goal oriented adaptivity in control constrained optimal control of partial differential equations dual weighted goal oriented error estimates for a class of pointwise control constrained optimal control problems for second order elliptic partial differential equations are derived it is demonstrated that the constraints give rise to a primal dual weighted error term representing the mismatch in the complementarity system due to discretization the paper also contains a posteriori error estimators for the l 2 norm of the error in the state and in the adjoint state

on the asymptotic exactness of error estimators for linear triangular finite elements this paper deals with a posteriori error estimates for piecewise linear finite element approximations of elliptic problems we analyze two estimators based on recovery operators for the gradient of the approximate solution by using superconvergence results we prove their asymptotic exactness under regularity assumptions on the mesh and the solution one of the estimators can be easily computed in terms of the jumps of the gradient of the finite element approximation this estimator is equivalent to the error in the energy norm under rather general conditions however we show that for the asymptotic exactness the regularity assumption on the mesh is not merely technical while doing this we analyze the relation between superconvergence and asymptotic exactness for some particular examples

weighted pseudo almost periodic mild solutions of semilinear evolution equations with nonlocal conditions in this paper applying the theory of semigroups of operators to evolution families and banach fixed point theorem we prove the existence and uniqueness of the weighted pseudo almost periodic mild solution of the semilinear evolution equation x t a t x t f t x t with nonlocal conditions x 0 x 0 g x in banach space x under some suitable hypotheses

on a class of second order anti periodic boundary value problems of concern is the existence and uniqueness of anti periodic solutions to a class of abstract nonlinear second order differential equations the investigation relies on the theory of m accretive operators in banach or hilbert spaces examples of boundary value problems for ordinary and partial differential equations are also discussed

weighted pseudo antiperiodic solutions for fractional integro differential equations in banach spaces in this paper we prove the existence of weighted pseudo antiperiodic mild solutions for fractional integro differential equations in the form d u t au t t a t s au s ds f t u t t r where f u is stepanov like weighted pseudo antiperiodic and a generates a resolvent family of bounded and linear operators on a banach space x a l loc 1 r and 0 here the fractional derivative is considered in the sense of weyl also we give a short proof to show that the vector valued space of stepanov like weighted pseudo antiperiodic functions is a banach space

augemented lagrangian techniques for elliptic state constrained optimal control problems we propose augmented lagrangian methods to solve state and control constrained optimal control problems the approach is based on the lagrangian formulation of nonsmooth convex optimization in hilbert spaces developed in k ito and k kunisch augmented lagrangian methods for nonsmooth convex optimization in hilbert spaces preprint 1994 we investigate a linear optimal control problem with a boundary control function as in m bergounioux numer funct anal optim 14 1993 pp 515 543 both the equation and the constraints are augmented the proposed methods are general and can be adapted to a much wider class of problems

on the performance of exponential integrators for problems in magnetohydrodynamics exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems while the results of these investigations showed that exponential integrators can provide significant computational savings the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages resistive magnetohydrodynamic mhd modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas in many problems numerical solution of mhd equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest in this paper we evaluate the performance of exponential integrators on large mhd problems and compare them to a state of the art implicit time integrator both the variable and constant time step exponential methods of epirk type are used to simulate magnetic reconnection and the kevin helmholtz instability in plasma performance of these methods which are part of the epic software package is compared to the variable time step variable order bdf scheme included in the cvode part of sundials library we study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as reynolds lundquist and prandtl numbers we find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for mhd problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as mhd

gradient recovery type a posteriori error estimates for finite element approximations on irregular meshes in this paper the gradient recovery type a posteriori error estimators for finite element approximations are proposed for irregular meshes both the global and the local a posteriori error estimates are derived moreover it is shown that the a posteriori error estimates is asymptotically exact on where the mesh is regular enough and the exact solution is smooth

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

convergence of finite element approximations to state constrained convex parabolic boundary control problems this paper is concerned with the numerical solution of state constrained parabolic boundary control problems by finite element approximations error estimates for the optimal values and in the coercive case for the optimal solutions are derived these estimates are used to prove convergence results under rather weak assumptions in the noncoercive case a bang bang principle is shown to obtain convergence results for the discrete controls also a discussion of several numerical examples concludes the paper

constrained lqr problems in elliptic distributed control systems with point observations in this paper we study bound constrained lqr problems in distributed control systems governed by the elliptic equation with point observations which are motivated by problems in corrosion engineering and contemporary smart materials several regularity and characterization theorems have been established in particular three decomposition formulas are obtained to characterize the optimal control and optimal layer density and are used to direct the numerical computations these results cannot be obtained by the traditional galerkin variational method in the process several useful lemmas are established which are of independent interest we point out that the classical lagrangian multiplier method lmm may fail to provide a reliable numerical algorithm based on our characterization results and the boundary element method two algorithms are proposed to carry out numerical computations it has been shown by our numerical experiments that both algorithms are efficient and insensitive to the partition number of the boundary an adaptive local refinement scheme has also been designed to handle the rough behavior of the optimal solution around sensor locations

weighted pseudo almost periodic solutions to a neutral delay integral equation of advanced type abstract this paper studies suitable sufficient conditions to ensure the existence and uniqueness of weighted pseudo almost periodic solutions to a neutral delay integral equation of advanced type introduced by t a burton in the literature the abstract results are then utilized to characterize weighted pseudo almost periodic solutions to the well known logistic equation

weighted pseudo almost automorphic solution of an integro differential equation with weighted stepanov like pseudo almost automorphic forcing term in this paper we ensure existence and uniqueness of weighted pseudo almost automorphic solution of an integrodifferential equation with weighted s p pseudo almost automorphic forcing term in a banach space x we use fractional powers of operators and the banach fixed point theorem in order to obtain our result very recently introduced concept of weighted s p pseudo almost automorphy has been used here

pointwise accuracy of a finite element method for nonlinear variational inequalities consider the following quasilinear elliptic pde which is equivalent to a nonlinear variational inequality divf u u f here is a singular maximal monotone graph and the nonlinear differential operator is only assumed to be monotone surfaces of prescribed mean curvature over obstacles may thus be viewed as relevant examples the numerical approximation proposed in this paper consists of combining continuous piecewise linear finite elements with a preliminary regularization of the resulting scheme is shown to be quasi optimally accurate inl the underlying analysis makes use of both a topological technique and a sharpl p duality argument

preconditioned implicit exponential integrators imexp for stiff pdes abstract we propose two new classes of time integrators for stiff des the implicit exponential imexp and the hybrid exponential methods in contrast to the existing exponential schemes the new methods offer significant computational advantages when used with preconditioners any preconditioner can be used with any of these new schemes this leads to a broader applicability of exponential methods the proof of convergence of these integrators and numerical demonstration of their efficiency are presented

comparative performance of exponential implicit and explicit integrators for stiff systems of odes exponential integrators have enjoyed a resurgence of interest in recent years but there is still limited understanding of how their performance compares with that of state of the art integrators most notably the commonly used newton krylov implicit methods in this paper we present comparative performance analysis of krylov based exponential implicit and explicit integrators on a suite of stiff test problems and demonstrate that exponential integrators have computational advantages compared to the other methods particularly as problems become larger and more stiff we argue that the faster convergence of the krylov iteration within exponential integrators accounts for the main proportion of the computational savings that they provide and illustrate how the structure of these methods ensures such efficiency in addition we demonstrate the computational advantages of the newly introduced tokman and loffeld 2010 17 exponential propagation runge kutta epirk fifth order methods the detailed analysis of the performance of the methods that is presented provides guidelines for the construction and implementation of efficient exponential methods and the quantitative comparisons inform the selection of appropriate schemes for other problems

nonhomogeneous boundary value problems the diffusion equation is solved under stochastic nonhomogeneity using eigen function expansion and the georges method the statistical moments of the solution process are computed through the two previously mentioned techniques and proved to be the same a general solution is obtained under general initial and boundary conditions a random source composed of deterministic and stochastic parts is taken into consideration the stochastic part is then restricted to a generalized gaussian field mainly modulated white noise a special case is considered under constant noise level and constant average noise a numerical case study concerning pollution in a stream is solved and a parametric study is achieved through various figures

a new class of exponential propagation iterative methods of runge kutta type epirk we propose a new class of the exponential propagation iterative methods of runge kutta type epirk the epirk schemes are exponential integrators that can be competitive with explicit and implicit methods for integration of large stiff systems of odes introducing the new more general than previously proposed ansatz for epirk schemes allows for more flexibility in deriving computationally efficient high order integrators recent extension of the theory of b series to exponential integrators 1 is used to derive classical order conditions for schemes up to order five an algorithm to systematically solve these conditions is presented and several new fifth order schemes are constructed several numerical examples are used to verify the order of the methods and to illustrate the performance of the new schemes

error analysis and applications of the fourier galerkin runge kutta schemes for high order stiff pdes an integrating factor mixed with runge kutta technique is a time integration method that can be efficiently combined with spatial spectral approximations to provide a very high resolution to the smooth solutions of some linear and nonlinear partial differential equations in this paper the novel hybrid fourier galerkin runge kutta scheme with the aid of an integrating factor is proposed to solve nonlinear high order stiff pdes error analysis and properties of the scheme are provided application to the approximate solution of the nonlinear stiff korteweg de vries the 3rd order pde dispersive equation kuramoto sivashinsky the 4th order pde dissipative equation and kawahara the 5th order pde equations are presented comparisons are made between this proposed scheme and the competing method given by kassam and trefethen it is found that for kdv ks and kawahara equations the proposed method is the best

existence of weighted pseudo anti periodic solutions to some non autonomous differential equations abstract in this paper we first introduce two new concepts called respectively weighted pseudo periodicity and weighted pseudo anti periodicity which generalize respectively the well known notions of periodicity and that of anti periodicity basic properties of these new functions are discussed furthermore the existence of weighted pseudo anti periodic solutions to some damped non autonomous second order differential equations will be investigated to illustrate our abstract results the existence of weighted pseudo anti periodic solutions to a plate like equation is discussed

adaptive finite element approximation for distributed elliptic optimal control problems in this paper sharp a posteriori error estimators are derived for a class of distributed elliptic optimal control problems these error estimators are shown to be useful in adaptive finite element approximation for the optimal control problems and are implemented in the adaptive approach our numerical results indicate that the sharp error estimators work satisfactorily in guiding the mesh adjustments and can save substantial computational work

a posteriori error estimates for elliptic variational inequalities we derive hierarchical a posteriori error estimates for elliptic variational inequalities the evaluation amounts to the solution of corresponding scalar local subproblems we derive some upper bounds for the effectivity rates and the numerical properties are illustrated by typical examples

adaptive finite element methods for optimal control of partial differential equations basic concept a new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations the lagrangian formalism yields the first order necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive galerkin finite element method the mesh design in the resulting reduced models is controlled by residual based a posteriori error estimates these are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error in this case the computed state and costate variables can be used as sensitivity factors multiplying the local cell residuals in the error estimators this results in a generic and simple algorithm for mesh adaptation within the optimization process this method is developed and tested for simple boundary control problems in semiconductor models

a posteriori error estimates for control problems governed by stokes equations in this paper we derive a posteriori error estimates for the finite element approximation of distributed optimal control problems governed by the stokes equations we obtain a posteriori error estimators for both the state and the control approximation in the l2 norm and the h1 norm these estimates can be used to construct reliable adaptive finite element approximation for the control problems

finite element approximation of parabolic time optimal control problems the numerical approximation of a parabolic time optimal control problem via piecewise linear splines is considered at each stage of the approximation a bang bang approximate control is selected by solving for its switching times as the solution of a constrained nonlinear least squares optimization problem the well posedness of the approximation scheme is shown and the rate of convergence to the exact solution investigated numerical results for some one and two dimensional parabolic control problems are given

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations this paper is concerned with the study of weighted pseudo almost periodic solutions to some classes of hyperbolic differential equations upon making some suitable assumptions the existence and uniqueness of a weighted pseudo almost periodic solution is obtained to illustrate our main result we study the existence of a weighted pseudo almost periodic solution to a system of partial differential equations arising in control systems described by abstract retarded functional differential equations with feedback control

analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls an optimal control problem for impressed cathodic systems in electrochemistry is studied the control in this problem is the current density on the anode a matching objective functional is considered we first demonstrate the existence and uniqueness of solutions for the governing partial differential equation with a nonlinear boundary condition we then prove the existence of an optimal solution next we derive a necessary condition of optimality and establish an optimality system of equations finally we define a finite element algorithm and derive optimal error estimates

ritz galerkin approximation of the time optimal boundary control problem for parabolic systems with dirichlet boundary conditions this paper deals with ritz galerkin approximations of the following two problems i boundary value problems with l 2 boundary data given in the form of dirichlet boundary conditions ii time optimal control problems for parabolic systems with control acting on the boundary in both cases optimal rates of convergence of approximation are obtained also the specialization is considered to approximating with nonconformal elements which are not required to satisfy boundary conditions corresponding rates of convergence are also proved

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

on the structure of lagrange multipliers for state constrained optimal control problems abstract state constrained optimal control problems represent severe analytical and numerical challenges it is shown that under certain conditions on the active set the measure representing the lagrange multiplier associated to the state constraint can be decomposed in a distributed part in l2 with support on the active set and a boundary measure concentrated on the interface between active and inactive sets

a posteriori error estimators for the stokes equations we present two a posteriori error estimators for the mini element discretization of the stokes equations one is based on a suitable evaluation of the residual of the finite element solution the other one is based on the solution of suitable local stokes problems involving the residual of the finite element solution both estimators are globally upper and locally lower bounds for the error of the finite element discretization numerical examples show their efficiency both in estimating the error and in controlling an automatic self adaptive mesh refinement process the methods presented here can easily be generalized to the navier stokes equations and to other discretization schemes

a posteriori error estimates for optimal control problems governed by parabolic equations in this paper we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation we obtain a posteriori error estimates for both the state and the control approximation such estimates which are apparently not available in the literature are an important step towards developing reliable adaptive finite element approximation schemes for the control problem

weighted pseudo almost automorphic functions and wpaa solutions to semilinear evolution equations abstract in this paper we reveal several basic properties about nonlinear vector valued weighted pseudo almost automorphic functions including equivalence completeness translation invariance composition theorem and convolution theorem of these functions we also give some concrete examples to illustrate our results finally we obtain a new existence theorem of nonlinear weighted pseudo almost automorphic solutions for semilinear evolution equations in banach spaces

weighted stepanov like pseudo almost automorphy and applications abstract in this paper we propose a new class of functions called weighted stepanov like pseudo almost automorphic functions which generalize in a natural fashion the concept of almost automorphy and its various extensions we systematically explore the properties of the weighted stepanov like pseudo almost automorphic functions in general banach space including a composition result as an application we establish some sufficient criteria for the existence uniqueness of the weighted pseudo almost automorphic solution to a class of partial neutral functional differential equations and also to a class of nonlinear volterra integral equations of convolution type with infinite delay in banach space some interesting examples are presented to illustrate the main findings

existence of weighted pseudo almost periodic solutions to some classes of differential equations with sp weighted pseudo almost periodic coefficients abstract in this paper we introduce the concept of weighted pseudo almost periodicity in the sense of stepanov also called s p weighted pseudo almost periodicity and study its properties next we present a result on the existence of weighted pseudo almost periodic solutions to the n dimensional heat equation with s p weighted pseudo almost periodic coefficients

new adaptive exponential propagation iterative methods of runge kutta type exponential integrators have emerged as an efficient alternative to commonly used time integrators recently a new class of exponential propagation iterative methods of runge kutta type epirk has been introduced m tokman j comput phys 230 2011 pp 8762 8778 these schemes possess a structure that makes them computationally advantageous compared to other exponential methods in addition the general epirk formulation offers flexibility that allows derivation of new efficient techniques in this paper we use this feature to derive new epirk methods which are particularly designed to take advantage of the adaptive krylov algorithm j niesen and w wright http arxiv org abs 0907 4631 2010 the adaptive krylov method significantly reduces the computational complexity of evaluating products of matrix varphi functions and vectors necessary for implementing an exponential integrator we present the derivation of the new adaptive epirk methods construct new schemes and illustrate the co

solution of time dependent pde through rapid estimation of block gaussian quadrature nodes abstract the stiffness of systems of odes that arise from spatial discretization of pdes causes difficulties for both explicit and implicit time stepping methods krylov subspace spectral kss methods present a balance between the efficiency of explicit methods and the stability of implicit methods by computing each fourier coefficient from an individualized approximation of the solution operator of the pde while kss methods are explicit methods that exhibit a high order of accuracy and stability similar to that of implicit methods their efficiency needs to be improved a previous asymptotic study of block lanczos iteration yielded estimates of extremal block gaussian quadrature nodes for each fourier component and led to an improvement in efficiency in this paper a more detailed asymptotic study is performed in order to rapidly estimate all nodes thus drastically reducing computational expense without sacrificing accuracy numerical results demonstrate that the new node estimation scheme does in fact accomplish these aims

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

weighted pseudo almost periodic solutions for some partial functional differential equations abstract in this work we give sufficient conditions for the existence and uniqueness of a weighted pseudo almost periodic solution for some partial functional differential equations to illustrate our main result we study the existence of a weighted pseudo almost periodic solution for some diffusion equation with delay

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

a new approach to constructing efficient stiffly accurate epirk methods the structural flexibility of the exponential propagation iterative methods of runge kutta type epirk enables construction of particularly efficient exponential time integrators while the epirk methods have been shown to perform well on stiff problems all of the schemes proposed up to now have been derived using classical order conditions in this paper we extend the stiff order conditions and the convergence theory developed for the exponential rosenbrock methods to the epirk integrators we derive stiff order conditions for the epirk methods and develop algorithms to solve them to obtain specific schemes moreover we propose a new approach to constructing particularly efficient epirk integrators that are optimized to work with an adaptive krylov algorithm we use a set of numerical examples to illustrate the computational advantages that the newly constructed epirk methods offer compared to previously proposed exponential integrators

exponential krylov methods for ordinary differential equations this paper develops a new family of exponential time discretization methods called exponential krylov expk the new schemes treat the time discretization and the krylov based approximation of exponential matrix vector products as a single computational process the classical order conditions theory developed herein accounts for both the temporal and the krylov approximation errors unlike traditional exponential schemes expk methods require the construction of only a single krylov space at each timestep the number of basis vectors that guarantee the temporal order of accuracy does not depend on the application at hand numerical results show favorable properties of expk methods when compared to current exponential schemes

antiperiodic solutions of semilinear integrodifferential equations in banach spaces abstract we are concerned in this paper with the existence of antiperiodic mild solutions of some semilinear integrodifferential equations in a banach space we study further properties of antiperiodic functions including a new composition theorem and introduce the new concept of asymptotically antiperiodic functions

solution of nonlinear time dependent pdes through componentwise approximation of matrix functions exponential propagation iterative epi methods provide an efficient approach to the solution of large stiff systems of odes compared to standard integrators however the bulk of the computational effort in these methods is due to products of matrix functions and vectors which can become very costly at high resolution due to an increase in the number of krylov projection steps needed to maintain accuracy in this paper it is proposed to modify epi methods by using krylov subspace spectral kss methods instead of standard krylov projection methods to compute products of matrix functions and vectors numerical experiments demonstrate that this modification causes the number of krylov projection steps to become bounded independently of the grid size thus dramatically improving efficiency and scalability as a result for each test problem featured as the total number of grid points increases the growth in computation time is just below linear while other methods achieved this only on selected test problems or not at all

optimal control for variational inequalities consider the problem of maximizing a functional which depends on a control function k and on the solution of an elliptic variational inequality with k appearing in the data the variational problem for k is nondiflerentiable and nonconvex we obtain necessary conditions on a maximizes k 0 and then use them to determine the structure of k 0 in some cases

a new class of split exponential propagation iterative methods of runge kutta type sepirk for semilinear systems of odes abstract in this paper the framework of the exponential propagation iterative methods of runge kutta type epirk is extended to construct split epirk sepirk integrators for semilinear systems of odes the structure of the sepirk methods possesses the flexibility and generality that allows construction of very efficient schemes we demonstrate how bicolored trees based b series can be used to derive the order conditions for the new integrators an algorithm is developed to solve the order conditions up to order three and several new schemes with desirable properties are proposed the numerical results illustrate the advantages offered by the new class of integrators the experiments also address the comparative performance of split vs non split epirk methods and the question of improving efficiency by optimizing coefficients of the sepirk schemes it is shown that specific schemes can be custom built to improve computational efficiency for a particular problem

on the existence of anti periodic solutions to a nonlinear evolution equation associated with odd subdifferential operators the existence of anti periodic solutions hence also that of periodic solutions to the nonlinear parabolic type evolution equation ddt u t u t f t u t 0 t r is shown under the assumptions that the subdifferential operator is odd and that f t t r is a family of monotone operators satisfying the anti periodicity condition f t t v f t v v d f t hence f t t v f t v v d d f t t r for a fixed t r here coerciveness of is not assumed an application to a heat equation is given without assuming boundedness of the domain of rxn

anti periodic solutions for semilinear evolution equations in this paper we use the homotopy method to establish the existence and uniqueness of anti periodic solutions for the nonlinear anti periodic problem a t x bx f t a e t r x t t x t where a t x is a nonlinear map and b is a bounded linear operator from r n to r n sufficient conditions for the existence of the solution set are presented also we consider the nonlinear evolution problems with a perturbation term which is multivalued we show that for this problem the solution set is nonempty and weakly compact in w 1 2 i r n for the case of convex valued perturbation and prove the existence theorems of anti periodic solutions for the nonconvex case all illustrative examples are provided

pricing the financial heston hull white model with arbitrary correlation factors via an adaptive fdm abstract this paper is concerned with the pricing procedure of one of the most challenging models known as the heston hull white partial differential equation pde in option pricing at which the model is a time dependent 3d linear pde including three mixed derivative terms the model comes from the fact that the price the volatility and the interest rate are assumed to be stochastic processes to contribute and avoid huge discretized systems an adaptive distribution of the nodes viz nonuniform nodes is taken into account with emphasis on the hot area of the solution curve new adaptive finite difference fd formulas of higher orders are constructed on these meshes then a set of semi discretized equations is attained which is tackled by a time stepping method several financial tests are discussed in detail to reveal the superiority of the proposed approach

local a posteriori error estimates for convex boundary control problems in this paper we derive some improved a posteriori error estimates for finite element approximation of neumann boundary control problems we first establish local upper a posteriori error estimates for both the state and the control approximation of the general convex boundary control problems we then derive local upper and lower a posteriori error estimates for a class of control problems that frequently appear in applications

exponential rosenbrock type methods we introduce a new class of exponential integrators for the numerical integration of large scale systems of stiff differential equations these so called rosenbrock type methods linearize the flow in each time step and make use of the matrix exponential and related functions of the jacobian in contrast to standard integrators the methods are fully explicit and do not require the numerical solution of linear systems we analyze the convergence properties of these integrators in a semigroup framework of semilinear evolution equations in banach spaces in particular we derive an abstract stability and convergence result for variable step sizes this analysis further provides the required order conditions and thus allows us to construct pairs of embedded methods we present a third order method with two stages and a fourth order method with three stages respectively the application of the required matrix functions to vectors are computed by krylov subspace approximations we briefly discuss these implementation issues and we give numerical examples that demonstrate the efficiency of the new integrators

finite dimensional approximation of a class of constrained nonlinear optimal control problems an abstract framework for the analysis and approximation of a class of nonlinear optimal control and optimization problems is constructed nonlinearities occur in both the objective functional and the constraints the framework includes an abstract nonlinear optimization problem posed on infinite dimensional spaces an approximate problem posed on finite dimensional spaces together with a number of hypotheses concerning the two problems the framework is used to show that optimal solutions exist to show that lagrange multipliers may be used to enforce the constraints to derive an optimality system from which optimal states and controls may be deduced and to derive existence results and error estimates for solutions of the approximate problem the abstract framework and the results derived from that framework are then applied to three concrete control or optimization problems and their approximation by finite element methods the first involves the von k arm an plate equations of nonlinear elasticity the second the ginzburg landau equations of superconductivity and the third the navier stokes equations for incompressible viscous flows

computing the action of the matrix exponential with an application to exponential integrators a new algorithm is developed for computing e ta b where a is an n times n matrix and b is n times n 0 with n 0 ll n the algorithm works for any a its computational cost is dominated by the formation of products of a with n times n 0 matrices and the only input parameter is a backward error tolerance the algorithm can return a single matrix e ta b or a sequence e t ka b on an equally spaced grid of points t k it uses the scaling part of the scaling and squaring method together with a truncated taylor series approximation to the exponential it determines the amount of scaling and the taylor degree using the recent analysis of al mohy and higham siam j matrix anal appl 31 2009 pp 970 989 which provides sharp truncation error bounds expressed in terms of the quantities a k 1 k for a few values of k where the norms are estimated using a matrix norm estimator shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems and analysis of rounding errors and of the conditioning of the problem provides theoretical support experimental comparisons with matlab codes based on krylov subspace chebyshev polynomial and laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy an important application of the algorithm is to exponential integrators for ordinary differential equations it is shown that the sums of the form sum k 0 p varphi k a u k that arise in exponential integrators where the varphi k are related to the exponential function can be expressed in terms of a single exponential of a matrix of dimension n p built by augmenting a with additional rows and columns and the algorithm of this paper can therefore be employed

a gaussian radial basis function finite difference technique to simulate the hcir equation abstract localized meshless radial basis functions rbfs which are due to the application of rbfs and finite difference fd methods at the same time are popular well resulted computational tools for tackling higher dimension partial differential equations pdes in this research the objective is not only to study a spatial discretization of the financial heston cox ingresoll ross hcir pde but also to apply a non uniform distribution of nodes for the application of gaussian rbfs the combination of these ideas simulate the hcir pde quickly and efficiently

existence of pseudo almost automorphic solutions to some abstract differential equations with sp pseudo almost automorphic coefficients abstract this paper introduces and studies the notion of s p pseudo almost automorphy or stepanov like pseudo almost automorphy which generalizes in a natural fashion the concept of pseudo almost automorphy that was recently introduced in the literature by liang xiao and zhang we next obtain the existence of pseudo almost automorphic solutions to some differential equations with s p pseudo almost automorphic coefficients