the regularization method for an obstacle problem we give a relatively complete analysis for the regularization method which is usually used in solving non differentiable minimization problems the model problem considered in the paper is an obstacle problem in addition to the usual convergence result and a priori error estimates we provide a posteriori error estimates which are highly desired for practical implementation of the regularization method

an upwind numerical approach for an american and european option pricing model the numerical solution of several mathematical models arising in financial economics for the valuation of both european and american call options on different types of assets is considered all the models are based on the black scholes partial differential equation in the case of european options a numerical upwind scheme for solving the boundary backward parabolic partial differential equation problem is presented when treating with american options an additional inequality constraint leads to a discretized linear complementarity problem in each case the numerical approximation of option values is computed by means of optimization algorithms in particular uzawa s method allows to compute the optimal exercise boundary which corresponds to the classical concept of moving boundary in continuum mechanics

error estimates for the numerical approximation of time dependent flow of bingham fluid in cylindrical pipes by the regularization method the flow of a bingham fluid in a cylindrical pipe can give rise to free boundary problems the fluid behaves like a viscous fluid if the shear stress expressed as a linear function of the shear rate exceeds a yield value and like a rigid body otherwise the surfaces dividing fluid and rigid zones are the free boundaries therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods considerable progress has been made in the development of numerical algorithms for bingham fluids 2 20 27 29 30 32 however very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution that is the error estimate of these numerical methods error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters this paper concerns the error estimates of a unsteady bingham fluid modeled as a variational inequality due to duvaut lions 16 and glowinski 30 the difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term a common technique called the regularization method is to replace the non differentiable term by a perturbed differentiable term which depends on a small regularization parameter the regularization method effectively reduces the variational inequality to an equation a regularized problem which is much easier to cope with this paper has achieved the following 1 error estimates are derived for a continuous time galerkin method in suitable norms 2 we give an estimate of the difference between the true solution and the regularized solution in terms of 3 some regularity properties for both regularized solution and the true solution are proved 4 the error estimates for full discretization of the regularized problem using piecewise linear finite elements in space and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t 5 we are able to improve our estimates in the one dimensional case or under stronger regularity assumptions on the true solution the estimates for the one dimensional case are optimal and confirmed by numerical experiment the estimates from 4 and 5 provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h t and in order to control the error see corollary 4 4 the above estimates extend the error bounds derived in glowinski lions and tremolieres 32 chapter 5 pp 348 404 for the stationary bingham fluid to the time dependent one which is the main contribution of this paper

a note on exponential rosenbrock euler method for the finite element discretization of a semilinear parabolic partial differential equation abstract in this paper we consider the numerical approximation of a general second order semilinear parabolic partial differential equation equations of this type arise in many contexts such as transport in porous media using finite element method for space discretization and the exponential rosenbrock euler method for time discretization we provide a convergence proof in space and time under only the standard lipschitz condition of the nonlinear part for both smooth and nonsmooth initial solutions this is in contrast to restrictive assumptions made in the literature where the authors have considered only approximation in time so far in their convergence proofs the main result reveals how the convergence orders in both space and time depend heavily on the regularity of the initial data in particular the method achieves optimal convergence order o h 2 t 2 when the initial data belongs to the domain of the linear operator numerical simulations to sustain our theoretical result are provided

dyson s crank of a partition holds he was thus led to conjecture the existence of some other partition statistic which he called the crank this unknown statistic should provide a combinatorial interpretation of p lln 6 in the same way that 1 1 and 1 2 treat the primes 5 and 7 in 4 5 one of us was able to find a crank relative to vector partitions as follows for a partition 7r let 7r be the number of parts of ir and cr n be the sum of the parts of ir or the number ir is partitioning with the convention 0 for the empty partition 0 of 0 let

far field boundary conditions for black scholes equations in this paper we consider the partial differential equations approach for valuing european style options in order to solve the equations numerically by finite difference or finite element methods it is necessary to introduce an artificial boundary in order to make the computational domain bounded we derive pointwise bounds for the error caused by various boundary conditions imposed on the artificial boundary the error estimates can be used to determine a priori a suitable location for the artificial boundary in terms of a given error tolerance

the finite element approximation of a coupled reaction diffusion problem with non lipschitz nonlinearities a coupled semilinear elliptic problem modelling an irreversible isothermal chemical reaction is introduced and discretised using the usual piecewise linear galerkin finite element approximation an interesting feature of the problem is that a reaction order of less than one gives rise to a dead core region initially one reactant is assumed to be acting as a catalyst and is kept constant it is shown that error bounds previously obtained for a scheme involving numerical integration can be improved upon by considering a quadratic regularisation of the nonlinear term this technique is then applied to the full coupled problem and optimal h 1 and l infty error bounds are proved in the absence of quadrature for a scheme involving numerical integration bounds similar to those obtained for the catalyst problem are shown to hold

exponential rosenbrock integrators for option pricing in this paper we are concerned with the time integration of differential equations modeling option pricing in particular we consider the black scholes equation for american options as an alternative to existing methods we present exponential rosenbrock integrators these integrators require the evaluation of the exponential and related functions of the jacobian matrix the resulting methods have good stability properties they are fully explicit and do not require the numerical solution of linear systems in contrast to standard integrators we have implemented some numerical experiments in matlab showing the reliability of the new method

order conditions for numerical methods for partitioned ordinary differential equations motivated by the consideration of runge kutta formulas for partitioned systems the theory of p series is studied this theory yields the general structure of the order conditions for numerical methods for partitioned systems and in addition for nystrom methods fory f y y for rosenbrock type methods with inexact jacobian w methods it is a direct generalization of the theory of butcher series 7 8 in a later publication the theory ofp series will be used for the derivation of order conditions for runge kutta type methods for volterra integral equations 1

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

a fast boundary element method for the two dimensional helmholtz equation abstract in this paper we present a fast method for solving boundary integral equations arising from the exterior dirichlet problem for the two dimensional helmholtz equation this method combines a quadrature method for discretizing the boundary integral equations with a preconditioned iterative method for solving the resulting dense nonsymmetric linear systems using this method a polynomial rate of convergence can be obtained by performing a finite number of iterations which yields high computational efficiency various numerical examples are presented

fast solution methods for fredholm integral equations of the second kind the main purpose of this paper is to describe a fast solution method for one dimensional fredholm integral equations of the second kind with a smooth kernel and a non smooth right hand side function let the integral equation be defined on the interval 1 1 we discretize by a nystrom method with nodes cos j n j 0 n this yields a linear system of algebraic equations with an n 1 n 1 matrixa generallyn has to be chosen fairly large in order to obtain an accurate approximate solution of the integral equation we show by fourier analysis thata can be approximated well by mathop a limits sim a low rank modification of the identity matrix replacinga by mathop a limits sim in the linear system of algebraic equations yields a new linear system of equations whose elements and whose solution tilde x can be computed ino n logn arithmetic operations if the kernel has two more derivatives than the right hand side function then tilde x is shown to converge optimally to the solution of the integral equation asn increases we also consider iterative solution of the linear system of algebraic equations the iterative schemes use botha anda they yield the solution ino n 2 arithmetic operations under mild restrictions on the kernel and the right hand side function finally we discuss discretization by the chebyshev galerkin method the techniques developed for the nystrom method carry over to this discretization method and we develop solution schemes that are faster than those previously presented in the literature the schemes presented carry over in a straightforward manner to fredholm integral equations of the second kind defined on a hypercube

finite element approximation of the transport of reactive solutes in porous media part 1 error estimates for nonequilibrium adsorption processes in this paper we analyze a fully practical piecewise linear finite element approximation involving numerical integration backward euler time discretization and possibly regularization of the following degenerate parabolic system arising in a model of reactive solute transport in porous media find u x t v x t such that begin eqnarray partial t u partial t v delta u f quad mbox in omega times 0 t qquad u 0 quad mbox on partial omega times 0 t partial tv k varphi u v quad mbox in omega times 0 t u cdot 0 g 1 cdot quad v cdot 0 g 2 cdot quad mbox in omega subset real d quad 1 le d le 3 end eqnarray for given data k in real f g 1 g 2 and a monotonically increasing varphi in c 0 real cap c 1 infty 0 cup 0 infty satisfying varphi 0 0 which is only locally holder continuous with exponent p in 0 1 at the origin e g varphi s equiv s p this lack of lipschitz continuity at the origin limits the regularity of the unique solution u v and leads to difficulties in the finite element error analysis

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

implicit explicit runge kutta methods for financial derivatives pricing models abstract implicit explicit runge kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach the methods are showed to be an alternative to other existing procedures for the numerical valuation of american type contracts we follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements finite differences and finite volume methods some numerical experiments are presented

high order computational methods for option valuation under multifactor models abstract many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second order discretization schemes a higher order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high order convergence rates for the solution of multi factor option problems these new schemes employ galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two asset option problems and time integration of the resulting semi discrete systems requires the computation of a single matrix exponential the computations indicate that this combination of high order finite elements and exponential time integration leads to efficient algorithms for multi factor problems highly accurate european prices are obtained with relatively coarse meshes and high order convergence rates are also observed for options with the american early exercise feature various numerical examples are provided for illustrating the accuracy of the option prices for heston s and bates stochastic volatility models and for two asset problems under merton s jump diffusion model

finite element error estimates in l 2 l2 for regularized discrete approximations to the obstacle problem this work is concerned with quasi optimal a priori finite element error estimates for the obstacle problem in the l 2 norm the discrete approximations are introduced as solutions to a finite element discretization of an accordingly regularized problem the underlying domain is only assumed to be convex and polygonally or polyhedrally bounded such that an application of pointwise error estimates results in a rate less than two in general the main ingredient for proving the quasi optimal estimates is the structural and commonly used assumption that the obstacle is inactive on the boundary of the domain then localization techniques are used to estimate the global l 2 error by a local error in the inner part of the domain where higher regularity for the solution can be assumed and a global error for the ritz projection of the solution which can be estimated by standard techniques we validate our results by numerical examples

a class of explicit multistep exponential integrators for semilinear problems a class of explicit multistep exponential methods for abstract semilinear equations is introduced and analyzed it is shown that the k step method achieves order k for appropriate starting values which can be computed by auxiliary routines or by one strategy proposed in the paper together with some implementation issues numerical illustrations are also provided

equivalence of nystrom s method and fourier methods for the numerical solution of fredholm integral equations on demontre un resultat d equivalence de la methode de nystrom et de la methode de fourier avec des fonctions orthogonales pour la solution numerique des equations integrales de fredholm on en deduit des estimations d erreur pour la methode de nystrom on decrit exactement la regle de quadrature pour l integration numerique

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

pointwise a posteriori error estimates for monotone semi linear equations we derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi linear equations the estimates hold for lagrange elements of any fixed order non smooth nonlinearities and take numerical integration into account the proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately and then applying the continuous maximum principle no geometric mesh constraints are thus required numerical experiments illustrate reliability and efficiency properties of the corresponding estimators and investigate the performance of the resulting adaptive algorithms in terms of the polynomial order and quadrature

explicit exponential runge kutta methods for semilinear parabolic problems the aim of this paper is to analyze explicit exponential runge kutta methods for the time integration of semilinear parabolic problems the analysis is performed in an abstract banach space framework of sectorial operators and locally lipschitz continuous nonlinearities we commence by giving a new and short derivation of the classical nonstiff order conditions for exponential runge kutta methods but the main interest of our paper lies in the stiff case by expanding the errors of the numerical method in terms of the solution we derive new order conditions that form the basis of our error bounds for parabolic problems we show convergence for methods up to order four and we analyze methods that were recently presented in the literature these methods have classical order four but they do not satisfy some of the new conditions therefore an order reduction is expected we present numerical experiments which show that this order reduction in fact arises in practical examples based on our new conditions we finally construct methods that do not suffer from order reduction

stirling probability and q bosons normal ordering of powers of the bosonic number operator can be used to define a discrete probability distribution associated with the number of elements of a random set we represent these random sets by vectors in a hilbert space and obtain q bosons

l error estimate for an approximation of a parabolic variational inequality almost optimall convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase stefan problem the discretization employs piecewise linear finite elements in space and the backward euler scheme in time by means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation the latter bound is obtained by using the smoothing property of the galerkin method

finite element approximation of a model reaction diffusion problem with a non lipschitz nonlinearity given 0 andp 0 1 we consider the following problem findu such that begin gathered delta u lambda u p 0in omega hfill u 1on partial omega hfill end gathered where 2 is a smooth convex domain we prove optimalh 1 andl error bounds for the standard continuous piecewise linear galerkin finite element approximation in addition we analyse a more practical approximation using numerical integration on the nonlinear term finally we consider a modified nonlinear sor algorithm which is shown to be globally convergent for solving the algebraic system derived from the more practical approximation

additive runge kutta methods for stiff ordinary differential equations certain pairs of runge kutta methods may be used additively to solve a system of n differential equations x j t x g t x pairs of methods of order p 4 where one method is semiexplicit and a stable and the other method is explicit are obtained these methods require the lu factorization of one n x n matrix and p evaluations of g in each step it is shown that such methods have a stability property which is similar to a stability property of perturbed linear differential equations 1 introduction in a recent article 2 the authors showed that certain pairs of methods may be used in an additive fashion to solve an initial value problem for a system of n differential equations x f t x x a xo a t b

a free boundary problem for semilinear elliptic equations etude de la nature de la frontiere libre u 0 pour un probleme de dirichlet semilineaire

parity of the partition function abstract although much is known about the partition function little is known about its parity for the polynomials d x d x 2 1 24 where d 23 mod 24 we show that there are infinitely many m resp n for which p d m is even resp p d n is odd if there is at least one such m resp n we bound the first m and n if any in terms of the class number h d for prime d we show that there are indeed infinitely many even values to this end we construct new modular generating functions using generalized borcherds products and we employ galois representations and locally nilpotent hecke algebras

numerical solution of the obstacle problem by the penalty method part ii time dependent problems in a previous paper we considered the possibility of using the penalty method to get approximations of the solution of elliptic variational inequalities in this note we show that the same idea can be used for the treatment of parabolic problems by coupling the penalty parameter e and the discretization parametersh and t quasi optimal error estimates are derived in suitable norms

on the approximation of optimal stopping problems with application to financial mathematics the determination of the value function associated with a given reward and stochastic process represents an important class of stochastic control problem in particular the expectations of such processes may be represented as solutions of variational inequalities of evolutionary type typically characterized by their high number of degrees of freedom unbounded domains and lack of natural boundary conditions in this paper we introduce two methodologies for computing the value function of optimal stopping associated with general stochastic processes our results are implemented utilizing finite elements and are validated using problems taken from financial mathematics

finite element approximation of a semilinear elliptic problem with a singular nonlinearity given nu in bbb r we consider the following problem find u 0 such that where omega subset bbb r d d 1 2 or 3 and c 0 in bar omega we prove h 1 and l infty error bounds for the standard continuous piecewise linear galerkin finite element approximation with a weakly acute triangulation our bounds are nearly optimal in addition for d 1 and 2 and c in bbb r we analyze a more practical scheme involving numerical integration on the nonlinear term we obtain nearly optimal h 1 and l infty error bounds for d 1 for this case we also present some numerical results

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

a fem approximation of a two phase obstacle problem and its a posteriori error estimate this paper is concerned with the two phase obstacle problem a type of a variational free boundary problem we recall the basic estimates of repin and valdman 2015 and verify them numerically on two examples in two space dimensions a solution algorithm is proposed for the construction of the finite element approximation to the two phase obstacle problem the algorithm is not based on the primal convex and nondifferentiable energy minimization problem but on a dual maximization problem formulated for lagrange multipliers the dual problem is equivalent to a quadratic programming problem with box constraints the quality of approximations is measured by a functional a posteriori error estimate which provides a guaranteed upper bound of the difference of approximated and exact energies of the primal minimization problem the majorant functional in the upper bound contains auxiliary variables and it is optimized with respect to them to provide a sharp upper bound a space density of the nonlinear related part of the majorant functional serves as an indicator of the free boundary

a numerical analysis of variational valuation techniques for derivative securities in this paper we consider the partial differential equations approach for valuing european and american style options on multiple assets we use a method of lines finite element implementation available in the software package femlab in order to solve the variational inequality that characterizes the american style option as well as the partial differential equation that defines the european style option for two and three state variables a detailed study of the approximation error is provided including a theoretical estimate an asymptotic analysis the space time distribution and the dependence on the size of the truncation domain

sieved partition functions and q binomial coefficients the q binomial coefficient is a polynomial in q given an integer t and a residue class r modulo t a sieved q binomial coefficient is the sum of those terms whose exponents are congruent to r modu o t in this paper explicit polynomial identities in qt are given for sieved q binomial coefficients as a limiting case generating functions for the sieved partition function are found as multidimensional theta functions a striking corollary of this representation is the proof of ramanujan s congruences mod 5 7 and 11 by exhibiting symmetry groups of orders 5 7 and 11 of explicit quadratic forms we also verify the subbarao conjecture for t 3 t 5 and t 10

finite element solution of diffusion problems with irregular data diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break down of the solution s regularity this may drastically reduce the accuracy of discretization schemes over the whole interval of integration unless certain precautions are taken the diagonal pade schemes of order 2 combined with a standard finite element discretization usually require an unnatural step size restriction in order to achieve even locally optimal accuracy it is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and in the case 1 does not increase the costs

shout options a framework for pricing contracts which can be modified by the investor a shout option may be broadly defined as a financial contract which can be modified by the holder according to specified rules in a simple example the holder could have the right to set the strike of an option equal to the current value of the underlying asset in such a case the holder effectively has the right to select when to take ownership of an at the money option more generally the holder could have multiple rights along these lines in some cases with a limit placed on the number of rights which may be exercised within a given time period e g four times per year the value of these types of contracts can be estimated by solving a system of interdependent linear complementarity problems this paper describes a general framework for the valuation of complex types of shout options numerical issues related to interpolation and choice of timestepping method are considered in detail some illustrative examples are provided

a modification of cauchy s method for quadratic equations the plan of this paper is to obtain one point iterative methods with any r order of convergence when they are applied to approximate solutions of quadratic equations in banach spaces to do this we consider real cauchy s method and under certain natural modifications it is extended to banach spaces some applications are also provided

multigrid methods for boundary integral equations multigrid methods are applied for solving algebraic systems of equations that occur to the numerical treatment of boundary integral equations of the first and second kind these methods originally formulated for partial differential equations of elliptic type combine relaxation schemes and coarse grid corrections the choice of the relaxation scheme is found to be essential to attain a fast convergent iterative process theoretical investigations show that the presented relaxation scheme provides a multigrid algorithm of which the rate of convergence increases with the dimension of the finest grid this is illustrated for the calculation of potential flow around an aerofoil

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

effect of numerical integration for elliptic obstacle problems an elliptic obstacle problem is approximated by piecewise linear finite elements with numerical integration on the penalty and forcing terms this leads to diagonal nonlinearities and thereby to a practical scheme optimal error estimates in the maximum norm are derived the proof is based on constructing suitable super and subsolutions that exploit the special structure of the penalization and using quite precise pointwise error estimates for an associated linear elliptic problem with quadrature via the discrete maximum principle

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence

multiple grid methods for the solution of fredholm integral equations of the second kind in this paper multiple grid methods are applied for the fast solution of the large nonsparse systems of equations that arise from the discretization of fredholm integral equations of the second kind various multiple grid schemes both with nystrom and with direct interpolation are considered for these iterative methods the rates of convergence are derived using the collectively compact operator theory by anselone and atkinson estimates for the asymptotic computational complexity are given which show that the multiple grid schemes result in e n2 arithmetic operations