surviving heat shock control strategies for robustness and performance molecular biology studies the cause and effect relationships among microscopic processes initiated by individual molecules within a cell and observes their macroscopic phenotypic effects on cells and organisms these studies provide a wealth of information about the underlying networks and pathways responsible for the basic functionality and robustness of biological systems at the same time these studies create exciting opportunities for the development of quantitative and predictive models that connect the mechanism to its phenotype then examine various modular structures and the range of their dynamical behavior the use of such models enables a deeper understanding of the design principles underlying biological organization and makes their reverse engineering and manipulation both possible and tractable the heat shock response presents an interesting mechanism where such an endeavor is possible using a model of heat shock we extract the design motifs in the system and justify their existence in terms of various performance objectives we also offer a modular decomposition that parallels that of traditional engineering control architectures

a class of implicit peer methods for stiff systems we present a class of s stage implicit two step peer methods for the solution of stiff differential equations using the function values from the previous step in addition to the new function values this allows to increase the order to p s and to ensure zero stability straightforwardly corresponding s stage methods for s 6 of order p s with optimal zero stability are presented and their stability is discussed under special conditions we prove that an optimally zero stable subclass of these methods is superconvergent of order p s 1 for variable step sizes numerical tests and comparison with ode15s show the high potential of this class of implicit peer methods

linearly stable orbits in 3 dimensional billiards we construct linearly stable periodic orbits in a class of billiard systems in 3 dimensional domains with boundaries containing semispheres arbitrarily far apart it shows that the results about planar billiard systems in domains with convex boundaries which have nonvanishing lyapunov exponents cannot be easily extended to 3 dimensions

automated generation of high order partial derivative models a standard problem in science and engineering consists of developing mathematics and sensitivity models for complex applications for optimizing a candidate design a chain rule based evaluation technique is presented for analytically evaluating partial derivatives of nonlinear functions and differential equations defined by a high level language a coordinate embedding strategy is introduced that replaces all scalar variables with higher dimensional objects the higher dimensional objects are defined by a concatenation of the original scalar and its jacobian hessian and higher order partials exact sensitivity models are recovered for arbitrarily complex mathematical models an operator overloading technique is used to define generalized operators for basic and standard library functions the generalized operators encode the chain rule of calculus and store the results of partial derivative calculations in the artificial dimensions used to redefine the scalar operations hidden operations automatically generate and evaluate exact first through fourth order partial derivative models which are accurate to the working precision of the machine the new algorithm replaces a normally complex error prone time consuming and labor intensive process for producing the partials with an automatic procedure module functions encapsulate new data types and extended mathematic and library functions for handling vector matrix and tensor operations matrix operations are shown to generalize easily the algorithm has broad potential for impacting the design and use of mathematical programming tools for applications in science and engineering several applications are presented that demonstrate the effectiveness of the methods

stability of moving mesh systems of partial differential equations moving mesh methods based on the equidistribution principle ep are studied from the viewpoint of stability of the moving mesh system of differential equations for fine spatial grids the moving mesh system inherits the stability of the original discretized partial differential equation pde unfortunately for some pdes the moving mesh methods require so many spatial grid points that they no longer appear to be practical failures and successes of the moving mesh method applied to three reaction diffusion problems are explained via an analysis of the stability and accuracy of the moving mesh pde

adjoint sensitivity analysis for differential algebraic equations algorithms and software an efficient numerical method for sensitivity computation of large scale differential algebraic systems is developed based on the adjoint method issues that are critical for the implementation are addressed complexity analysis and numerical results demonstrate that the adjoint sensitivity method is advantageous over the forward sensitivity method for applications with a large number of sensitivity parameters and few objective functions

review paper efficient numerical methods for the solution of stiff initial value problems and differential algebraic equations in recent years after a prolonged period of intense activity the study of numerical methods for solving stiff initial value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity there now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems in this paper we sketch some of the main theory which underpins stiff integration methods and we use this to describe and put into context some of the best codes currently available by referencing only codes which have been thoroughly tested and are widely available our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type an additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated

second order design sensitivity analysis of mechanical system dynamics dependence of dynamic response of nonlinear mechanical systems on design variables is analysed an adjoint variable method is used to derive first and second order derivatives of measures of dynamic response with respect to design variables a computational algorithm is presented for numerical calculation of first and second design derivatives a simple oscillator example is solved analytically and by the adjoint variable method with identical results a burst fire automatic weapon mechanism with linear and nonlinear damping is treated numerically it is shown that quadratic appriximations of dynamic response using results of second order design sensitivity analysis can be substantially better than conventional linear approximations

on the relations between b2v ms and runge kutta collocation methods the principal aim of this paper is a rigorous analysis of the relations between block boundary value methods b 2v ms with minimal blocksize defined over a suitable nonuniform finer mesh and well known runge kutta collocation methods moreover a further aspect that will be briefly investigated is the construction of an extended finer mesh for building b 2v ms with nonminimal blocksize some advantages that may arise from the use of the so obtained methods will be also discussed

modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in odes and daes abstract for many years the methods of choice for the numerical solution of stiff initial value problems and certain classes of differential algebraic equations have been the well known backward differentiation formulae bdf more recently however new classes of formulae which can offer some important advantages over bdf have emerged in particular some recent large scale independent comparisons have indicated that modified extended backward differentiation formulae mebdf are particularly efficient for general stiff initial value problems and for linearly implicit daes with index 3 in the present paper we survey some of the more important theory associated with these formulae discuss some of the practical applications where they are particularly effective e g in the solution of damped highly oscillatory problems and describe some significant recent extensions to the applicability of mebdf codes

adjoint sensitivity analysis for differential algebraic equations the adjoint dae system and its numerical solution an adjoint sensitivity method is presented for parameter dependent differential algebraic equation systems daes the adjoint system is derived along with conditions for its consistent initialization for daes of index up to two hessenberg for stable linear daes stability of the adjoint system for semi explicit daes or of an augmented adjoint system for fully implicit daes is shown in addition it is shown for these systems that numerical stability is maintained for the adjoint system or for the augmented adjoint system

adaptive algorithms for optimal control of time dependent partial differential algebraic equation systems summary this paper describes an adaptive algorithm for optimal control of time dependent partial dierential algebraic equation pdae systems a direct method based on a modied multiple shooting type technique and sequential quadratic programming sqp is used for solving the optimal control problem while an adaptive mesh renement amr algorithm is employed to dynamically adapt the spatial integration mesh issues of coupling the amr solver to the optimization algorithm are addressed for time dependent pdaes which can benet from the use of an adaptive mesh the resulting method is shown to be highly ecient copyright 2003 john wiley sons ltd

how high dimensional stadia look like we give the affirmative answer to the long standing question whether or not the mechanism of defocusing can produce a chaotic behavior in high dimensional hamiltonian systems to do this we prove that billiards in a class of regions in r n n 2 with focusing and flat boundary components have nonvanishing lyapunov exponents

an adaptive moving mesh method for two dimensional thin film flow equations with surface tension abstract in this paper we extend our previous work alharbi and naire 2017 on a one dimensional r adaptive moving mesh technique based on a mesh density function and moving mesh partial differential equations mmpdes to two dimensions as a test problem we consider the gravity driven thin film flow down an inclined and pre wetted plane including surface tension and a moving contact line this technique accurately captures and resolves the moving contact line and associated fingering instability moreover the computational effort is hugely reduced in comparison to a fixed uniform mesh

an equation based algorithmic differentiation technique for differential algebraic equations this work presents a novel algorithmic differentiation ad approach for sensitivity analysis of differential algebraic equation systems daes for the first time the algorithmic specification of a computationally memory efficient equation based ad technique is presented this approach is mainly targeting equation based modeling and simulation tools capable of constructing high level models using state of the art object oriented modeling principles the approach is based on fundamental tree algorithms that are even manually applicable on implicit equation systems of long formulas the main building blocks of model components by applying the presented forward differentiation scheme on a given dae efficient representation of sensitivity equation systems sess is computed parameter sensitivities are evaluated by direct integration of the derived sess to overcome the runtime performance drawback of direct numerical integration a system decomposition approach is recommended it is shown that the runtime performance using modern variable step integration methods tends to achieve the expected theoretical complexity of the forward differentiation scheme under few realistic assumptions

membrane systems with proteins embedded in membranes membrane computing is a biologically inspired computational paradigm motivated by brane calculi we investigate membrane systems which differ from conventional membrane systems by the following features 1 biomolecules proteins can move through the regions of the systems and can attach onto and de attach from membranes and 2 membranes can evolve depending on the attached molecules the evolution of membranes is performed by using rules that are motivated by the operation of pinocytosis the pino rule and the operation of cellular dripping the drip rule that take place in living cells we show that such membrane systems are computationally universal we also show that if only the second feature is used then one can generate at least the family of parikh images of the languages generated by programmed grammars without appearance checking which contains non semilinear sets of vectors if moreover the use of pino drip rules is non cooperative i e not dependent on the proteins attached to membranes then one generates a family of sets of vectors that is strictly included in the family of semilinear sets of vectors we also consider a number of decision problems concerning reachability of configurations and boundness

tensor product adaptive grids based on coordinate transformations in this paper we discuss a two dimensional adaptive grid method that is based on a tensor product approach adaptive grids are a commonly used tool for increasing the accuracy and reducing computational costs when solving both partial differential equations pdes and ordinary differential equations a traditional and widely used form of adaptivity is the concept of equidistribution which is well defined and well understood in one space dimension the extension of the equidistribution principle to two or three space dimensions however is far from trivial and has been the subject of investigation of many researchers during the last decade besides the nonsingularity of the transformation that defines the nonuniform adaptive grid the smoothness of the grid or transformation plays an important role as well we will analyse these properties and illustrate their importance with numerical experiments for a set of time dependent pde models with steep moving pulses fronts and boundary layers

finite difference methods and spatial a posteriori error estimates for solving parabolic equations in three space dimensions on grids with irregular nodes adaptive methods for solving systems of partial differential equations have become widespread much of the effort has focused on finite element methods in this paper modified finite difference approximations are obtained for grids with irregular nodes the modifications are required to ensure consistency and stability asymptotically exact a posteriori error estimates of the spatial error are presented for the finite difference method these estimates are derived from interpolation estimates and are computed using central difference approximations of second derivatives of the solution at grid nodes the interpolation error estimates are shown to converge for irregular grids while the a posteriori error estimates are shown to converge for uniform grids computational results demonstrate the convergence of the finite difference method and a posteriori error estimates for cases not covered by the theory

billiards with pesin region of measure one we give a description of a large class of plane billiards with pesin region of measure one open conditions including properly those founded by wojtkowski w1 forc4 focusing boundaries are obtained lyapunov s forms introduced by lewowicz are used

an improved algorithm and its parallel implementation for solving a general blood tissue transport and metabolism model fast algorithms for simulating mathematical models of coupled blood tissue transport and metabolism are critical for the analysis of data on transport and reaction in tissues here by combining the method of characteristics with the standard grid discretization technique a novel algorithm is introduced for solving a general blood tissue transport and metabolism model governed by a large system of one dimensional semilinear first order partial differential equations the key part of the algorithm is to approximate the model as a group of independent ordinary differential equation ode systems such that each ode system has the same size as the model and can be integrated independently thus the method can be easily implemented in parallel on a large scale multiprocessor computer the accuracy of the algorithm is demonstrated for solving a simple blood tissue exchange model introduced by sangren and sheppard w c sangren c w sheppard a mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment bull math biophys 15 1953 387 394 which has an analytical solution numerical experiments made on a distributed memory parallel computer an hp linux cluster and a shared memory parallel computer a sgi origin 2000 demonstrate the parallel efficiency of the algorithm

implicit interpolation error based error estimation for reaction diffusion equations in two space dimensions several authors have proposed an error estimation strategy for the finite element method applied to linear reaction diffusion equations in two space dimensions based on an odd even order dichotomy principle for odd order approximations the estimates are computed directly for even order approximations a second solution is computed although both estimators are asymptotically exact the latter are more robust than the former herein the even order method is extended to all orders greater than one thereby recovering robustness for odd orders proofs of asymptotic exactness are extended to nonlinear reaction diffusion equations computational results demonstrating their effectiveness are presented

a transversal fundamental theorem for semi dispersing billiards for billiards with a hyperbolic behavior fundamental theorems ensure an abundance of geometrically nicely situated and sufficiently large stable and unstable invariant manifolds a transversal fundamental theorem has recently been suggested by the present authors to proveglobal ergodicity and then as an easy consequence the k property of semidispersing billiards in particular the global ergodicity of systems ofn 3 elastic hard balls conjectured by the celebratedboltzmann sinai ergodic hypothesis in fact the suggested transversal fundamental theorem has been successfully applied by the authors in the casesn 3 and 4 the theorem generalizes the fundamental theorem of chernov and sinai that was really the fundamental tool to obtainlocal ergodicity of semi dispersing billiards our theorem however is stronger even in their case too since its conditions are simpler and weaker moreover a complete set of conditions is formulated under which the fundamental theorem and its consequences like the zig zag theorem are valid for general semi dispersing billiards beyond the utmost interesting case of systems of elastic hard balls as an application we also give conditions for the ergodicity and consequently the k property of dispersing billiards transversality means the following instead of the stable and unstable foliations occurring in the chernov sinai formulation of the stable version of the fundamental theorem we use the stable foliation and an arbitrary nice one transversal to the stable one

semi focusing billiards hyperbolicity in this paper we answer affirmatively the question concerning the existence of hyperbolic billiards in convex domains of 3 we also prove that a related class of semi focusing billiards has mixed dynamics i e their phase space is an union of two invariant sets of positive measure such that the dynamics is integrable on one set and is hyperbolic on the other these billiards are the first rigorous examples of billiards in domains of 3 with divided phase space

a moving mesh study for diffusion induced effects in avascular tumour growth abstract studying the dynamical evolution of complex systems as biological ones from the continuum point of view requires monitoring several parameters involved whose modelling leads to system of non linear coupled partial differential equations the interaction of the urokinase plasminogen activator system with a model for cancer cell in the avascular phase is faced with the moving mesh partial differential equation numerical technique monitoring the dynamical evolution of the system as a function of the diffusion properties of cancer cells and of cell proliferation factor over a one dimensional biological domain the computations are consistent with previous results confirming that cancer proliferation in the very early stage of invasion occurs through highly irregular spatio temporal pattern which depends essentially on cancer motility characteristics but non obvious effects are observed which depend on the model proliferation parameters

hybrid dynamics of stochastic programs we provide stochastic concurrent constraint programming sccp a stochastic process algebra based on ccp with a semantics in terms of hybrid automata we associate with each sccp program both a stochastic and a non deterministic hybrid automaton then we compare such automata with the standard stochastic semantics given by a continuous time markov chain and the one based on ordinary differential equations obtained by a fluid flow approximation technique we discuss in detail two case studies repressilator and the circadian clock with particular regard to the robustness exhibited by the different semantic models and to the effect of discreteness in dynamical evolution of such systems

conditional proof of the boltzmann sinai ergodic hypothesis we consider the system of n 2 elastically colliding hard balls of masses m1 mn and radius r on the flat unit torus mathbb t nu 2 we prove the so called boltzmann sinai ergodic hypothesis i e the full hyperbolicity and ergodicity of such systems for every selection m1 mn r of the external parameters provided that almost every singular orbit is geometrically hyperbolic sufficient i e the so called chernov sinai ansatz is true the present proof does not use the formerly developed rather involved algebraic techniques instead it employs exclusively dynamical methods and tools from geometric analysis

nowhere dispersing 3d billiards with non vanishing lyapunov exponents we describe a class of 3 dimensional regions with focusing components that generate a billiard system with non vanishing lyapunov exponents to do this we answer affirmatively the long standing question whether or not the chaotic motion caused by defocusing can be produced in more than two dimensions

an aggregation based domain decomposition preconditioner for groundwater flow we consider theoretical and computational issues associated with an aggregation based domain decomposition preconditioner applied to a bi cgstab iterative solver used to solve both laplace s equation and an important nonlinear model from hydrology used to simulate unsaturated flow richards equation theoretical results for laplace s equation provide estimates of the condition number and the rate of convergence for a two level schwarz domain decomposition preconditioner computational results for laplace s equation and richards equation show excellent scalability although no theory is yet available to support the results for the difficult nonlinear problem

pryce pre analysis adapted to some dae solvers abstract the ode solver hbt 12 5 of order 12 t nguyen ba h hao h yagoub r vaillancourt one step 5 stage hermite birkho taylor ode solver of order 12 appl math comput 211 2009 313 328 doi 10 1016 j amc 2009 01 043 which combines a taylor series method of order 9 with a runge kutta method of order 4 is expanded into the dae solver hbt 12 5dae of order 12 dormand prince s dp 8 7 13m is also expanded into the dae solver dp 8 7 dae pryce structural pre analysis extended odes and ode first order forms are adapted to these dae solvers with a stepsize control based on local error estimators and a modified pryce algorithm to advance integration hbt 12 5dae uses only the first nine derivatives of the unknown variables as opposed to the first 12 derivatives used by the taylor series method t12dae of order 12 numerical results show the advantage of hbt 12 5dae over t12dae dp 8 7 dae and other known dae solvers

an incomplete assembly with thresholding algorithm for systems of reaction diffusion equations in three space dimensions iat for reaction diffusion systems solving systems of reaction diffusion equations in three space dimensions can be prohibitively expensive both in terms of storage and cpu time herein i present a new incomplete assembly procedure that is designed to reduce storage requirements incomplete assembly is analogous to incomplete factorization in that only a fixed number of nonzero entries are stored per row and a drop tolerance is used to discard small values the algorithm is incorporated in a finite element method of lines code and tested on a set of reaction diffusion systems the effect of incomplete assembly on cpu time and storage and on the performance of the temporal integrator daspk algebraic solver gmres and preconditioner ilut is studied

finding elliptic fekete points sets two numerical solution approaches abstract the objective of this work is to provide a methodology for approximating globally optimal fekete point configurations this problem is of interest in numerical mathematics and scientific modeling following a brief discussion of the analytical background lipschitz global optimization lgo is applied to determine i e to numerically approximate fekete point configurations next to this optimization approach an alternative strategy by formulating a set of differential algebraic equations daes of index 2 will be considered the steady states of the daes coincide with the optima of the function to be minimized illustrative numerical results with configurations of up to 150 fekete points are presented to show the viability of both approaches

strategies for solving index one dae with non negative constraints liquid liquid extraction modeling leads to solve an index one dae system for the sake of robustness it is desirable to account for non negative constraints based on the dassl architecture a classical index one dae solver we propose and compare three different strategies to implement these bound constraints each of these strategies corresponds to a different newton modification clipping damping or interior point method the comparisons are made on two test cases the robertson ode problem and an example from liquid liquid extraction modeling

simulation techniques for the calculus of wrapped compartments the modelling and analysis of biological systems has deep roots in mathematics specifically in the field of ordinary differential equations odes alternative approaches based on formal calculi often derived from process algebras or term rewriting systems provide a quite complementary way to analyse the behaviour of biological systems these calculi allow to cope in a natural way with notions like compartments and membranes which are not easy sometimes impossible to handle with purely numerical approaches and are often based on stochastic simulation methods recently it has also become evident that stochastic effects in regulatory networks play a crucial role in the analysis of such systems actually in many situations it is necessary to use stochastic models for example when the system to be described is based on the interaction of few molecules when we are at the presence of a chemical instability or when we want to simulate the functioning of a pool of entities whose compartmentalised structure evolves dynamically in contrast stable metabolic networks involving a large number of reagents for which the computational cost of a stochastic simulation becomes an insurmountable obstacle are efficiently modelled with odes in this paper we define a hybrid simulation method combining the stochastic approach with odes for systems described in the calculus of wrapped compartments cwc a calculus on which we can express the compartmentalisation of a biological system whose evolution is defined by a set of rewrite rules

gip integrators for matrix riccati differential equations matrix riccati differential equations mrdes are initial value problems of the form x a 2 1 xa 1 1 a 2 2x xa 1 2x x 0 x 0 these equations arise frequently throughout applied mathematics science and engineering it can happen that even when the a i j are smooth functions of t or constant the solution x may have a singularity or even infinitely many singularities this paper shows several classes of numerical algorithms which we call gip integrators that can solve for x past its singularities furthermore none of the algorithms require knowledge of the placement or even existence of singularities in x also it is shown how embedded runge kutta methods can be used to construct gip integrators to not only approximate x past singularities but also provide for error estimation to allow efficient time stepping finally several examples are shown to validate the theory

software and algorithms for sensitivity analysis of large scale differential algebraic systems sensitivity analysis for dae systems is important in many engineering and scientific applications the information contained in the sensitivity trajectories is useful for parameter estimation optimization model reduction and experimental design in this paper we present algorithms and software for sensitivity analysis of large scale dae systems of index up to two the new software provides for consistent initialization of the solutions and the sensitivities interfaces seamlessly with automatic differentiation for the accurate evaluation of the sensitivity equations and is capable via mpi of exploiting the natural parallelism of sensitivity analysis as well as providing an efficient solution in sequential computations

an adaptive moving mesh method for thin film flow equations with surface tension we present an adaptive moving mesh method for the numerical solution of thin liquid film spreading flows with surface tension we follow the r adaptive moving mesh technique which utilises a mesh density function and moving mesh partial differential equations mmpdes to adapt and move the mesh coupled to the pde s describing the thin film flow problem numerical experiments are performed on two one dimensional thin film flow equations to test the accuracy and efficiency of the method this technique accurately resolves the multiple one dimensional structures observed in these test problems moreover it reduces the computational effort in comparison to the numerical solution using the finite difference scheme on a fixed uniform mesh we are the first to implement r adaptive schemes to higher order parabolic pdes this method accurately resolves the solution and reduces the computational effort we introduce a new mesh density function that resolves multiple solution structures this mesh density function is also adapted to include multiple solution components

a trust region method for parabolic boundary control problems in this paper we develop a trust region algorithm for constrained parabolic boundary control problems for the computation of a trust region step we propose an iterative scheme which is a projected form of the steihaug trust region conjugate gradient method to ensure the good local convergence properties in the terminal phase a smoothing step at each iteration is added this step and the projection require the modification of the standard trust region algorithm and its convergence proof the algorithm has sup norm convergence in the terminal phase and l2 convergence in the global phase the results are illustrated for a parabolic boundary control problem

using krylov methods in the solution of large scale differential algebraic systems in this paper a new algorithm for the solution of large scale systems of differential algebraic equations is described it is based on the integration methods in the solver dassl but instead of a direct method for the associated linear systems which arise at each time step we apply the preconditioned gmres iteration in combination with an inexact newton method the algorithm along with those in dassl is implemented in a new solver called daspk we outline the algorithms and strategies used and discuss the use of the solver we develop and analyze some preconditioners for a certain class of dae stems and finally demonstrate the application of daspk on two example problems

a real time framework for model predictive control of continuous time nonlinear systems a new formulation of model predictive control mpc for continuous time nonlinear systems is developed which allows for the use of ldquoreal timerdquo rt optimization techniques in which the solution to the finite horizon optimal control problem opc evolves within the same timescale as the process dynamics the computational savings of the rt solver are enhanced by the unique framework within which the opc is posed enabling significant reduction in the dimensionality of the search for situations where computational speed takes priority over optimality of the solutions this framework and its associated proof of stability encompasses results on sampled data sd nonlinear model predictive control nmpc implementation as a special case

using integrability to produce chaos billiards with positive entropy a new open class of convex 2 dimensional planar billiards with positive lyapunov exponent almost everywhere is constructed we introduce the notion of a focusing arc and show that such arcs can be used to build billiard systems with positive lyapunov exponents we prove that under smallc6 perturbations focusing arcs remain focusing and thereby show that perturbations of the bunimovich stadium billiard have positive lyapunov exponents

a theorem on ergodicity of two dimensional hyperbolic billiards ergodicity of two dimensional billiards which satisfy some general conditions is proved this theorem is applied to one concrete class of billiards that contains in particular billiards in the stadium

preconditioned implicit solution of linear hyperbolic equations with adaptivity this paper describes a method for solving hyperbolic partial differential equations using an adaptive grid the spatial derivatives are discretised with a finite volume method on a grid which is structured and partitioned into blocks which may be refined and derefined as the solution evolves the solution is advanced in time via a backward differentiation formula the discretisation used is second order accurate and stable on cartesian grids the resulting system of linear equations is solved by gmres at every time step with the convergence of the iteration being accelerated by a semi toeplitz preconditioner the efficiency of this preconditioning technique is analysed and numerical experiments are presented which illustrate the behaviour of the method on a parallel computer

efficient computation of sensitivities for ordinary differential equation boundary value problems for models described by ordinary differential equation boundary value problems ode bvps we derive adjoint equations for sensitivity analysis giving explicit forms for the boundary conditions of the adjoint boundary value problem the solutions of the adjoint equations are used to efficiently compute gradients of both integral form and pointwise constraints existence and stability results are given for the adjoint system and its numerical solution the use of the method is demonstrated for a simple example where it is seen that the method is particularly advantageous for problems with more than a few parameters

an adaptive finite element method for parabolic differential systems some algorithmic considerations in solving in three space dimensions an adaptive h refinement method is described for solving systems of parabolic partial differential equations in three space dimensions on hexahedral grids these grids typically have irregular hanging nodes solutions are calculated using galerkin s method with a piecewise trilinear basis in space and a backward difference formula bdf code in time new a posteriori error indicators based on interpolation error estimates for irregular grids are used to control refinement and coarsening a more efficient algorithm for assembling banded portions of the jacobian is introduced a simple strategy for dealing with storage limitations by limiting the level of refinement is developed computational results demonstrate the effectiveness of the adaptive method on linear and nonlinear problems

a parallelizable preconditioner for the iterative solution of implicit runge kutta type methods the main diculty in the implementation of most standard implicit runge kutta irk methods applied to sti ordinary dierential equations odes is to eciently solve the nonlinear system of equations in this article we propose the use of a preconditioner whose decomposition cost for a parallel implementation is equivalent to the cost for the implicit euler method the preconditioner is based on the w transformation of the rk coecient matrices discovered by hairer and wanner for sti odes the preconditioner is by construction asymptotically exact for methods with an invertible rk coecient matrix the methodology is particularly useful when applied to super partitioned additive runge kutta spark methods the nonlinear system can be solved by inexact simplied newton iterations at each simplied newton step the linear system can be approximately solved by an iterative method applied to the preconditioned linear system c 1999 published by elsevier science b v all rights reserved

many dimensional nowhere dispersing billiards with chaotic behavior abstract we consider some examples of domains in the d dimensional euclidean space d 3 that generate billiards with chaotic behavior time correlations in these billiards tend to zero in the limit t the rate of decay of the correlations is a power law for billiards in discrete time and could be a power law or quasiexponential for billiards with continuous time

rkc an explicit solver for parabolic pdes the fortran program rkc is intended for the time integration of parabolic partial differential equations discretized by the method of lines it is based on a family of runge kutta chebyshev formulas with a stability bound that is quadratic in the number of stages remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size moreover they make it possible to evaluate the explicit formulas in just a few vectors of storage these characteristics of the program make it especially attractive for problems in several spatial variables rkc is compared to the bdf solver vodpk on two test problems in three spatial variables

robust solution of richards equation for nonuniform porous media capillary pressure saturation relative permeability relations described using the van genuchten 1980 and mualem 1976 models for nonuniform porous media lead to numerical convergence difficulties when used with richards equation for certain auxiliary conditions these difficulties arise because of discontinuities in the derivative of specific moisture capacity and relative permeability as a function of capillary pressure convergence difficulties are illustrated using standard numerical approaches to simulate such problems we investigate constitutive relations interblock permeability nonlinear algebraic system approximation methods and two time integration approaches an integral permeability approach approximated by hermite polynomials is recommended and shown to be robust and economical for a set of test problems which correspond to sand loam and clay loam media