Indefinite Operator Research Articles

Characterizations of monotone [formula omitted]-regularly varying functions by means of indefinite eigenvalue problems and HELP type inequalities

Connections between different areas of mathematical analysis are obtained: regular variation, indefinite operator theory and HELP type inequalities. Using a result by Parfyonov, the nondecreasing and so-called positively increasing functions induce precisely those indefinite Kreĭn–Feller eigenvalue problems such that the eigenfunctions have the Riesz basis property. Furthermore, these functions induce precisely those Lebesgue–Stieltjes measures such that the associated HELP type inequalities are valid. The so-called dual Kreĭn–Feller eigenvalue problems allow a similar characterization of the class of nondecreasing O-regularly varying functions.

P-Adic Mikusinski calculus and its applications to the fourier and the mahler expansions of locally constant functions

We consider the p-adic counterpart of Mikusinski’s operational calculus based on the algebra C(ℤp) of continuous functions on ℤp taking values in ℂp and equipped with the discrete Laplace convolution. Elements of the field (hyperfunctions) corresponding to shift operators, difference operators, and the indefinite sum operator are considered. A notion of p-adic exponent is generalized. Applications to the Fourier and the Mahler expansions of the indicator function of a ball and the convolution of two indicator functions are provided. Two ways of applying the p-adic analog of Mikusinski’s operational calculus lead us to the Fourier expansion for the fractional part of a p-adic number.

Stability of operator splitting methods for systems with indefinite operators: Advection–diffusion–reaction systems

This brief paper presents an A-stability result for operator splitting type time integration methods applied to advection–diffusion–reaction equations with possibly indefinite source terms. These results extend our earlier work on diffusion–reaction systems [D.L. Ropp, J.N. Shadid, Stability of operator splitting methods for systems with indefinite operators: reaction–diffusion systems, J. Comput. Phys. 203 (2) (2005) 449–466]. The A-stability result presents sufficient conditions that control both low and high wave number instabilities. A corollary shows that if L-stable methods are used for the diffusion term the high wave number instability will be controlled more easily. Numerical results are presented that verify second-order convergence for the operator splitting methods and demonstrate control of instabilities on a chemotaxis problem by use of an L-stable diffusion integrator.

A note on four-point partial difference equations with variable coefficients

The aim of this note is to find the explicit solution of four-point linear partial difference equations with variable coefficients. Three-point and four-point cases in quadrant and cone domains are considered and the boundary conditions, allowing the existence and the uniqueness of the solution, are detailed. The adopted procedure is a double iteration and the solution consists of a product of series, whose number of factors may be reduced successively by means of the indefinite sum operator of the difference calculus.

Multilevel Preconditioners for Non--self-adjoint or Indefinite Orthogonal Spline Collocation Problems

Efficient numerical algorithms are developed and analyzed that implement symmetric multilevel preconditioners for the solution of an orthogonal spline collocation (OSC) discretization of a Dirichlet boundary value problem with a non--self-adjoint or an indefinite operator. The OSC solution is sought in the Hermite space of piecewise bicubic polynomials. It is proved that the proposed additive and multiplicative preconditioners are uniformly spectrally equivalent to the operator of the normal OSC equation. The preconditioners are used with the preconditioned conjugate gradient method, and numerical results are presented that demonstrate their efficiency.

Stability of operator splitting methods for systems with indefinite operators: reaction-diffusion systems

In this paper numerical results are reviewed [D.L. Ropp, J.N. Shadid, C.C. Ober, Studies of the accuracy of time integration methods for reaction-diffusion equations, J. Comput. Phys. 194 (2) (2004) 544–574] that demonstrate that common second-order operator-splitting methods can exhibit instabilities when integrating the Brusselator equations out to moderate times of about seven periods of oscillation. These instabilities are manifested as high wave number spatial errors. In this paper, we further analyze this problem, and present a theorem for stability of operator-splitting methods applied to linear reaction-diffusion equations with indefinite reaction terms which controls both low and high wave number instabilities. A corollary shows that if L-stable methods are used for the diffusion term the high wave number instability will be controlled more easily. In the absence of L-stability, an additional time step condition that suppresses the high wave number modes appears to guarantee convergence at the asymptotic order for the operator-splitting method. Numerical results for a model problem confirm this theory, and results for the Brusselator problem agree as well.

Analysis of a Multigrid Algorithm for Time Harmonic Maxwell Equations

This paper considers a multigrid algorithm suitable for efficient solution of indefinite linear systems arising from finite element discretization of time harmonic Maxwell equations. In particular, a backslash multigrid cycle is proven to converge at rates independent of refinement level if certain indefinite block smoothers are used. The method of analysis involves comparing the multigrid error reduction operator with that of a related positive definite multigrid operator. This idea has previously been used in multigrid analysis of indefinite second order elliptic problems. However, the Maxwell application involves a nonelliptic indefinite operator. With the help of a few new estimates, the earlier ideas can still be applied. Some numerical experiments with lowest order Nedelec elements are also reported.

A Preconditioned Conjugate Gradient Method for Nonselfadjoint or Indefinite Orthogonal Spline Collocation Problems

We study the computation of the orthogonal spline collocation solution of a linear Dirichlet boundary value problem with a nonselfadjoint or an indefinite operator of the form $Lu=\sum a_{ij}(x)u_{x_ix_j}+\sum b_i(x) u_{x_i}+c(x)u$. We apply a preconditioned conjugate gradient method to the normal system of collocation equations with a preconditioner associated with a separable operator, and prove that the resulting algorithm has a convergence rate independent of the partition step size. We solve a problem with the preconditioner using an efficient direct matrix decomposition algorithm. On a uniform N × N partition, the cost of the algorithm for computing the collocation solution within a tolerance $\epsilon$ is O$(N^{2}\ln N |\ln \epsilon|)$.

Kernel polynomials for the solution of indefinite and ill-posed problems

We introduce a new family of semiiterative schemes for the solution of ill-posed linear equations with selfadjoint and indefinite operators. These schemes avoid the normal equation system and thus benefit directly from the structure of the problem. As input our method requires an enclosing interval of the spectrum of the indefinite operator, based on some a priori knowledge. In particular, for positive operators the schemes are mathematically equivalent to the so-called ν-methods of Brakhage. In a way, they can therefore be seen as appropriate extensions of the ν-methods to the indefinite case. This extension is achieved by substituting the orthogonal polynomials employed by Brakhage in the definition of the ν-methods by appropriate kernel polynomials. We determine the rate of convergence of the new methods and establish their regularizing properties.

Response to “Comment on ‘Generalized W∞ symmetry algebra of the conditionally integrable nonlinear evolution equation’ ” [J. Math. Phys. 40, 3685 (1999)

To verify the correctness of the high order symmetries for the Jimbo–Miwa–Kadomtsev–Petviashvilli system (JMKP) and the potential JMKP (PJMKP) system given in our paper [J. Math. Phys. 36, 3492–3497 (1995)], Ma’s definition of ∂x−1 may be used only for the general nonkernel solutions of the models. For some types of special solutions which are the kernels of some differential operators, one has to use ∂x−1 as the indefinite operator and selected the integral functions appropriately.

A tutorial in elementary finite element error analysis: A systematic presentation of a priori and a posteriori error estimates

This paper provides a tutorial on the derivation of a priori and explicit residual-based a posteriori error estimates for Galerkin finite element discretizations of general linear elliptic operators. For simplicity, the presentation is in one dimension, although extensions to multidimensions should be straightforward. The structures of explicit residual-based a posteriori error estimators and a priori error estimators are very similar—they are each in the form of an upper bound on the error measured in some norm. The structures of their derivations also follow similar lines. It is therefore appropriate to present and discuss a priori and a posteriori error estimators in parallel. In this paper error bounds are obtained for the error measured in the energy norm and the L 2 norm. In addition, symmetric, nonsymmetric, positive definite and indefinite operators are discussed. A priori estimates for nonsymmetric operators are treated with a novel approach involving the introduction of the skew norm. Special attention is given to the advection-diffusion equation (a nonsymmetric operator) and the Helmholtz equation (an indefinite operator). We systematically describe the necessary steps in deriving a priori and a posteriori error estimates, and provide a simplified understanding of their differences as well as their similarities.

Evaluation of Queries under Closed-World Assumption

The minimal entailment vMin has been characterized elsewhere by P vMin} p iff Cn(P ∪ {p}) ∩ Pos ⊆ Cn(P) where Cn is the first-order consequence operation, P is a set of clauses (indefinite deductive data base; in short: a data base), p is a clause (a query), and Pos is the set of positive (that is, bodiless) ground clauses. In this paper, we address the problem of the computational feasibility of criterion (1). Our objective is to find a query evaluation algorithm that decides P vMin p by what we call indefinite modeling, without actually computing all ground positive consequences of P and P ∪ {p}. For this purpose, we introduce the concept of minimal indefinite Herbrand model MP of P, which is defined as the set of subsumption-minimal ground positive clauses provable from P. The algorithm first computes MP by finding the least fixed-point of an indefinite consequence operator μ TIP. Next, the algorithm verifies whether every ground positive clause derivable from MP ∪ {p} by one application of the parallel positive resolution rule (in short: the PPR rule) is subsumed by an element of MP. We prove that the PPR rule, which can derive only positive clauses, is positively complete, that is, every positive clause provable from a data base P is derivable from P by means of subsumption and finitely many applications of PPR. From this we conclude that the presented algorithm is partially correct and that it eventually halts if both P and MP are finite. Moreover, we indicate how the algorithm can be modified to handle data bases with infinite indefinite Herbrand models. This modification leads to a concept of universal model that allows for nonground clauses in its Herbrand base and appears to be a good candidate for representation of indefinite deductive data bases.

The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems

The appearance of unphysical "spurious solutions" in addition to the true physical modes of a resonator or waveguide is a problem which pertains to a variety of numerical methods for the electromagnetic field eigenvalue problem e.g. finite-elements, boundary-elements, finite-differences, point-matching and the spectral domain method. The common cause of the problem is shown to be a misconception of the discretization of a parameter dependent indefinite operator. In particular the application of the method of moments (MoM) and equivalent procedures to problems which they are not suited for. The method of least squares with intermediate projection (MLSIP) is described as one reliable approach for discretization of parameter dependent indefinite operators.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Boundary modifications of the dissipation operators for the three-dimensional Euler equations

Explicit methods for the solution of fluid flow problems are of considerable interest in supercomputing. These methods parallelize well. The treatment of the boundaries is of particular interest with respect to both the numeric behavior of the solution and the computational efficiency. We have solved the three-dimensional Euler equations for a twisted channel using second-order, centered difference operators, and a three-stage Runge-Kutta method for the integration. Three different fourth-order dissipation operators were studied for numeric stabilization: one positive definite, one positive semidefinite, and one indefinite. The operators only differ in the treatment of the boundary. For computational efficiency all dissipation operators were designed with a constant bandwidth in matrix representation, with the bandwidth determined by the operator in the interior. The positive definite dissipation operator results in a significant growth in entropy close to the channel walls. The other operators maintain constant entropy. Several different implementations of the semidefinite operator obtained through factoring of the operator were also studied. We show the difference both in convergence rate and robustness for the different dissipation operators, and the factorizations of the operator due to Eriksson. For the simulations in this study one of the factorizations of the semidefinite operator required 70%–90% of the number of iterations required by the positive definite operator. The indefinite operator was sensitive to perturbations in the inflow boundary conditions. The simulations were performed on a 8,192 processor Connection Machine system CM-2. Full processor utilization was achieved, and a performance of 135 Mflops/sec in single precision was obtained. A performance of 1.1 Gflops/sec for a fully configured system with 65,536 processors was demonstrated.

Variational calculations for scattering lengths

Approximations for scattering lengths are calculated using variational methods. In most cases bounds for the scattering are obtained. The calculations extend the usual theory to deal with indefinite operators and potentials and in all cases good approximations are achieved.