Robust univariate spline models for interpolating interval data

We consider spline interpolation problems where information about the approximated function is given by means of interval estimates for the function values over ranges of x-values instead of specific knots. We propose two robust univariate spline models formulated as convex semi-infinite optimization problems. We present simplified equivalent formulations of both models as finite explicit convex optimization problems for splines of degrees up to 3. This makes it possible to use existing convex optimization algorithms and software.     

Applications of some formulas for finite Markov chains

We present some explicit matrix formulas for a finite state Markov chain. The first gives sums of probabilities along some general subsets of paths. Another formula yields the probability mass function (pmf) of the random variable which adds costs along subsets of paths. We then discuss how these formulas can be used to efficiently compute expected values of a function of the sum of costs along paths, as well as related applications. We conclude by describing a procedure allowing us to avoid using Monte Carlo simulation in stochastic approaches to solving some general boundary value problems. Instead, we show how to evaluate the relevant expected values exactly for discretizations of the original continuous problem.     

A sand surface element for efficient modeling of residual stress in castings

This article describes a method to efficiently predict residual stresses in foundry castings. We replace explicit modeling of the sand mold during stress analysis by using a locally applied boundary condition element. Methods are given for assigning appropriate values to the individual elements, and we present several example problems. In these test problems, we also evaluate several constitutive models for the sand. We find that plastic strains and displacements in the casting do not depend strongly on the sand constitutive model. Further, the stress states produced, when using the new surface elements, are generally in good agreement with those found using a full mold. The computational advantages of the new element are demonstrated.     

A parametric symmetric linear four-step method for the efficient integration of the Schrödinger equation and related oscillatory problems

In this article, we develop an explicit symmetric linear phase-fitted four-step method with a free coefficient as parameter. The parameter is used for the optimization of the method in order to solve efficiently the Schrödinger equation and related oscillatory problems. We evaluate the local truncation error and the interval of periodicity as functions of the parameter. We reveal a direct relationship between the periodicity interval and the local truncation error. We also measure the efficiency of the new method for a wide range of possible values of the parameter and compare it to other well known methods from the literature. The analysis and the numerical results help us to determine the optimal values of the parameter, which render the new method highly efficient.     

Diffusions on graphs, Poisson problems and spectral geometry

We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. We associate to certain subgraphs (domains) a pair of sequences, each of which is invariant under the action of the automorphism group of the underlying graph. We prove that these invariants differ by an explicit combinatorial factor given by Stirling numbers of the first and second kind. We prove that for any domain with a natural weighting, these invariants determine the eigenvalues of the Laplace operator corresponding to eigenvectors with nonzero mean. As a specific example, we investigate the relationship between our invariants and heat content asymptotics, expressing both as special values of an analog of a spectral zeta function.

     

Values of zeta functions at negative integers, Dedekind sums and toric geometry

We study relations among special values of zeta functions, invariants of toric varieties, and generalized Dedekind sums. In particular, we use invariants arising in the Todd class of a toric variety to give a new explicit formula for the values of the zeta function of a real quadratic field at nonpositive integers. We also express these invariants in terms of the generalized Dedekind sums studied previously by several authors. The paper includes conceptual proofs of these relations and explicit computations of the various zeta values and Dedekind sums involved.

     

On Harmonic Continuation of Differentiable Functions Defined on Part of the Boundary

We establish an explicit formula for reconstruction of a harmonic function in a domain from its values and the values of its normal derivative on part of the boundary; i.e., we give an explicit solution to the Cauchy problem for the Laplace equation.     

Nonoverlapping domain decomposition and additive splitting up methods for multidimensional parabolic problem

The explicit implicit domain decomposition methods are noniterative types of methods for nonoverlapping domain decomposition but due to the use of the explicit step for the interface prediction, the methods suffer from inaccuracy of the usual explicit scheme. In this article a specific type of first‐ and second‐order splitting up method, of additive type, for the dependent variables is initially considered to solve the two‐ or three‐dimensional parabolic problem over nonoverlapping subdomains. We have also considered the parallel explicit splitting up algorithm to define (predict) the interface boundary conditions with respect to each spatial variable and for each nonoverlapping subdomains. The parallel second‐order splitting up algorithm is then considered to solve the subproblems defined over each subdomain; the correction step will then be considered for the predicted interface nodal points using the most recent solution values over the subdomains. Finally several model problems will be considered to test the efficiency of the presented algorithm. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On Harmonic Continuation of Differentiable Functions Defined on Part of the Boundary

We establish an explicit formula for reconstruction of a harmonic function in a domain from its values and the values of its normal derivative on part of the boundary; i.e., we give an explicit solution to the Cauchy problem for the Laplace equation.     

Nonlinear wave analysis of geochemical injection for multicomponent two phase flow in porous media

We study nonlinear waves in two phase geochemical flow in one spatial dimension in porous media. We assume that each chemical species may flow in one or both phases with concentrations obeying thermodynamical equilibrium. We present a new methodology for reducing a number of equations applicable to injection problems for general systems of conservation laws in Geochemistry. This reduction is achieved by solving a nonlinear inverse problem. Nevertheless, we are able to perform a complete and explicit characteristic analysis, obtaining rarefaction and shock waves that are used to solve the representative Riemann problems, besides the main bifurcations structures appearing in the phase space are derived with explicit expressions. We illustrate the methodology by means of an example with four equations.     

A Carleman Function and the Cauchy Problem for the Laplace Equation

We suggest an explicit formula for reconstruction of a harmonic function in a domain from its values and the values of its normal derivative on part of the boundary; i.e., we give an explicit continuation formula and a regularization procedure for a solution to the Cauchy problem for the Laplace equation.     

Lattice games without rational strategies

We show that the lattice games of Guo and Miller support universal computation, disproving their conjecture that all lattice games have rational strategies. We also state an explicit counterexample to that conjecture: a three dimensional lattice game whose set of winning positions does not have a rational generating function.     

Singular value expansion for the Green function of Helmholtz operator

We consider the integral operator defined on a circular disk, and with kernel the Green function of the Helmholtz operator. We present an analytic framework for the explicit computation of the singular system of this kernel. In particular, the main formulas of this framework are given by a characteristic equation for the singular values and explicit expressions for the corresponding singular functions. We provide also a property of the singular values, that gives an important information for the numerical evaluation of the singular system. Finally, we present a simple numerical experiment, where the singular system computed by a simple implementation of these analytic formulas is compared with the singular system obtained by a discretization of the Green function of the Helmholtz operator.     

Non convex control problems in Banach spaces

We consider a general control problem which includes, as particular cases, Bolza, Lagrange and Mayer problems. We show that it can be reduced to a free problem and we give sufficient conditions for the existence of a minimum over all absolutely continuous arcs with values in a reflexive, separable Banach space. A regularization result is also proved and an application to explicit control problems is considered.This work was supported by the Laboratorio per la Matematica Applicata del C. N. R.-Istituto di Matematica della Università di Genova.     

A Variational Approach to Nonlinear Singularly Perturbed Boundary-Value Problems

Carrier and Pearson introduced a nonlinear singularly perturbed boundary value problem that has served as a paradigm for problems where the method of matched asymptotic expansions (MAE) apparently fails. The “failure” of MAE is its inability to select the location of possible internal layers, though their structure is determined. Thus, a straightforward application of MAE leaves the positions of any internal layers arbitrary, though the asymptotic expansion of the exact solution to the problem exhibits internal layers only at specific locations. For this reason the solutions produced by MAE have been referred to as spurious solutions. We resolve the question of finding the positions of the interior layers by employing the variational approach of Grasman and Matkowsky. In addition, we show that this method tells how solutions bifurcate as the boundary values are varied, and give an alternative motivation for the variational approach via Newton”s method.     

Spectral Deferred Correction Methods for Ordinary Differential Equations

We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For non-stiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a state-of-the-art extrapolation code (at least, at moderate to high precision).Deferred correction methods based on the Picard equation appear to be promising candidates for further investigation.     

Using multiple sources of information to recognize and classify objects

In this paper we consider the problem of selecting an object or a course of action from a set of possible alternatives. To give the paper focus, we concentrate initially on an object recognition problem in which the characteristic features of the object are reported by remote sensors. We then extend the method to a more general class of selection problems and consider several different scenarios.

Information is provided by a set of knowledge system reports on a single feature, and the output from these systems is not totally explicit but provides posible values for the observed feature along with a degree of certitude.We use fuzzy sets to represent this vague information. Information from independent sources is combined using the Dempster-Shafer approach adapted to the situation in which the focal elements are fuzzy as in the recent paper by J. Yen [7]. We base our selection rule on the belief and plausibility functions generated by this approach to accessing evidence.

For situations in which the information is too sparse and/or too vague to make a single selection, we construct a preference relationship based on the concept of averaged subsethood for fuzzy sets as discussed by B. Koskoin [4]. We also define an explicit metric upon which to base our selection mechanism for situations in which the Dempster-Shafer rule of combination is inappropriate     

On Unique Solutions of Multiple-State Optimal Design Problems on an Annulus

We study the uniqueness and explicit derivation of the relaxed optimal solutions, corresponding to the minimization of weighted sum of potential energies for a mixture of two isotropic conductive materials on an annulus. Recently, it has been shown by Burazin and Vrdoljak that even for multiple-state problems, if the domain is spherically symmetric, then the proper relaxation of the problem by the homogenization method is equivalent to a simpler relaxed problem, stated only in terms of local proportions of given materials. This enabled explicit calculation of a solution on a ball, while problems on an annulus appeared to be more tedious. In this paper, we discuss the uniqueness of a solution of this simpler relaxed problem, when the domain is an annulus and we use the necessary and sufficient conditions of optimality to present a method for explicit calculation of the unique solution of this simpler proper relaxation, which is demonstrated on an example.     

Continuous extensions to Nyström methods for second order initial value problems

In this work we consider interpolants for Nyström methods, i.e., methods for solving second order initial value problems. We give a short introduction to the theory behind the discrete methods, and extend some of the work to continuous, explicit Nyström methods. Interpolants for continuous, explicit Runge-Kutta methods have been intensively studied by several authors, but there has not been much effort devoted to continuous Nyström methods. We therefore extend some of the work by Owren.     

On the Solution of Some Non-Local Problems

This paper deals with two types of non-local problems for the Poisson equation in the disc. The first of them deals with the situation when the function value on the circle is given as a combination of unknown function values in the disc. The other type deals with the situation when a combination of the value of the function and its derivative by radius on the circle are given as a combination of unknown function values in the disc. The existence and uniqueness of the classical solution of these problems is proved. The solutions are constructed in an explicit form.