N-dimensional Problem Research Articles

Selections of shape functions for dimensional reduction to Helmholtz's equation

The boundary value problem of Helmholtz's equation on a n + 1 dimensional thin slab is approximated by appropriate systems of the n-dimensional boundary value problem. The very detailed estimates for modeling error in the H1-norm demonstrate convergence when the thickness of the slab approaches 0 as well as when the size of the systems approaches infinity. Shape functions through the thickness are first selected by finitely many eigenfunctions, and the tail is then selected to consist of polynomials. The presence of two types of functions gives rise to a certain choice in the selection of a particular set of shape functions. Numerical results provide a good illustration of the effect of different choices for specific problems. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 169–190, 1999

Duality, triality and complementary extremum principles in non-convex parametric variational problems with applications

This paper presents a reasonably complete duality theory and a nonlinear dual transformation method for solving the fully nonlinear, non-convex parametric variational problem inf{W(Λu - μ) - F(u)}, and associated nonlinear boundary value problems, where Λ is a nonlinear operator, W is either convex or concave functional of p = Λu, and μ is a given parameter. Detailed mathematical proofs are provided for the complementary extremum principles proposed recently in finite deformation theory. A method for obtaining truly dual variational principles (without a dual gap and involving the dual variable p* of Λu only) in n-dimensional problems is proposed. It is proved that for convex W(p), the critical point of the associated Lagrangian Lμ(u, p*) is a saddle point if and only if the so-called complementary gap function is positive. In this case, the system has only one dual problem. However, if this gap function is negative, the critical point of the Lagrangian is a so-called super-critical point, which is equivalent to the Auchmuty's anomalous critical point in geometrically linear systems. We discover that, in this case, the system may have more than one primal-dual set of problems. The critical point of the Lagrangian either minimizes or maximizes both primal and dual problems. An interesting triality theorem in non-convex systems is proved, which contains a minimax complementary principle and a pair of minimum and maximum complementary principles. Applications in finite deformation theory are illustrated. An open problem left by Hellinger and Reissner is solved completely and a pure complementary energy principle is constructed. It is proved that the dual Euler-Lagrange equation is an algebraic equation, and hence, a general analytic solution for non-convex variational-boundary value problems is obtained. The connection between nonlinear differential equations and algebraic geometry is revealed. Keywords:duality theory; triality; complementary energy; nonlinear variational problem; non-convex optimization; fully nonlinear system; finite deformation theory; partial differential equations; analytic solution; phase transitions; variational inequality

On behavior of perfect elastoplasticity under rectilinear paths

A perfect elastoplastic model for the relation of generalized stresses and their conjugate generalized strain rates is investigated in the paper. The exact solutions are solved for the model subjected to rectilinear generalized strain paths1 and the concepts of response subspace and visualization are introduced such that the n-dimensional problem is reduced to a two-dimensional problem in the response subspace. The response path is found to be, geometrically speaking, a geodesic in the n-dimensional closed ball Bn of generalized stresses. The phase portrait of dissipation is studied and it is a remarkable fact that in the response subspace the coordinate x is exactly the dissipation power per unit generalized strain rate when x ⩾ xon [see eqn (63)]. The generalized stress-strain curves are classified by geometrical shapes into ten types, in contrast to the superficial impression of only one type of shape, an inclined straight line followed by a horizontal line. The existence of a limit strength vector and a limit power of dissipation is demonstrated by the long-term behavior based on the exact solutions, and the existence together with the response's stability is further verified by Lyapunov's direct method. The closeness to the limit strength point is also estimated and the exact formula accounted for prestress is derived. The dependence of the responses on the product space of initial values, input paths, property constants and closeness to the limit is thoroughly investigated with control parameters identified.

Optimal Control of a Coercive Dirichlet Problem

In this paper, the maximum principle for some n-dimensional coercive Dirichlet problem of the second order is proved and sufficient conditions for the existence of an optimal solution are given. The results obtained generalize, in the sense of the dimension of the state space, some special case of the maximum principle for the one-dimensional Dirichlet problem, derived in [M. Goebel and V. Raitums, J. Global Optim., 4 (1994), 367--395].

On supernode transformation with minimized total running time

With the objective of minimizing the total execution time of a parallel program on a distributed memory parallel computer, this paper discusses how to find an optimal supernode size and optimal supernode relative side lengths of a supernode transformation (also known as tiling). We identify three parameters of supernode transformation: supernode size, relative side lengths, and cutting hyperplane directions. For algorithms with perfectly nested loops and uniform dependencies, for sufficiently large supernodes and number of processors, and for the case where multiple supernodes are mapped to a single processor, we give an order n polynomial whose real positive roots include the optimal supernode size. For two special cases, 1) two-dimensional algorithm problems and 2) n-dimensional algorithm problems, where the communication cost is dominated by the startup penalty and, therefore, can be approximated by a constant, we give a closed form expression for the optimal supernode size, which is independent of the supernode relative side lengths and cutting hyperplanes. For the case where the algorithm iteration index space and the supernodes are hyperrectangular, we give closed form expressions for the optimal supernode relative side lengths. Our experiment shows a good match of the closed form expressions with experimental data.

Multilayer neural networks and Bayes decision theory

There are many applications of multilayer neural networks to pattern classification problems in the engineering field. Recently, it has been shown that Bayes a posteriori probability can be estimated by feedforward neural networks through computer simulation. In this paper, Bayes decision theory is combined with the approximation theory on three-layer neural networks, and the two-category n-dimensional Gaussian classification problem is studied. First, we prove theoretically that three-layer neural networks with at least 2 n hidden units have the capability of approximating the a posteriori probability in the two-category classification problem with arbitrary accuracy. Second, we prove that the input–output function of neural networks with at least 2 n hidden units tends to the a posteriori probability as Back-Propagation learning proceeds ideally. These results provide a theoretical basis for the study of pattern classification by computer simulation.

Dubins' problem is intrinsically three-dimensional

In his 1957 paper [1] L. Dubins considered the problem of finding shortest differentiable arcs in the plane with curvature bounded by a constant and prescribed initial and terminal positions and tangents. One can generalize this problem to non-euclidean manifolds as well as to higher dimensions (cf. [15]). Considering that the boundary data - initial and terminal position and tangents - are genuinely three-dimensional, it seems natural to ask if the n-dimensional problem always reduces to the three-dimensional case. In this paper we will prove that this is true in the euclidean as well as in the noneuclidean case. At first glance one might consider this a trivial problem, but we will also give an example showing that this is not the case.

A Fundamental Study On the Statistical Evaluation of Receiver Response for an Electromagnetic Wave Environment in Multi-Dimensional Signal Space- Theory and Basic Experiment

In this paper, a statistical electromagnetic (abbr., EM) interference study is constructed mainly from a theoretical viewpoint on the basis of N-dimensional random walk problem in multi-dimensional signal space. First, a characteristic function of the Hankel transform type matched to this EM environmental study is introduced especially in an extended form of D. Middleton's basic result. Then, the probability density function (abbr., pdf) is explicitly derived for the EM random fluctuation with a power scaled variable (or an effective value) measured by a receiver with a mean square operation. The proposed theory is verified by showing the agreement with one of D. Middleton's great researches as a special case of the proposed expressions. Moreover, the effectiveness of the proposed method is experimentally confirmed through an acoustic type simulation of EM environment and the application to a simplified actual EM environment with joint space-time non-Poisson interference as one of the same wave motion type environment as EM environment.

A fast and accurate imaging algorithm in optical/diffusion tomography

An n-dimensional (n = 2,3) inverse problem for the parabolic/diffusion equation , , , is considered. The problem consists of determining the function a(x) inside of a bounded domain given the values of the solution u(x,t) for a single source location on a set of detectors , where is the boundary of . A novel numerical method is derived and tested. Numerical tests are conducted for n = 2 and for ranges of parameters which are realistic for applications to early breast cancer diagnosis and the search for mines in murky shallow water using ultrafast laser pulses. The main innovation of this method lies in a new approach for a novel linearized problem (LP). Such a LP is derived and reduced to a well-posed boundary-value problem for a coupled system of elliptic partial differential equations. A principal advantage of this technique is in its speed and accuracy, since it leads to the factorization of well conditioned, sparse matrices with non-zero entries clustered in a narrow band near the diagonal. The authors call this approach the elliptic systems method (ESM). The ESM can be extended to other imaging modalities.

A genetic algorithm based multi-dimensional data association algorithm for multi-sensor—multi-target tracking

The central problem in multitarget-multisensor tracking is the data association problem of partitioning the observations into tracks and false alarms so that an accurate estimate of true tracks can be found. The data association problem is formed as an N-dimensional ( N-D) assignment problem, which is a state-of-the-art method and is NP-hard for N ≥ 3 sensor scans. This paper proposes a new genetic algorithm for solving the above problem which is typically encountered in the application of target tracking. The data association capacities of the genetic algorithm have been studied in different environments, and the results are presented.

Bifurcations of hyperbolic fixed points for explicit Runge-Kutta methods

University of Trier, Department of Mathematics, D-54286 Trier, Germany[Received 20 April 1994 and in revised form 27 March 1996]Discretization of autonomous ordinary differential equations by numerical methodsmight, for certain step sizes, generate solution sequences not corresponding tothe underlying flow—so-called 'spurious solutions' or 'ghost solutions'. In thispaper we explain this phenomenon for the case of explicit Runge-Kutta methodsby application of bifurcation theory for discrete dynamical systems. An importanttool in our analysis is the domain of absolute stability, resulting from theapplication of the method to a linear test problem. We show that hyperbolicfixed points of the (nonlinear) differential equation are inherited by the differencescheme induced by the numerical method while the stability type of theseinherited genuine fixed points is completely determined by the method's domainof absolute stability. We prove that, for small step sizes, the inherited fixed pointsexhibit the correct stability type, and we compute the corresponding limit stepsize. Moreover, we show in which way the bifurcations occurring at the limitstep size are connected to the values of the stability function on the boundary ofthe domain of absolute stability, where we pay special attention to bifurcationsleading to spurious solutions. In order to explain a certain kind of spurious fixedpoints which are not connected to the set of genuine fixed points, we interpretethe domain of absolute stability as a Mandelbrot set and generalize this approachto nonlinear problems.1. IntroductionIn order to solve an n-dimensional initial value problem for the autonomousordinary differential equation*(0) = i?x = f(x) (/6C'(R,R)) (1.1)numerically, the continuous differentia] equation is replaced by a differenceequation. It is a major question whether the corresponding (discrete) approximatesolution has the same asymptotic behaviour as the exact (continuous) solution.For several years it has been known that this is not always the case (Brezzi etal (1984), Dekker & Verwer (1984), Iserles (1989), Priifer (1985), Yamaguti &Ushiki (1981)), and in recent years a theory has been developed to understandthese phenomena (Griffiths et al (1992), Hairer et al (1990), Humphries (1993),Iserles (1990), Iserles et al (1991), Iserles & Stuart (1992)).In the present paper, we restrict ou r attention to the class of explicit Runge-Kuttamethods in order to obtain insight into the main features of the problem. Consider

Bessel basis with applications:N-dimensional isotropic polynomial oscillators

The efficient technique of expanding the wave function into a Fourier–Bessel series to solve the radial Schrödinger equation with polynomial potentials, , in two dimensions is extended to N-dimensional space. It is shown that the spectra of two- and three-dimensional oscillators cover the spectra of the corresponding N-dimensional problems for N. Extremely accurate numerical results are presented for illustrative purposes. The connection between the eigenvalues of the general anharmonic oscillators and the confinement potentials of the form is also discussed. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 935–947, 1997

Design of practically stable n-dimensional feedback systems: a state-space approach

This paper investigates, by using the state-space approach, some basic properties and state feedback compensation problem of n-dimensional discrete systems in the practical sense that the input and output signals of the systems are unbounded in, at most, one dimension. Notions of practical controllability, practical observability, practical stabilizability by state feedback and practical detectability are introduced and corresponding necessary and sufficient conditions are derived. Moreover, a connection between the state-space representation and doubly coprime matrix fractional description on the ring of practically stable rational functions is shown. The obtained results clarify that all the n-dimensional problems considered in the practical sense are in fact equivalent to the corresponding problems of n one-dimensional systems, hence can be essentially solved by using one-dimensional methods.

Optimised design of Jaumann radar absorbing materials using a genetic algorithm

The design of wideband, multilayer radar absorbing materials involves the solution of an N-dimensional optimisation problem. Genetic algorithms appear to offer significant advantages over conventional optimisation techniques for this type of problem owing to their robustness and independence of performance function derivatives. To illustrate their use, the paper considers two problems: first the optimum design of wideband, multilayer, passive Jaumann radar absorbers with or without an outer high-permittivity skin; and secondly, an examination of the potential of such absorbers when configured as active i.e. frequency agile, dynamic structures.

An alternative way of solving secular equations for the Hamiltonian matrices of regular quasi‐one‐dimensional systems

An alternative approach to secular problems for Hamiltonian matrices H of regular quasi-one-dimensional systems is suggested. The essence of this approach consists of the inverted order of operations against that of the traditional solid-state theory, viz., taking into account the local structure of the system is followed by regarding the translational symmetry of the whole chain. The first step is performed by reducing the initial system of secular equations into an effective N × N-dimensional secular problem, wherein a single equation corresponds to each of N elementary fragments of the initial chain. An implicit form of the dispersion relation and the level density function follow directly from the reduced problem without passing into the delocalized description of the system. The resulting eigenfunctions of the matrix H prove to be expressed as the Bloch sums of N nonorthogonal eigenvalue-dependent local-structure-determined orbitals of algebraic form, each of them corresponding to a definite elementary fragment of the chain. © 1996 John Wiley & Sons, Inc.

Improved two-point function approximations for design optimization

Two-point approximations are developed by utilizing both the function and gradient information of two data points. The objective of this work is to build a high-quality approximation to realize computational savings in solving complex optimization and reliability analysis problems. Two developments are proposed in the new twopoint approximations: 1) calculation of a correction term by matching with the previous known function value and supplementing it to the first-order approximation for including the effects of higher order terms and 2) development of a second-order approximation without the actual calculation of second-order derivatives by using an approximate Hessian matrix. Several highly nonlinear functions and structural examples are used for demonstrating the new two-point approximations that improved the accuracy of existing first-order methods. EVELOPMENTof new and improved constraint approximations has been an active area of research in the mathematical optimization community over the last two decades.13 More accurate function approximations reduce the finite element analysis cost and increase the search domain in each iteration of an optimization run. Recent developments in constraint approximations addressed methods for eigenvalues, dynamic response, static forces, structural reliability, etc. Some of the typical methods are summarized in the following. Kirsch4 developed a scaling factor for a stiffness matrix and approximated the displacements, stresses, and forces with respect to sizing and topology variables. Vanderplaats and Salajegheh5 used two levels of approximations for discrete sizing and shape design of structures, first for solving the continuous problem and next the discrete one using a dual theory. Canfield6 developed an approach for eigenvalues by approximating the modal strain and kinetic energies independently. Thomas et al.7 presented an approximation for the frequency response of damped structures by using the concept of Ref. 6. A multivariate spline approximation was developed by Wang and Grandhi8 making use of the previously generated exact finite element analyses. Similarly, a Hermite approximation for n-dimensional problems used several data points and had the property of reproducing the function and gradient values at the known data points.9 Canfield10 used multipoint data in building an approximate Hessian matrix and constructed a second-order approximation for improving the accuracy. Toropov et al.11 approximated functions as a linear combination or products of variables, and the coefficients of the polynomial were computed by applying a least squared approach on multiple data points. Fadel et al. 12 considered intermediate variables in terms of exponentials and they were computed by matching the approximate function gradients with the previous point's exact values. This approach did not compare the function values of the previous point. Wang and Grandhi13 used a single exponential for all of the variables and calculated it by matching the approximate function with the previous point exact value, and the gradient information of the previous point was not utilized. The proposed paper develops an improved two-point approximation utilizing both function and gradient information of two data points. In the new two-point approximation, two developments are proposed: 1) calculation of a correction term based on two function values and

Identification of transmissivity coefficients by mollification techniques. Part II: n-dimensional elliptic problems

We analyze the numerical identification of the transmissivity coefficient in the n-dimensional model for linear elliptic equations. This problem arises in a number of important practical problems in science. The identification of this coefficient is an inverse problem which is highly ill-posed and difficult to solve. In order to address the ill-posedness of this parameter estimation problem, we use mollification techniques combined with finite differences to get a numerical method which is easy to implement in rectangular domains. Some numerical examples of interest for n = 2 are presented.

A general solution of the n-dimensional B-tree problem

We present a generic solution to a problem which lies at the heart of the unpredictable worst-case performance characteristics of a wide class of multi-dimensional index designs: those which employ a recursive partitioning of the data space. We then show how this solution can produce modified designs with fully predictable and controllable worst-case characteristics. In particular, we show how the recursive partitioning of an n-dimensional dataspace can be represented in such a way that the characteristics of the one-dimensional B-tree are preserved in n dimensions, as far as is topologically possible i.e. a representation guaranteeing logarithmic access and update time, while also guaranteeing a one-third minimum occupancy of both data and index nodes.

Multiple Sequence Comparison and Consistency on Multipartite Graphs

Calculation of dot-matrices is a widespread tool in biological sequence comparison. As a visual aid they are used in pairwise sequence comparison but so far have been of little help in the simultaneous comparison of several sequences. Viewing dot-matrices as projections of unknown n-dimensional points we consider the multiple alignment problem (for n sequences) as an n-dimensional image reconstruction problem with noise. We model this situation using a multipartite graph and introduce a notion of "consistency" on such a graph. From this perspective we introduce and develop the filtering method due to Vingron and Argos ( J. Mol. Biol. 218 (1991), 33-43). We discuss a conjecture of theirs regarding the number of iterations their algorithm requires and demonstrate that this number may be large. An improved version of the original algorithm is introduced that avoids costly dot-matrix multiplications and runs in O( n 3 · L 3) time ( L is the length of the longest sequence and n is the number of sequences). This is equivalent to only one iteration of the original algorithm. We further consider the relationship between consistency and transitivity and introduce a hierarchy of notions linking consistency and transitivity. Finally applications to biological sequence comparison will be presented.

Motion planning for many degrees of freedom: sequential search with backtracking

Gupta (1990) presented a sequential framework to develop practical motion planners for manipulator arms with many degrees of freedom. The crux of this framework is to sequentially plan the motion of each link, starting from the base link, thereby solving n single-link problems (each of which is solved as a 2-D planning problem) instead of one n-dimensional problem. The solution of each single-link problem is based on the search of a visibility graph (Vgraph) constructed from a polygonal representation of forbidden regions as seen in successive 2-D t/sub i//spl times/q/sub i/ spaces. In this paper, the authors present a backtracking mechanism within this sequential framework to make it more effective in planning collision-free paths in cluttered situations. The essence of the backtracking mechanism is based on an edge deletion mechanism that modifies the Vgraph in t/sub i-1//spl times/q/sub i-1/ space if no path is found in t/sub i//spl times/q/sub i/ space. The level of backtracking, b, i.e., the number of links the planner backtracks is an adjustable parameter that can trade off computational speed versus the relative completeness of the planner. Incorporating such a backtracking mechanism has significantly improved the performance of planners developed within this framework. The authors present extensive experimental results with up to eight degree-of-freedom manipulators in quite cluttered 3-D environments. Although the planner is not complete, the authors' empirical results are very encouraging. These empirical results indicate that b can be chosen small-typically 2 or 3-in over 90% of the cases. These results show that the authors' approach would be useful in practice.