

Inverse Problem for a Differential Operator with Nonseparated Boundary Conditions

Uniqueness theorems for the solution of an inverse problem for a fourth-order differential operator with nonseparated boundary conditions are proved. The spectral data for the problem is specified as the spectrum of the problem itself (or its three eigenvalues) and the spectral data of a system of three problems.     

Nonlocal boundary problems for a third-order one-dimensional nonlinear pseudoparabolic equation

We consider initial boundary value problems for a third-order nonlinear pseudoparabolic equation with one space dimension. The boundary condition is given by an integral; the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet or Neumann counterparts. By means of appropriate elliptic estimates we are able to seek solutions not only in the weighted spaces but also in the usual Sobolev spaces. The procedure is carried out in a unified way. Our results characterize a regularity of the pseudoparabolic operator that is different from that of the parabolic operator.     

Neumann problem with the integro-differential operator in the boundary condition

The Neumann problem for a second-order parabolic equation with integro-differential operator in the boundary condition is considered. A well-posedness theorem is proved, in particular, the integral representation of the solution is obtained, estimates for the derivatives of the solution are established, and the kernel of the inverse operator of the problem is explicitly expressed.