the adiabatic mode parabolic equation an efficient method is developed for solving problems involving three dimensional acoustic propagation in a thin spherical shell the method is based on the adiabatic mode approximation which assumes that energy coupling between modes is negligible and the parabolic equation pe method which efficiently handles spatial variations in the acoustic properties the pe method is used to solve two dimensional wave equations for the adiabatic mode coefficients over latitude and longitude due to this reduction in dimension the adiabatic mode pe should be practical for solving global scale ocean acoustics problems these types of problems which are currently of great interest heaney et al j acoust soc am 90 2586 2594 1991 baggeroer and munk phys today 45 9 22 30 1992 are too large to solve with other existing three dimensional models

an energy conserving parabolic equation for elastic media the accuracy of the parabolic equation pe method is improved for range dependent elastic media the energy conserving elastic pe is a generalization of the energy conserving acoustic pe a range dependent waveguide is approximated by a sequence of range independent regions a linear approximation is derived for the nonlinear condition corresponding to conservation of compressional energy flux at the vertical interfaces between range independent regions the rotated elastic pe is applied to generate reference solutions for demonstrating accuracy the energy conserving elastic pe is applied to solve problems involving the propagation of an interface wave along a sloping ocean bottom and a problem involving mode cutoff and coupling into shear wave beams in the ocean bottom the latter problem is analogous to a cutoff problem that was first solved with the acoustic pe f b jensen and w a kuperman j acoust soc am 67 1564 1566 1980

an approach for finding the most preferred alternative in the presence of multiple criteria abstract an interactive approach for discrete multiple criteria decision making problems is developed the approach requires the decision maker to compare pairs of presented alternatives these alternatives may be existing alternatives or dummy alternatives created from existing alternatives inferior alternatives are sequentially eliminated by either using the responses of the decision maker directly or by constructing cones of inferior solutions based on these responses the approach converges to the most preferred solution so long as the decision maker has a nondecreasing quasiconcave value function the aim is to keep the number of pairwise comparisons required of the decision maker as small as possible computational experience on randomly generated problems is reported and the results are compared with the results of similar approaches

a multiple criteria method for choosing among discrete alternatives abstract an interactive method is presented for solving a discrete alternative multiple criteria problem under certainty the method is based on the approach of zionts and wallenius a computer program embodying the method has been written and the results to date have been favorable extensions to the case in which some of the criteria are not readily quantifiable are also considered

finite difference solution to the parabolic wave equation in this article an implicit finite difference ifd scheme is introduced for solving underwater acoustic wave propagation problems this scheme is applied to the parabolic wave equation model for the acoustic problem with a free surface and arbitrary bottom and bottom boundary conditions the mathematical theory of the implicit finite difference method as applied to the parabolic equation is developed and a computer model to implement the ifd is introduced representative problems are presented to demonstrate the capability of the ifd model advantages and limitations of the ifd are discussed

heterogeneous domain decomposition for singularly perturbed elliptic boundary value problems a heterogeneous domain decomposition method is presented for the numerical solution of singularly perturbed elliptic boundary value problems the method which is parallelizable at various levels uses several ideas of asymptotic analysis the subdomains match the domains of validity of the local inner and outer asymptotic expansions and cut off functions are used to match solutions in neighboring subdomains the positions of the interfaces as well as the mesh widths depend on the small parameter varepsilon on the subdomains iterative solution techniques are used which may vary from one subdomain to another the global convergence rate depends on varepsilon it generally increases like some power of log varepsilon 1 1 as varepsilon downarrow 0 the method is illustrated on several two dimensional singular perturbation problems

a two way parabolic equation for acoustic backscattering in the ocean the parabolic equation pe method is generalized to handle backscattered acoustic energy in the ocean the two way pe is based on the single scattering approximation and the approach of two way coupled modes in which range dependent environments are approximated by a sequence of range independent regions at the vertical boundaries between regions the solution of the two way pe is required to satisfy two continuity conditions the range derivative in one of the conditions is replaced by a higher order pe depth operator the reflected and transmitted fields that satisfy these conditions are computed with an efficient iteration scheme the outgoing and incoming fields are propagated by two way range marching the two way pe which is presently implemented for two dimensional problems is a practical method for solving large scale reverberation problems the accuracy of the two way pe is demonstrated by comparisons with reference solutions the two way pe is applied to simulate the localization of a source of backscattering using the method of back propagation

scalar electromagnetic propagation modelling using parabolic equations and the split step pade approximation this paper presents a solution of 2d electromagnetic wave propagation problems in complicated terrestrial domains scalar 2d parabolic approximations are derived from maxwell s equations for both vertical and horizontal polarization the parabolic equations are then solved using a new technique involving pade rational function approximations of the macroscopic operator this method allows for larger than normal range stepping speeding up computational time significantly the pade approximation and the numerical implementation are fully discussed the discontinuity at the earth s surface is handled directly by using classical continuity conditions and deriving exact interface conditions for linking the fields in the atmosphere to those in the terrain the interface conditions are then implemented using the concept of virtual points preliminary benchmark tests show the interface treatment to work well finally several example runs are presented illustrating results

solving the discrete multiple criteria problem using convex cones an interactive method employing pairwise comparisons of attainable solutions is developed for solving the discrete deterministic multiple criteria problem assuming a single decision maker who has an implicit quasi concave increasing utility or value function the method chooses an arbitrary set of positive multipliers to generate a proxy composite linear objective function which is then maximized over the set of solutions the maximizing solution is compared with several solutions using pairwise judgments asked of the decision maker responses are used to eliminate alternatives using convex cones based on expressed preferences and then a new set of weights is found that satisfies the indicated preferences the requisite theory and proofs as well as a detailed numerical example are included in addition the results of some computational experiments to test the effectiveness of the method are described

overlapping schwarz methods for a singularly perturbed semilinear elliptic problem and their parallel implementation domain decomposition algorithms based on the schwarz alternating method are applied to the numerical solution of a singularly perturbed semilinear elliptic problem the emphasis is given to the description of the algorithms multidomain schwarz alternating method and two level schwarz method as well as their computer realization on a distributed memory multiprocessor computer numerical experiments related to a model problem illustrate the performance of the algorithms on parallel environment as well as their behavior with respect to the critical parameters such as the perturbation parameter the amount of overlapping and sizes of subdomains application of the two level schwarz method to a model semiconductor problem is also discussed

a higher order parabolic equation for wave propagation in an ocean overlying an elastic bottom a higher order elastic parabolic equation hepe is derived for wave propagation in depth dependent and weakly range dependent fluid solid media galerkin s method is used to discretize the depth operators in the hepe within layers in which depth variations in the lame constants and density are continuous discontinuities in material properties are handled with centered differences for the interface conditions between layers the numerical solution of the hepe also involves the method of alternating directions and crank nicolson integration the hepe is applied to underwater wave propagation problems involving a water column over an elastic bottom including a weakly range dependent problem the accuracy of the hepe is demonstrated with benchmark calculations

asymptotic numerical study of supersensitivity for generalized burgers equations this article addresses some asymptotic and numerical issues related to the solution of burgers equation varepsilon u xx u t u u x 0 on 1 1 subject to the boundary conditions u 1 1 delta u 1 1 and its generalization to two dimensions varepsilon delta u u t u u x u u y 0 on 1 1 times pi pi subject to the boundary conditions u x 1 1 delta u x 1 1 with 2 pi periodicity in y the perturbation parameters delta and varepsilon are arbitrarily small positive and independent when they approach 0 they satisfy the asymptotic order relation delta o s rm e a varepsilon for some constant a in 0 1 the solutions of these convection dominated viscous conservation laws exhibit a transition layer in the interior of the domain whose position as t to infty is supersensitive to the boundary perturbation algorithms are presented for the computation of the position of the transition layer at steady state the algorithms generalize to viscous conservation laws with a convex nonlinearity and are scalable in a parallel computing environment

a new parallel solver for the nonperiodic incompressible navier stokes equations with a fourier method we present a specific use of domain decomposition and decomposition in function space combined with asymptotic analytical qualitative results to obtain on parallel computers efficient and accurate solvers 3 for rapidly varying quasi planar unsteady combustion fronts in liquids in particular we give anew parallel direct solverof the unsteady incompressible navier stokes equations in the stream function formulation this solver is based on an embedding technique that allows us to generalize our previous results from the case with periodic boundary conditions 6 7 to thenonperiodiccase with wall boundary conditions in a direction perpendicular to front propagation the solution is decomposed into a particular solution suitable for a fourier method and the general homogeneous solution calculated from an analytic solution with high precision to satisfy the boundary conditions the algorithm is implemented for parallel computers and results in a very effective code results on the effect of the convection onto the front propagation are provided

dual cone approach to convex cone dominance in multiple criteria decision making abstract in a paper published in management science in 1984 korhonen wallenius and zionts presented the idea and method based on convex cone dominance in the discrete multiple criteria decision making framework in our current paper we revisit the old idea from a new standpoint and provide the mathematical theory leading to a dual cone based approach to solving such problems our paper makes the old results computationally more tractable the results provided in the present paper also help extend the theory

characterization of nonsmooth quasiconvex and pseudoconvex functions this paper characterizes the nonsmooth quasiconvex and pseudoconvex functions using the properties of limiting subdifferentials

interactive approaches for discrete alternative multiple criteria decision making with monotone utility functions in this paper we develop interactive approaches for the discrete alternative multiple criteria decision making problem we develop an algorithm that finds the most preferred alternative of a decision maker dm assuming only that the dm has a monotonic utility function the algorithm divides the criteria space into a number of smaller subspaces and then uses the ideal points of these subspaces to eliminate alternatives we also develop a more efficient version of the algorithm for the more restrictive case of a monotonic quasiconcave utility function we present favorable computational results in terms of the required number of pairwise comparisons for both versions of the algorithm we then develop a general algorithm that first identifies the type of the dm s utility function and then employs the approach that is compatible with the identified utility function type we also present computational results for the general algorithm

on some applications of the superposition principle with fourier basis this paper presents several applications of local fourier basis combined with corrector techniques via the superposition principle to compute solutions of boundary value problems our methodology is inspired by the well known corrector technique used in asymptotic singular perturbation theory see for example w eckhaus asymptotic analysis of singular perturbations north holland amsterdam 1979 we build solvers for time dependent boundary value problems and or singular perturbation problems using fourier expansions combined to additional convenient set of time independent basis functions to enforce the boundary conditions the algorithms are very well suited for parallel computing because they rely mainly on ffts and basis functions given analytically or computed in parallel once and for all our method can be spectral accurate for simple geometry we obtain in addition interesting new results for boundary value problems with complex geometry we describe in this paper the implementation of the method and the numerical results for many linear and nonlinear problems our goal is to test the advantages and the limits of our approach in order to motivate further theoretical investigations our method applied to steady one dimensional problems is similar to the work of israeli vozovoi and averbuch j sci comput 8 1993 pp 135 149 on domain decomposition with local fourier basis for the helmotz problem however our methodology for time dependent problem and or two space dimensions geometry is rather different in its spirit and complements previous investigations of israeli vozovoi and averbuch

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

domain decomposition to solve transition layers and asymptotics the author considers the numerical computation of stiff nonlinear partial differential equations pdes that can be studied by the methods of singular perturbation two domain decomposition methods that solve numerically the layers of the singular perturbation problem are presented these numerical methods use at different stages the information given by the asymptotic analysis the simplified model of reacting flow of majda siam j sci statist comput 7 1986 pp 1059 1080 a singular perturbation problem with a turning point l abrahamsson and s osher siam j numer anal 19 1982 pp 979 992 and a combustion problem j pelaez siam j appl math 47 1987 pp 781 799 that models a sequence of two chemical reactions will be considered as test problems

a split step pade solution for the parabolic equation method a split step pade solution is derived for the parabolic equation pe method higher order pade approximations are used to reduce both numerical errors and asymptotic errors e g phase errors due to wide angle propagation this approach is approximately two orders of magnitude faster than solutions based on pade approximations that account for asymptotic errors but not numerical errors in contrast to the split step fourier solution which achieves similar efficiency for some problems the split step pade solution is valid for problems involving very wide propagation angles large depth variations in the properties of the waveguide and elastic ocean bottoms the split step pade solution is practical for global scale problems

use of convex cones in interactive multiple objective decision making one approach for solving decision problems involving multiple objectives is interactive optimization methods based on this approach assess the decision maker s preference structure interactively typically based on pairwise comparisons and tradeoffs and guide the search process toward identifying improved solutions a desirable feature of such approaches that is based on minimizing the preference information requirements is fast convergence toward this end the use of convex cones as a preference structure representation has been proposed in the literature in this work new theory is developed that aids in further reducing preference information requirements and improving convergence new cones termed p cones are developed the efficiencies of solution alternatives are evaluated with respect to the p cones and these are termed p cone efficiencies acceleration and early termination procedures that are based on these efficiencies are proposed the procedures are presented within a solution framework for solving multiple objective linear programming molp problems along with computational results

a higher order energy conserving parabolic equqation for range dependent ocean depth sound speed and density outgoing solutions of the wave equation including parabolic equation pe and normal mode solutions are usually formulated so that pressure is continuous with range for range dependent problems the accuracy of normal mode solutions has been improved by conserving energy rather than maintaining continuity of pressure porter et al the problem of energy conservation in one way equations j acoust soc am 89 1058 1067 1991 this approach is applied to derive a higher order energy conserving pe that provides improved accuracy for problems involving large ocean bottom slopes and large range and depth variations in sound speed and density a special numerical approach and complex pade coefficients are applied to suppress gibbs oscillations the back propagated half space field an improved pe starter is applied to handle wide propagation angles reference solutions generated with a complex ray model and with the rotated pe are used for comparison

evaluation functions and decision conditions of three way decisions with game theoretic rough sets abstract three way decisions have been used over the years in many application areas the rough sets and its extensions provide useful approaches for three way decisions typically these approaches employ a single evaluation function or criterion to induce three way decisions when extending the rough set based three way decisions to multiple criteria decision making mcdm two issues are encountered the first issue is related to the construction and definition of aggregation mechanisms for dealing with differences in results of evaluation functions the second issue is related to the setting of choice structure for selecting the three types of decision choices in this article we consider the role and use of game theoretic rough set gtrs model to resolve and address these two issues the issue related to differences in evaluation functions is addressed with gtrs by implementing a game that considers multiple evaluation functions as game players the game theoretic analysis in the gtrs is employed to resolve the differences by determining a tradeoff between evaluation functions the issue related to choice structure is addressed by considering the conditions under which different game outcomes could constitute a game solution in particular the equilibrium analysis within games is used to construct the rules for three way decisions a demonstrative example is used to explain the use of the proposed approach the relationship between the proposed approach and the probabilistic rough sets is also discussed