on polyhedra of perron frobenius eigenvectors abstract this paper deals with the positive eigenvectors of nonnegative irreducible matrices which are merely characterized by a given upper bound u on their spectral radius and by a given matrix l of lower bounds for their elements for any such matrix the normalized positive left right eigenvector is shown to belong to the polyhedron the vertices of which are given by the normalized rows columns of the matrix u i l 1 this polyhedron is proven to be also the smallest closed set which is guaranteed to contain the positive left right normalized eigenvector its vertices are therefore the best componentwise bounds one can obtain on the positive eigenvectors of these matrices a less general result has also been obtained for the symmetrical case when the matrices are only characterized by a given lower bound l on their spectral radius and by a given matrix u of upper bounds for their elements

an adjoint method for the inverse design of solidification processes with natural convection this paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes in particular the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved this is the first time that an infinite dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse design problems with moving boundaries and boussinesq incompressible flow the present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary while overspecified boundary conditions are supplied in another part of the boundary here the freezing interface the solidification design problem is mathematically posed as a whole time domain optimization problem the gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem the minimization process is realized by the conjugate gradient method via the solutions of the direct adjoint and sensitivity sub problems the proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold the non uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized the inverse design problem is then posed as finding the appropriate spatial temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity the numerical example demonstrates the accuracy and convergence of the adjoint formulation finally open related research design problems are discussed 1998 john wiley sons ltd

on a direct method for the solution of nearly uncoupled markov chains this note is concerned with the accuracy of the solution of nearly uncoupled markov chains by a direct method based on the lu decomposition it is shown that plain gaussian elimination may fail in the presence of rounding errors a modification of gaussian elimination with diagonal pivoting and correction of small pivots is proposed and analyzed it is shown that the accuracy of the solution is affected by two condition numbers associated with aggregation and the coupling respectively

aggregation of the policy iteration method for nearly completely decomposable markov chains a steady state optimal control problem is considered for nearly completely decomposable markov chains in order to apply the policy iteration method of r a howard dynamic programming and markov processes cambridge ma mit press 1960 a high dimensional ill conditioned system of algebraic equations must be solved in the value determination step although algorithms exist for aggregation of the steady state probability distribution problem they only provide methods for computing the cost not the dual variables using a singular perturbation approach an aggregation method for the value determination equation is developed the aggregation method is developed in three steps first a class of similarity transformations that transform the system into a singularly perturbed form is developed second an aggregation method to compute the steady state probability distribution is derived third this aggregation method is applied to the value determination step of howard s method 1960

lu decompositions of generalized diagonally dominant matrices using the simple vehicle ofm matrices the existence and stability oflu decompositions of matricesa which can be scaled to diagonally dominant possibly singular matrices are investigated bounds on the growth factor for gaussian elimination ona are derived motivation for this study is provided in part by applications to solving homogeneous systems of linear equationsax 0 arising in markov queuing networks input output models in economics and compartmental systems wherea or a is an irreducible singularm matrix this paper extends earlier work by funderlic and plemmons and by varga and cai

a fixed grid finite element technique for modelling phase change in steady state conduction advection problems a new fixed grid finite element technique for modelling phase change effects in steady state conduction advection problems is described the method is appropriate for materials that transform at a fixed critical temperature and is based on the assumption that the enthalpy experiences a jump discontinuity at this temperature yielding a singular specific heat this singularity is described using a dirac delta function and the appropriate finite element matrix equations are developed for isoparametric elements a simple first order upwinding scheme is used to avoid numerical oscillations in advection dominated problems the utility of the method is illustrated through the solution of a one dimensional test problem using eight node hexahedral isoparametric finite elements the technique may be applied to a wide range of finite element types and offers a number of advantages over other schemes for modelling phase change effects

a regularized sampling method for solving three dimensional inverse scattering problems the inverse scattering problem under consideration is to determine the shape of an obstacle in re 3 from a knowledge of the time harmonic incident acoustic wave and the far field pattern of the scattered wave with frequency in the resonance region a method for solving this nonlinear and improperly posed problem is presented which is based on solving a linear integral equation of the first kind and avoids the use of nonlinear optimization methods numerical examples are given showing the practicality of this new approach

comparison of finite element techniques for solidification problems the accuracies of the computed temperatures of a liquid in a corner region under freezing conditions are compared for various fixed grid finite element techniques using the analytical solution for this problem as a reference in the finite element formulation of the problem different time stepping schemes are compared the implicit euler backward algorithm combined with an iterative scheme and two three time level methods the lees algorithm and a dupont algorithm which are both applied as non iterative schemes furthermore different methods for handling the evolution of latent heat are examined an approximation method suggested by lemmon and one suggested by del giudice both using the enthalpy formulation as well as a fictitious heat flow method presented by rolph and bathe results of calculations performed with the consistent heat capacity matrix are compared with those performed with a lumped heat capacity matrix

deflated decomposition of solutions of nearly singular systems when solving the linear system ax b where a may be nearly singular and b is not consistent with a one is often interested in computing a deflated solution i e a unique solution to a nearby singular but consistent system a s x d tilde b kelley 14 15 has considered deflated solutions with a s corresponding to setting a small pivot of the lu factorization of a to zero stewart 22 proposed an iterative algorithm for computing a deflated solution with a s corresponding to setting the smallest singular value of a to zero kelley s approach explicitly uses submatrices of the lu factors whereas stewart s approach is implicit in that it only involves solving systems with a in this paper we generalize the concept of a deflated solution to that of a deflated decomposition which expresses the solution x in terms of x d and the null vectors of a s we treat such decompositions in a uniform framework that includes the approaches of kelley and stewart and introduce some new deflated

transient heat conduction analysis of 3d solids with fiber inclusions using the boundary element method a boundary element formulation is presented in this work for transient heat conduction analysis of three dimensional 3d fiber reinforced materials the cylindrical shaped fibers in a 3d matrix are idealized by a system of curvilinear line elements with a prescribed diameter the variations in the temperature and flux fields in the circumferential direction are represented in terms of a trigonometric shape function together with a linear or quadratic variation in the longitudinal direction this approach significantly reduces the modeling effort and the computing cost the storage requirement for the convolution integrals is eliminated by adopting an accurate integration based convolution algorithm for the surface of the hole and the fibers as well as a fast convolution algorithm for the outer boundary recently developed by the present authors numerical examples are presented to demonstrate the accuracy and applicability of the proposed method of analysis of fiber reinforced materials copyright 2007 john wiley sons ltd

the range test and the no response test for oseen problems the reconstruction of unknown shapes and inclusions is an important task for many applied sciences in electromagnetics and acoustics waves are used to probe some unknown region however often different types of measurements are available in industrial or natural applications such as measurements of fluid flow at some boundary of a region in space then the task is to detect the location and shape of an unknown inclusion from these remote flow measurements the goal of this work is to develop the theory to establish two new methods for finding unknown inclusions embedded in some fluid which is governed by the oseen equation we will introduce and analyze a range test based on ideas of kusiak potthast and sylvester 2003 and the no response test as suggested by luke and potthast 2003 for acoustic and electromagnetic problems in this work we will provide a new approach to show convergence of the no response test by showing its equivalence to a regularized version of the range test so in addition to two novel methods in inverse fluid dynamics this also provides further basic steps of methodological development on sampling and probing methods

highly efficient oscillation free solution of the transport equation over long times and large spaces transport problems with sharp transitions or high convection often stress even the most advanced computational methods beyond feasibility limits one promising line of attack involves transforming problems to a moving deforming coordinate system mdcs using mdcs as recently developed one may concentrate computational attention when and where it is most appropriate also possibly greatly reducing the effective magnitude of terms which cause trouble in principle mdcs may be used with finite difference finite element or other numerical methods according to preference in the particular finite element system advanced here mesh movement may be variable in time and space so that physically fixed boundaries may remain fixed in the transformed coordinate system a simple finite difference system is used in time when the method is applied to the standard transport equation test case great efficiency and accuracy are obtained in early time even very steep fronts may be represented well with no oscillations using space and time step sizes well beyond conventional peclet courant number constraints more important problems are solved involving transport over much longer times and larger spaces when diffusion has some chance to operate and hence a parabolic model is more warranted the same accuracy and stability are achieved as during early time using no more nodes and time steps overall than during early time runs this constitutes orders of magnitude savings in effort over what the application of conventional constraints would require relative efficiency from application of mdcs to other problems would depend on the particulars of each case

iterative aggregation disaggregation procedures for discounted semi markov reward processes the equation v q mv where m is a matrix with nonnegative elements and spectral radius less than one arises in markovian decision processes and input output models in this paper we solve the equation using an iterative aggregation disaggregation procedure that alternates between solving an aggregated problem and disaggregating the variables one block at a time in terms of the aggregate variables of the other blocks the disaggregated variables are then used to guide the choice of weights in the subsequent aggregation computational experiments on randomly generated and inventory problems indicate that this algorithm is significantly faster than successive approximations when the spectral radius of m is near one and is slower in unstructured problems with spectral radii in the neighborhood of 0 8 the algorithm appears promising for large structured problems where it can often reduce computational time and main memory storage requirements and offer greater robustness to initial values

boundary integral equation solution of moving boundary phase change problems boundary integral equation methods are presented for the solution of some two dimensional phase change problems convection may enter through boundary conditions but cannot be considered within phase boundaries a general formulation based on space time green s functions is developed using the complete heat equation followed by a simpler formulation using the laplace equation the latter is pursued and applied in detail an elementary noniterative system is constructed featuring linear interpolation over elements on a polygonal boundary nodal values of the temperature gradient normal to a phase change boundary are produced directly in the numerical solution the system performs well against basic analytical solutions using these values in the interphase jump condition with the simplest formulation of the surface normal at boundary vertices because the discretized surface changes automatically to fit the scale of the problem the method appears to offer many of the advantages of moving mesh finite element methods however it only requires the manipulation of a surface mesh and solution for surface variables in some applications coarse meshes and very large time steps may be used relative to those which would be required by fixed grid domain methods computations are also compared to original lab data describing two dimensional soil freezing with a time dependent boundary condition agreement between simulated and measured histories is good

a functional optimization approach to an inverse magneto convection problem a formulation and an fem implementation for the solution of an ill posed inverse design magneto convection problem is proposed in particular an incompressible viscous electrically conducting liquid material occupying a given domain is considered convection is driven by buoyancy effects as well as a lorentz force generated due to the applied magnetic field thermal boundary conditions are only prescribed in the part h0 of the boundary in addition the temperature distribution is also prescribed in the part i of the boundary h1 where the heat flux is known i e i is a boundary with overspecified thermal boundary conditions the inverse magneto convection problem is posed as an optimization problem in l2 i 0 tmax for the calculation of the boundary heat flux qo x t with x t h0 0 tmax the optimization scheme minimizes the discrepancy m x t x t qo l2 i 0 tmax between the temperature x t qo calculated from the solution of a direct problem for each flux qo and the desired or measured temperature m x t on the boundary i the exact l2 gradient of the cost functional is obtained from appropriately defined adjoint fields the adjoint problem is defined from the sensitivity operators obtained by linearization of the equations governing the direct problem the standard supg pspg stabilized fem formulation for incompressible fluid flow simulation is here extended to thermo magnetically driven flows the proposed fem formulation is used for the calculation of the direct adjoint and sensitivity thermal fluid flow and electric potential fields the entire optimization algorithm is solved using the conjugate gradient method finally the method is demonstrated through the solution of a few inverse problems with known results the need for regularization is identified in one of the example problems with uniformly distributed random errors in the temperature data m x t and a h1 regularized formulation is introduced to obtain stable solutions finally the numerical results are elucidated and potential applications are addressed

entrywise perturbation theory and error analysis for markov chains grassmann taksar and heyman introduced a variant of gaussian climination for computing the steady state vector of a markov chain in this paper we prove that their algorithm is stable and that the problem itself is well conditioned in the sense of entrywise relative error thus the algorithm computes each entry of the steady state vector with low relative error even the small steady state probabilities are computed accurately the key to our analysis is to focus on entrywise relative error in both the data and the computed solution rather than making the standard assessments of error based on norms our conclusions do not depend on any condition numbers for the problem

a solver for the stochastic master equation applied to gene regulatory networks an important driver of gene regulatory networks is noise arising from the stochastic nature of interactions of genes their products and regulators thus such systems are stochastic and can be modelled by the chemical master equations a major challenge is the curse of dimensionality which occurs when one attempts to integrate these equations while stochastic simulation techniques effectively address the curse many repeated simulations are required to provide precise information about stationary points bifurcation phenomena and other properties of the stochastic processes an alternative way to address the curse of dimensionality is provided by sparse grid approximations the sparse grid methodology is applied and the application demonstrated to work efficiently for up to 10 proteins as sparse grid methods have been developed for the approximation of smooth functions a variant for infinite sequences had to be developed together with a multiresolution analysis similar to haar wavelets error bounds are provided which confirm the effectiveness of sparse grid approximations for smooth high dimensional probability distributions

perturbation results for nearly uncoupled markov chains with applications to iterative methods the standard perturbation theory for linear equations states that nearly uncoupled markov chains numcs are very sensitive to small changes in the elements indeed some algorithms such as standard gaussian elimination will obtain poor results for such problems a structured perturbation theory is given that shows that numcs usually lead to well conditioned problems it is shown that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve these structured error bounds a variant of gaussian elimination due to grassman taksar and heyman was recently shown by o cinneide to achieve such bounds

moving boundary moving mesh analysis of phase change using finite elements with transfinite mappings two dimensional heat conduction phase change problems are solved using a moving boundary moving mesh approach a transfinite mapping technique successfully controls interior mesh motion and numerical results compare well with analytical solutions calculations also agree well with two dimensional laboratory data for cases featuring time dependent boundary conditions

on a rayleigh ritz refinement technique for nearly uncoupled stochastic matrices abstract an analysis is presented for the convergence of an iterative technique for computing the dominant left eigenvector of a nearly uncoupled nearly completely decomposable stochastic matrix the technique combines the power method and rayleigh ritz refinement to force rapid convergence to the stationary probability vector of the associated markov chain

a new numerical algorithm for 2d moving boundary problems using a boundary element method phase change problems are of practical importance and can be found in a wide range of engineering applications in the present paper two proposed numerical algorithms are developed the first one is general for phase change problems while the second one is for ablation problems the boundary elements method is used as a mathematical tool in conjunction with the proposed algorithms two test examples were solved and the results agree with the physics of the problems

two and three dimensional transient heat conduction and thermoelastic analyses by bem via efficient time convolution abstract efficient re integration based convolution algorithms are presented for a large class of problems involving transient two and three dimensional heat conduction and uncoupled thermoelastic analyses based on the earlier works of dargush and banerjee g f dargush p k banerjee application of boundary element method to transient heat conduction int j numer methods engrg 31 1991 1231 1248 g f dargush p k banerjee development of a boundary element method for time dependent planar thermoelasticity int j solids struct 25 1989 999 1021 the main motivation behind the current development is the fact that the storage based convolution approach used in earlier developments was not found to be very suitable for solving large scale practical problems because of substantial computer disk space requirement and the inability to handle time dependent boundary conditions in this paper a re integration based full convolution as well as an efficient and simplified re integration based fast convolution algorithms are presented in which all of the time convolution steps have been calculated by preserving the energy balance as expressed in the convolution while assuming equivalent values of variables to represent the total effect the accuracy and efficiency of the developed algorithms are demonstrated via numerical examples

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

two dimensional unsteady solidification problem calculated by using the boundary fitted coordinate system abstract using the boundary fitted coordinate system we calculate the unsteady solidification problem the model for calculation is constructed of cylindrical coordinates with a symmetrical axis the gauss seidel scheme with second order accuracy of time and space is used as a solution method for the governing equations and the boundary fitted coordinate system enables us to calculate the moving interface easily the numerical results are compared with the analysis of a one dimensional unsteady solidification problem and are in good agreement with it from the calculations we find a simple form of the crystal growth rate

triangular and skew symmetric splitting method for numerical solutions of markov chains in this paper a theorem is presented to indicate that there exists a nonnegative constant e 0 such that the matrix a q email protected is a positive definite matrix where email protected r n x n is an identity matrix and q email protected r n x n is a matrix with positive diagonal and nonpositive off diagonal elements then a class of triangular and skew symmetric splitting tss iteration method is applied to solve the positive definite linear system ax b for obtaining the stationary probability vector of an irreducible markov chain theoretical analyses show that the tss iteration method converges unconditionally to the unique solution of the linear system with the upper bound of its contraction factor dependent only on the spectrum of the triangular part and independent of the eigenvectors of the matrices involved moreover the inexact triangular and skew symmetric splitting itss iteration method which employs certain krylov subspace methods as the inner iteration processes at each step of the outer tss iteration method is proposed to accelerate the convergence of the tss iteration method numerical experiments are used to illustrate the effectiveness of the tss and itss iteration methods

block iterative algorithms for stochastic matrices abstract the theory of regular splittings for singular m matrices is used to derive the necessary and sufficient conditions for the convergence of iterative decomposition and aggregation techniques in the computation of the perron frobenius eigenvector of a stochastic irreducible matrix these conditions turn out to be less restrictive than those which must be satisfied by the block jacobi or the block gauss seidel method

aggregation disaggregation methods for computing the stationary distribution of a markov chain we implement and analyse aggregation disaggregation procedures constructed to accelerate the convergence of successive approximation methods suitable for computing the stationary distribution of a finite markov chain we define six of these methods and analyse them in detail in particular we show that some existing procedures lie in the aggregation disaggregation framework we set and hence can be considered as special cases also for all described methods we identify cases where they are promising numerical examples for the applications of some of the methods for nearly completely decomposable stochastic matrices are given as well

on a two level multigrid solution method for finite markov chains abstract we study a two level algebraic multigrid scheme for computing the stationary distribution of a homogeneous markov chain with a finite state space we reveal its relationship to an iterative aggregation disaggregation method provide an error analysis and prove its local convergence moreover we discuss the stochastic meaning of a block sor procedure that can be used as iterative smoother of the multigrid scheme

a boundary element method for multiple moving boundary problems a boundary element formulation for the solution of multiple moving boundary problems is presented and tested herein a heat transfer problem involving heating of solid melting of solid and partial vaporisation of liquid is considered numerical results show that the boundary element method is more suitable and more accurate than both finite difference and finite element methods for this kind of problem

the no response test for the reconstruction of polyhedral objects in electromagnetics we develop a no response test for the reconstruction of a polyhedral obstacle from two or few time harmonic electromagnetic incident waves in electromagnetics the basic idea of the test is to probe some region in space with waves which are small on some test domain and thus do not generate a response when the scatterer is inside of this test domain we will prove that the no response test checks analytic continuability of a time harmonic field from the far field pattern into the domain b email protected r email protected email protected for a non vibrating test domain b we show that two incident waves defined by one incident direction and two appropriately chosen directions of polarization are enough to recover the convex hull of polyhedrals based on this uniqueness result we build up the no response test and we prove convergence in the sense that it fully reconstructs a convex polyhedral scatterer d or the convex hull of an arbitrary polyhedral scatterer further we will describe the algorithmic realization of the no response test and show the feasibility of the method by reconstruction of convex polyhedral objects in three dimensions this is the first formulation of the no response test for electromagnetics

residual minimization least squares method for inverse heat conduction abstract a numerical method is systematically developed for resolving an inverse heat conduction problem in the presence of noisy discrete data this paper illustrates the effect of imposing constraints on the unknown function of interest a volterra integral equation of the first kind is derived and used as the starting point for residual minimization least squares methodology symbolic manipulation is exploited for purposes of augmenting the computational methodology preliminary indications suggest that the imposition of physical constraints on the system drastically reduces the level of mathematical sophistication needed for accurately approximating the unknown function of interest these constraints are actually available in many design studies or from models which are derived by physical processes

aggregation disaggregation for eigenvalue problems convergence of the power method for the algebraic eigenvalue problem is accelerated by an aggregation disaggregation process a lower dimensional aggregated eigenvalue problem is created from time to time during the course of the power iteration and the solution of this problem is used to produce acceleration analysis and numerical experiments show the effectiveness of the method the more so the more poorly conditioned is the original eigenvalue problem

the linear sampling method for solving the electromagnetic inverse scattering problem we consider the inverse scattering problem of determining the shape of an obstacle in mathbb r from knowledge of the time harmonic incident electromagnetic wave and the far field pattern of the scattered wave with frequency in the resonance region the approach used is the linear sampling method which does not require a priori knowledge of either the boundary condition or the connectivity of the scattering obstacle numerical examples are given for the case of both a simply connected perfect conductor and multiply connected obstacles satisfying an impedance boundary condition on the boundary of one component and satisfying a perfect conducting boundary condition on the boundary of another component

the adjoint method for an inverse design problem in the directional solidification of binary alloys this paper presents a systematic procedure based on the adjoint method for solving a class of inverse directional alloy solidification design problems in which a desired growth velocityvfis achieved under stable growth conditions to the best of our knowledge this is the first time that a continuum adjoint formulation is proposed for the solution of an inverse problem with simultaneous heat and mass transfer thermo solutal convection and phase change in this paper the interfacial stability is considered to imply a sharp solid liquid freezing interface this condition is enforced using the constitutional undercooling criterion in the form of an inequality constraint between the thermal and solute concentration gradients gandgc respectively at the freezing front the main unknowns of the design problem are the heating and or cooling boundary conditions on the mold walls the inverse design problem is formulated as a functional optimization problem the cost functional is defined by the square of thel2norm of the deviation of the freezing interface temperature from the temperature corresponding to thermodynamic equilibrium a continuum adjoint system is derived to calculate the adjoint temperature concentration and velocity fields such that the gradient of the cost functional can be expressed analytically the cost functional minimization process is realized by the conjugate gradient method via the finite element method solutions of the continuum direct sensitivity and adjoint problems the developed formulation is demonstrated with an example of designing the directional solidification of a binary aqueous solution in a rectangular mold such that a stable vertical interface advances from left to right with a desired growth velocity

analytical perturbations in markov chains abstract the problem of recomputing the steady state probabilities in a markov chain is considered after a small deviation is introduced to the original infinitesimal generator approximate expressions are developed to calculate the corresponding perturbed vector of probabilities computational stability and accuracy issues associated with the approximation are discussed as an application the perturbation approach is incorporated into an optimization scheme to identify workloads that maximize the output rate in a production line where qualitative and quantitative measures are introduced to control the accuracy of the perturbation

high order finite element methods for moving boundary problems with prescribed boundary evolution abstract we introduce a framework for the design of finite element methods for two dimensional moving boundary problems with prescribed boundary evolution that have arbitrarily high order of accuracy both in space and in time at the core of our approach is the use of a universal mesh a stationary background mesh containing the domain of interest for all times that adapts to the geometry of the immersed domain by adjusting a small number of mesh elements in the neighborhood of the moving boundary the resulting method maintains an exact representation of the prescribed moving boundary at the discrete level or an approximation of the appropriate order yet is immune to large distortions of the mesh under large deformations of the domain the framework is general making it possible to achieve any desired order of accuracy in space and time by selecting a preferred and suitable finite element space on the universal mesh for the problem at hand and a preferred and suitable time integrator for ordinary differential equations we illustrate our approach by constructing a particular class of methods and apply them to a prescribed boundary variant of the stefan problem we present numerical evidence for the order of accuracy of our schemes in one and two dimensions

local convergence of the exact and inexact iterative aggregation method for linear systems and markov operators the iterative aggregation method for the solution of linear systems is extended in several directions to operators on banach spaces to the method with inexact correction i e to methods where the inner linear system is in turn solved iteratively and to the problem of finding stationary distributions of markov operators local convergence is shown in all cases convergence results apply to the particular case of stochastic matrices moreover an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled markov chains as well as for markov chains with other zero nonzero structures

a sharp interface isogeometric solution to the stefan problem abstract in the present paper the stefan problem is solved by enriching an underlying nurbs based isogeometric approximation with an explicitly defined sharp interface on which a hybrid function derivative condition is isoparametrically described since the geometry of the enrichment is explicitly defined normals and curvatures are explicitly computed at any point on the interface thus the enriched approximation naturally captures the interfacial discontinuity in temperature gradient and naturally enables the imposition of gibbs thomson condition the blending of the enrichment with the underlying approximation requires an estimate of distance to the enriching geometry from a quadrature point and the parametric value of the footpoint on the enriching geometry these quantities are computed efficiently in the present paper using an algebraic estimate of distance coupled with an algebraic point projection method these algebraic schemes rely on implicitization of the parametric curve and are shown to be more efficient and robust than newton raphson iterations procedures for adaptive time stepping refinement and coarsening of geometry are developed to increase the stability and efficiency of the developed methodology several numerical examples of classical and dendritic stefan problem are presented to demonstrate the methodology

blockwise perturbation theory for nearly uncoupled markov chains and its application abstract let p be the transition matrix of a nearly uncoupled markov chain the states can be grouped into aggregates such that p has the block form p p ij i j 1 k where p ii is square and p ij is small for i j let t be the stationary distribution partitioned conformally as t 1 t k t in this paper we bound the relative error in each aggregate distribution i t caused by small relative perturbations in p ij the error bounds demonstrate that nearly uncoupled markov chains usually lead to well conditioned problems in the sense of blockwise relative error as an application we show that with appropriate stopping criteria iterative aggregation disaggregation algorithms will achieve such structured backward errors and compute each aggregate distribution with high relative accuracy

unified approach to simulation on deforming elements with application to phase change problems abstract several different approaches to simulation on deforming finite elements are shown to generate the same weighted residuals formulation for the evolution of the dependent variables control of node motion by means of mesh stretching in two dimensions is illustrated in the context of phase change problems simulation of problems with large phase boundary excursions shows good agreement with analytic solutions

modelization of phase changes by fictitious heat flow this paper presents an efficient algorithm to solve phase change problems the first part gives the definition of the problem and mentions some practical applications the second part analyses how to deal with phase changes by means of concentrated fictitious heat flows the principles of the method are described and compared to those of other methods the concept of the area associated with a node is developed and some practical values for the weight coefficients are recommended conclusions are given about the convergence and the stability of the algorithm and about the improvements obtained by prescribing the temperatures and using relaxation techniques theoretical and practical examples illustrate the validity and efficiency of the proposed method copyright 1984 john wiley sons ltd

the scattering support we discuss inverse problems for the helmholtz equation at fixed energy specifically the inverse source problem and the inverse scattering problem from a medium or an obstacle we introduce the convex scattering support of a far field a set which will be a subset of the convex hull of the support of any source which can produce it we give several theorems which explain how to compute the convex scattering support and how to relate it to the actual support of a source medium or obstacle 2003 wiley periodicals inc

bounds for the positive eigenvectors of nonnegative matrices and for their approximations by decomposition cet article traite des vecteurs propres positifs des matrices irreductibles non reparties qui sont simplement caracterisees par une limite superieure donnee sur leur rayon spectral et par une matrice l donnee de limites inferieures pour leurs elements analyse du cas particulier des matrices stochastiques et on donne un exemple numerique

application of infinite elements to phase change situations on deforming meshes in recent years progress has been made in applying moving and deforming mesh systems to phase change problems this allows the numerical attention where it is needed near the migrating phase change zone in spatially unbounded problems one hopes that numerically finite outer boundaries either escape significant activity or are automatically pushed further away as activity nears not infrequently this approach fails temperature activity often spreads more rapidly than phase change thereby reaching far boundaries stretching of the mesh by movement of far boundaries can challenge mesh control and cause ill conditioning in this paper the advantages of time dependent mesh adaption are enhanced by the joining of a new formulation for infinite elements to far boundaries this is accomplished through a co ordinate transformation within the framework of conventional 2 d quadratic biquadratic and linear quadratic elements standard 2 by 2 gauss legendre quadrature suffices throughout and normal galerkin finite element features are undisturbed including strict conservation of energy the formulation is independent of global co ordinates entails no restrictions on the unknown function and should be applicable to other problem types all test cases on quadrilateral and triangular grids show very significant improvements with infinite elements relative to comparable solution systems using strictly finite grids

the no response test a sampling method for inverse scattering problems we describe a novel technique which we call the no response test to locate the support of a scatterer from knowledge of a far field pattern of a scattered acoustic wave the method uses a set of sampling surfaces and a special test response to detect the support of a scatterer without a priori knowledge of the physical properties of the scatterer specifically the method does not depend on information about whether the scatterer is penetrable or impenetrable nor does it depend on any knowledge of the nature of the scatterer absorbing reflecting etc in contrast to previous sampling algorithms the techniques described here enable one to locate obstacles or inhomogeneities from the far field pattern of only one incident field the no response test is a one wave method we investigate the theoretical basis for the no response test and derive a one wave uniqueness proof for a region containing the scatterer we show how to find the object within this region we demonstrate the applicability of the method by reconstructing sound soft sound hard impedance and inhomogeneous medium scatterers in two dimensions from one wave with full and limited aperture far field data

stochastic complementation uncoupling markov chains and the theory of nearly reducible systems a concept called stochastic complementation is an idea which occurs naturally although not always explicitly in the theory and application of finite markov chains this paper brings this idea to the forefront with an explicit definition and a development of some of its properties applications of stochastic complementation are explored with respect to problems involving uncoupling procedures in the theory of markov chains furthermore the role of stochastic complementation in the development of the classical simon ando theory of nearly reducible system is presented

an analysis of solidification problems by the boundary element method the problem of interest in this paper is the calculation of the motion of the solid liquid interface and the time dependent temperature field during solidification of a pure metal an iterative implicit algorithm has been developed for this purpose using the boundary element method bem with time dependent green s functions and convolution integrals the bem approach requires discretization of only the surface of the solidifying body thus the numerical method closely follows the physics of the problems and is intuitively very appealing the formulation and the numerical scheme presented here are general and can be applied to a broad range of moving boundary problems emphasis is given to two dimensional problems comparison with existing semi analytical solutions and other numerical solutions from the literature reveals that the method is fast accurate and without major time step limitations

computable error bounds for aggregated markov chains abstract this paper describes a method for computing the steady state probability vector of a nearly completely decomposable markov chain the method is closely related to one proposed by simon and ando and developed by courtois however the method described here does not require the determination of a completely decomposable stochastic approximation to the transition matrix and hence it is applicable to matrices other than stochastic an error analysis of the procedure is given which results in effectively computable error bounds author

a bootstrap algebraic multilevel method for markov chains this work concerns the development of an algebraic multilevel method for computing state vectors of markov chains we present an efficient bootstrap algebraic multigrid amg method for this task in our proposed approach we employ a multilevel eigensolver with interpolation built using ideas based on compatible relaxation algebraic distances and least squares fitting of test vectors our adaptive variational strategy for computation of the state vector of a given markov chain is then a combination of this multilevel eigensolver and an associated additive multilevel preconditioned correction process we show that the bootstrap amg eigensolver by itself can efficiently compute accurate approximations to the steady state vector an additional benefit of the bootstrap approach is that it yields an accurate interpolation operator for many other eigenmodes this in turn allows for the use of the resulting multigrid hierarchy as a preconditioner to accelerate the generalized minimal residual gmres iteration for computing an additive correction equation for the approximation to the steady state vector unlike other existing multilevel methods for markov chains our method does not employ any special processing of the coarse level systems to ensure that stochastic properties of the fine level system are maintained there the proposed approach is applied to a range of test problems involving nonsymmetric m matrices arising from stochastic matrices and showing promising results

aggregation and disaggregation techniques and methodology in optimization a fundamental issue in the use of optimization models is the tradeoff between the level of detail and the ease of using and solving the model aggregation and disaggregation techniques have proven to be valuable tools for manipulating data and determining the appropriate policies to employ for this tradeoff furthermore aggregation and disaggregation techniques offer promise for solving large scale optimization models supply a set of promising methodologies for studying the underlying structure of both univariate and multivariate data sets and provide a set of tools for manipulating data for different levels of decision makers in this paper we develop a general framework for aggregation and disaggregation methodology survey previous work regarding aggregation and disaggregation techniques for optimization problems illuminate the appropriate role of aggregation and disaggregation methodology for optimization applications and propose future research directions

the extended finite element method xfem for solidification problems an enriched finite element method for the multi dimensional stefan problems is presented in this method the standard finite element basis is enriched with a discontinuity in the derivative of the temperature normal to the interface the approximation can then represent the phase interface and the associated discontinuity in the temperature gradient within an element the phase interface can be evolved without re meshing or the use of artificial heat capacity techniques the interface is described by a level set function that is updated by a stabilized finite element scheme several examples are solved by the proposed method to demonstrate the accuracy and robustness of the method copyright 2001 john wiley sons ltd

a functional optimization formulation and implementation of an inverse natural convection problem abstract a formulation and a fem implementation for the solution of an ill posed inverse design natural convection problem is proposed in particular we consider an incompressible viscous liquid material occupying a given domain where convection is driven by buoyancy thermal boundary conditions are only prescribed in the part h 0 of the boundary in addition the temperature distribution is also prescribed in the part i of the boundary h 1 where the heat flux is known i e i is a boundary with over specified thermal boundary conditions the inverse convection problem is posed as an optimization problem in l 2 i 0 t max for the calculation of the boundary heat flux q o x t with x t h 0 0 t max the optimization scheme minimizes the discrepancy m x t x t q o l 2 1 0 t max between the temperature x t q o calculated from the solution of a direct problem for each flux q 0 and the measured or desired temperature m x t at the boundary i the gradient of the cost functional is obtained from the adjoint fields the adjoint problem is defined from the sensitivity operators obtained by linearization of the equations governing the direct problem the conjugate gradient method is employed to perform the optimization process the fem is used for the calculation of the direct adjoint and sensitivity thermal and fluid flow fields finally the method is demonstrated with the solution of some two dimensional inverse problems with known solutions the numerical results are discussed and future work with potential applications is addressed

an efficient direct bem numerical scheme for phase change problems using fourier series abstract due to the moving boundary nature of the solidification and melting problems the application of the standard time domain boundary element formulation of the problem presents the difficulty of the numerical evaluation of the history dependence representing more and more work as time proceeds in that regard we present an algorithm based on a green s function which allows a fourier representation of the domain integral in the reinitialization scheme of the direct boundary element method formulation of the problem from which it is possible to carry out the history integration in a recursive manner to obtain the reinitialization approach in the integral representation formula the green s function corresponding to zero temperature in a box containing the original domain will be used instead of using the classical free space fundamental solution this green s function is given in terms of the original fundamental solution plus a regular solution of the heat equation inside the domain in consideration it can therefore be used in the integral representation formula of the heat equation direct formulation to obtain the solution of a heat problem in such domain the mentioned green s function can be obtained by the images method and the resulting source series can also be rewritten in terms of a double fourier series that will be used in the domain integral of the integral representation formula to transform such integral into equivalent surface integrals in this work a new reinitialization scheme has been applied to solve moving boundary problems to detect the location of the phase change interface in solidification and melting problems the numerical results were compared with analytical solutions in the case where they were available showing a good agreement between the results giving special emphasis to the interface location and the temperature distribution

bounding availability of repairable systems markov models are widely used for the analysis of availability of computer communication systems realistic models often involve state space cardinalities that are so large that it is impractical to generate the transition rate matrix let alone solve for availability measures an approximation technique is presented for determining steady state availability of particular interest is that the method also provides bounds on the error examples are given to illustrate the method

on the solution of an ill posed design solidification problem using minimization techniques in finite and infinite dimensional function spaces this paper provides a comparative study of two alternative methodologies for the solution of an inverse design solidification problem it is the one dimensional solidification problem of calculating the boundary heat flux history that achieves a desired freezing front velocity and desired heat fluxes at the freezing front the front velocity h t and flux history qms t on the solid side of the front control the obtained cast structure as such the potential applications of the proposed methods to the control of casting processes are enormous the first technique utilizes a finite dimensional approximation of the unknown boundary heat flux function q0 t the second technique uses the adjoint method to calculate in l2 the derivative of the cost functional tm t h t t q0 that expresses the square error between the calculated t h t t q0 and the given freezing front temperature tm both steepest descent sdm and conjugate gradient methods cgm are examined a front tracking fem technique is used for the discretization of the state space a detailed numerical analysis of the space and time discretization of the parameter and state spaces of the effect of the end condition of the adjoint problem and of other parameters in the solution are examined

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods