alternative techniques for solving systems of nonlinear equations abstract in this paper a selection of familiar iterative techniques are compared for their ability to solve the large nonlinear equation systems typically encountered in econometrics the general convergence theory of methods based on jacobi and gauss seidel iterations including a new comer fast gauss seidel is taken as far as possible where extensions are impossible numerical comparisons are made on some representative econometric models the winner is fast gauss seidel previous convergence and comparison theorems have concerned only certain specialised equation systems not encountered in econometrics a shortcoming of econometric theory in that neither convergence analysis nor evaluation of these techniques has been available to guide the applied economist is thereby overcome

a stable runge kutta methods for semilinear evolution equations we consider semilinear evolution equations for which the linear part generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces in this setting we prove the existence of solutions which are temporally smooth in the norm of the lowest rung of the scale for an open set of initial data on the highest rung of the scale under the same assumptions we prove that a class of implicit a stable runge kutta semidiscretizations in time of such equations are smooth as maps from open subsets of the highest rung into the lowest rung of the scale under the additional assumption that the linear part of the evolution equation is normal or sectorial we prove full order convergence of the semidiscretization in time for initial data on open sets our results apply in particular to the semilinear wave equation and to the nonlinear schrodinger equation 2012 elsevier inc

on the solution sets of particular classes of linear interval systems we characterize the solution set s of real linear system ax b by a set of inequalities if b lies between some given bounds b b and if the n n coefficient matrix a varies similarly between two bounds a and a in addition we restrict a to a particular class of matrices for instance the class of the symmetric the skew symmetric the persymmetric the toeplitz and the hankel matrices respectively in this way we generalize the famous oettli prager criterion numer math 6 1964 405 results by hartfiel numer math 35 1980 355 and the contents of the papers in r b kearfott v kreinovich eds applications of interval computations kluwer boston ma 1996 pp 61 79 and siam j matrix anal appl 18 1997 693

on the convergence of rational approximations of semigroups on intermediate spaces we generalize a result by brenner and thomee on the rate of convergence of rational approximation schemes for semigroups using abstract interpolation techniques we obtain convergence on a continuum of intermediate spaces between the banach space x and the domain of a certain power of the generator of the semigroup the sharpness of the results is also discussed

maximal inner boxes in parametric ae solution sets with linear shape we consider linear systems of equations a p x b p where the parameters p are linearly dependent and come from prescribed boxes and the sets of solutions defined in various ways which have linear boundary one fundamental problem is to compute a box being inside a parametric solution set we first consider parametric tolerable solution sets being convex polyhedrons for such solution sets we prove that finding a maximal inner box is an np hard problem this justifies our exponential linear programming methods for computing maximal inner boxes we also propose a polynomial heuristic that yields a large but not necessarily the maximal inner box next we discuss how to apply the presented linear programming methods for finding large inner estimations of general parametric ae solution sets with linear shape numerical examples illustrate the properties of the methods and their application

off spectral analysis of bergman kernels the asymptotic analysis of bergman kernels with respect to exponentially varying measures near emergent interfaces has attracted recent attention such interfaces typically occur when the associated limiting bergman density function vanishes on a portion of the plane the off spectral region this type of behavior is observed when the metric is negatively curved somewhere or when we study partial bergman kernels in the context of positively curved metrics in this work we cover these two situations in a unified way for exponentially varying weights on the complex plane we obtain a uniform asymptotic expansion of the coherent state of depthn rooted at an off spectral point which we also refer to as the root function at the point in question the expansion is valid in the entire off spectral component containing the root point and protrudes into the spectrum as well this allows us to obtain error function transition behavior of the density of states along the smooth interface previous work on asymptotic expansions of bergman kernels is typically local and valid only in the bulk region of the spectrum which contrasts with our non local expansions

convergence analysis of strang splitting for vlasov type equations a rigorous convergence analysis of the strang splitting algorithm for vlasov type equations in the setting of abstract evolution equations is provided it is shown that under suitable assumptions the convergence is of second order in the time step tau as an example it is verified that the vlasov poisson equations in 1 1 dimensions fit into the framework of this analysis further numerical experiments for the latter case are presented

on the optimal complex extrapolation of the complex cayley transform abstract the cayley transform f f a i a 1 i a with a c n n and 1 a where denotes spectrum and its extrapolated counterpart f a c 0 and 1 a are of significant theoretical and practical importance see e g a hadjidimos m tzoumas on the principle of extrapolation and the cayley transform linear algebra appl in press in this work we extend the theory in 8 to cover the complex case specifically we determine the optimal extrapolation parameter c 0 for which the spectral radius of the extrapolated cayley transform f a is minimized assuming that a h where h is the smallest closed convex polygon and satisfies o 0 h as an application we show how a complex linear system with coefficient a certain class of indefinite matrices which the adi type method of hermitian skew hermitian splitting fails to solve can be solved in a best way by the aforementioned method

the principle of extrapolation and the cayley transform abstract the cayley transform f i a 1 i a with a c n n and 1 a where denotes spectrum is of significant theoretical importance and interest and has many practical applications e g in the solution of the linear complementarity problem lcp in the solution of linear systems arising from the discretization of model problems elliptic pdes by alternating direction implicit adi iterative methods in the solution of complex linear systems by adi type methods of hermitian skew hermitian or normal skew hermitian splittings etc in the present work we apply the principle of extrapolation to generalize the cayley transform and determine in an optimal sense the extrapolation parameter involved so that problems in many practical applications are solved much more efficiently

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

on the equivalence of extrapolation and richardson s iteration and its applications abstract sometime ago two articles g opfer g schober richardson s iteration for nonsymmetric matrices linear algebra appl 58 1984 343 361 a hadjidimos the optimal solution to the problem of complex extrapolation of a first order scheme linear algebra appl 62 1984 241 261 appeared the same year in this journal solving the problem of optimizing the convergence factor of the stationary richardson s iteration i e x m 1 x m a ax m b m 0 1 2 for the solution of the system ax b a c n n det a 0 b c n in opfer and schober g opfer g schober richardson s iteration for nonsymmetric matrices linear algebra appl 58 1984 343 361 when the spectrum of a was contained in a certain compact set s 1 not including the origin the optimal parameter a was determined in hadjidimos a hadjidimos the optimal solution to the problem of complex extrapolation of a first order scheme linear algebra all 62 1984 241 261 for the solution of ax b the consistent linear scheme x m 1 tx m c m 0 1 2 for which the spectrum of t was contained in a certain compact convex set s 2 not including the point 1 was considered and the optimal extrapolation parameter of its extrapolation scheme i e x m 1 1 i t x m c m 0 1 2 was determined because the approach in obtaining the optimal parameters in the two articles was quite different it seems it was difficult for any one to notice their equivalence and therefore to use them even in trivial cases in this article we show the equivalence of the optimal results in opfer and schober g opfer g schober richardson s iteration for nonsymmetric matrices linear algebra appl 58 1984 343 361 and hadjidimos a hadjidimos the optimal solution to the problem of complex extrapolation of a first order scheme linear algebra all 62 1984 241 261 explore them further obtain new results and give applications for cases which may occur in practice finally based on observations regarding the optimal two cyclic successive overrelaxation sor and accelerated overrelaxation aor methods in two real case problems we solve the problem of optimal two cyclic aor in the more general complex case

runge kutta convolution quadrature methods for well posed equations with memory runge kutta based convolution quadrature methods for abstract well posed linear and homogeneous volterra equations non necessarily of sectorial type are developed a general representation of the numerical solution in terms of the continuous one is given the error and stability analysis is based on this representation which for the particular case of the backward euler method also shows that the numerical solution inherits some interesting qualitative properties such as positivity of the exact solution numerical illustrations are provided

safe bounds in linear and mixed integer linear programming current mixed integer linear programming solvers are based on linear programming routines that use floating point arithmetic occasionally this leads to wrong solutions even for problems where all coefficients and all solution components are small integers an example is given where many state of the art milp solvers fail it is then shown how using directed rounding and interval arithmetic cheap pre and postprocessing of the linear programs arising in a branch and cut framework can guarantee that no solution is lost at least for mixed integer programs in which all variables can be bounded rigorously by bounds of reasonable size

a fresh view on the tolerance approach to sensitivity analysis in linear programming abstract the tolerance approach to sensitivity analysis in linear programming aims at finding a unique numerical value tolerance representing the maximum absolute perturbation which can be applied simultaneously and independently on each right hand side or objective coefficient without affecting the optimality of the given basis some extensions have been proposed in the literature which allow for individual tolerances for each coefficient thus enlarging the tolerance region in this paper we review the main results concerning the approach giving new and simpler proofs and we propose an efficient geometric algorithm returning a tolerance region that is maximal with respect to inclusion we compare our method with the existing ones on two examples showing how a priori information can be naturally exploited by our algorithm to further enlarge individual tolerances

smooth invariant manifolds in banach spaces with nonuniform exponential dichotomy we establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v a t v f t v in banach spaces assuming that the linear equation v a t v admits a nonuniform exponential dichotomy our proof of the ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of ck functions with holder continuous kth derivative this is a closed subset of the space of continuous functions with the supremum norm by an apparently not so well known lemma of henry see proposition 3 the estimates showing that the functions maintain the original bounds when transformed under the fixed point operator are obtained through a careful application of the faa di bruno formula for the higher derivatives of the compositions see 31 and 35 as a consequence we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class ck

optimization of extrapolated cayley transform with non hermitian positive definite matrix abstract for the extrapolated cayley transform we give necessary and sufficient conditions for guaranteeing its convergence and contraction in the euclidean norm we derive upper bounds for the convergence and the contraction factors and compute the optimal parameters minimizing these upper bounds and the corresponding optimal values of these upper bounds numerical computations show that these upper bounds are reasonably sharp compared with the exact convergence and the exact contraction factors of the extrapolated cayley transform respectively

richardson s iteration for nonsymmetric matrices abstract to solve the linear n n system 1 ax a for any nonsingular matrix a richardson s iteration 2 x j 1 x j j ax j a j 1 2 n which is applied in a cyclic manner with cycle length n is investigated where the j are free parameters the objective is to minimize the error x n 1 x where x is the solution of 1 if the spectrum of a is known to lie in a compact set s one is led to the chebyshev type approximation problem 3 min p 1 v n max z s p z where v n is the linear span of z z 2 z n if p solves 3 then the reciprocals of the zeros of p are optimal iteration parameters j it is shown that for a real problem 1 the iteration 2 can be carried out with real arithmetic alone even when there are complex j the stationary case n 1 is solved completely i e for all compact sets s the problem 3 is solved explicitly as a consequence the converging stationary iteration processes can be characterized for arbitrary closed disks s the problem 3 can be solved for all n n and a simple proof is provided the lemniscates associated with s are introduced they appear as an important tool for studying the stability of the iteration 2

improved enclosure for some parametric solution sets with linear shape consider linear algebraic systems where the elements of the matrix and the right hand side vector depend linearly on a number of interval parameters we prove some sufficient conditions for the united parametric solution set of such a system to have linear boundary these conditions imply an equivalent representation of the parametric system where each parameter appears once in a diagonal matrix the latter representation allows us to expand the scope of applicability of the best known so far interval method developed by a neumaier and a pownuk for enclosing the parametric solution set and to generalize the method for systems where the parameter dependencies connect the matrix and the right hand side vector some examples demonstrate that parametric solution sets with linear boundary appear in various application domains the generalized method improves the solution enclosure and the proven sufficient conditions can be helpful for solving various other interval problems

on splitting methods for schrodinger poisson and cubic nonlinear schrodinger equations we give an error analysis of strang type splitting integrators for nonlinear schrodinger equations for schrodinger poisson equations with an h 4 regular solution a first order error bound in the h 1 norm is shown and used to derive a second order error bound in the l 2 norm for the cubic schrodinger equation with an h 4 regular solution first order convergence in the h 2 norm is used to obtain second order convergence in the l 2 norm basic tools in the error analysis are lie commutator bounds for estimating the local error and h m conditional stability for error propagation where m 1 for the schrodinger poisson system and m 2 for the cubic schrodinger equation

time averaging under fast periodic forcing of parabolic partial differential equations exponential estimates the phases of a large class of parabolic partial differential equations with rapid time periodic forcing can be separated up to exponential small errors the originally nonautonomous equation is transformed such that the nonautonomous terms are exponentially small in the period h of the forcing this is a counterpart for partial differential equations of the theorem by a neishtadt 1984 j appl math mech 48 134 139 for ordinary differential equations in our case the exponential rate depends on time t and the estimates have the form h exp c t h 1 3

comparison of fast boundary element methods on parametric surfaces we compare fast black box boundary element methods on parametric surfaces in r3 these are the adaptive cross approximation the multipole method based on interpolation and the wavelet galerkin scheme the surface representation by a piecewise smooth parameterization is in contrast to the common approximation of surfaces by panels nonetheless parametric surface representations are easily accessible from computer aided design cad and are recently topic of the studies in isogeometric analysis especially we can apply two dimensional interpolation in the multipole method a main feature of this approach is that the cluster bases and the respective moment matrices are independent of the geometry this results in a superior compression of the far field compared to other cluster methods

rigorous lower and upper bounds in linear programming we consider the computation of rigorous lower and upper error bounds for the optimal value of linear programming problems the input data of the lp problem may be exactly given or may vary between given lower and upper bounds the results are then verified for the family of lp problems with input data inside these bounds in many cases only a small computational effort is required for problems with finite simple bounds the rigorous lower bound of the optimal value can be computed with o n2 operations the error bounds can be used as well to perform a sensitivity analysis provided the width of the uncertainties is not too large some numerical examples are presented

runge kutta methods for parabolic equations and convolution quadrature we study the approximation properties of runge kutta time discretizations of linear and semilinear parabolic equations including incompressible navier stokes equations we derive asymptotically sharp error bounds and relate the temporal order of convergence which is generally noninteger to spatial regularity and the type of boundary conditions the analysis relies on an interpretation of runge kutta methods as convolution quadratures in a different context these can be used as efficient computational methods for the approximation of convolution integrals and integral equations they use the laplace transform of the confolution kernal via a discrete operational calculus 23 refs 2 tabs

stability under galerkin truncation of a stable runge kutta discretizations in time we consider semilinear evolution equations for which the linear part is normal and generates a strongly continuous semigroup and the nonlinear part is sufficiently smooth on a scale of hilbert spaces we approximate their semiflow by an implicit a stable runge kutta discretization in time and a spectral galerkin truncation in space we show regularity of the galerkin truncated semiflow and its time discretization on open sets of initial values with bounds that are uniform in the spatial resolution and the initial value we also prove convergence of the space time discretization without any condition that couples the time step to the spatial resolution then we estimate the galerkin truncation error for the semiflow of the evolution equation its runge kutta discretization and their respective derivatives showing how the order of the galerkin truncation error depends on the smoothness of the initial data our results apply in particular to the semilinear wave equation and to the nonlinear schr odinger equation

runge kutta time semidiscretizations of semilinear pdes with non smooth data we study semilinear evolution equations frac mathrm du mathrm dt au b u dudt au b u posed on a hilbert space mathcal y y where a is normal and generates a strongly continuous semigroup b is a smooth nonlinearity from mathcal y ell d a ell yl d al to itself and ell in i subseteq 0 l l i 0 l l ge 0 l 0 0 l in i 0 l i in particular the one dimensional semilinear wave equation and nonlinear schrodinger equation with periodic neumann and dirichlet boundary conditions fit into this framework we discretize the evolution equation with an a stable runge kutta method in time retaining continuous space and prove convergence of order o h p ell p 1 o hpl p 1 for non smooth initial data u 0 in mathcal y ell u0 yl where ell le p 1 l p 1 for a method of classical order p extending a result by brenner and thomee for linear systems our approach is to project the semiflow and numerical method to spectral galerkin approximations and to balance the projection error with the error of the time discretization of the projected system numerical experiments suggest that our estimates are sharp

solving over constrained systems of non linear interval equations and its robotic application abstract this paper presents and describes in details an original method developed to solve over constrained systems of non linear interval equations that arise namely in parameter identification problems deriving from physical models and uncertain measurements our approach consists of computing an interval enclosure of the least square solution set and an inner box of tolerable solution set this method is applied in a detailed example and some interesting results obtained for the calibration of a cable driven robot are shown

the optimal solution to the problem of complex extrapolation of a first order scheme abstract given the linear stationary first order iterative scheme x m 1 tx m c for the solution of the linear complex system i t x c its extrapolated complex scheme x m 1 t x m c t 1 i t is considered the problem which is studied and solved is that of determining an optimum value for over the set of complex numbers such that the extrapolated scheme considered converges asymptotically as fast as possible

the tolerance approach to sensitivity analysis in linear programming in constrast to ordinary sensitivity analysis in linear programming the tolerance approach considers simultaneous and independent changes in the objective function coefficients and in the right hand side terms this approach yields a maximum tolerance percentage such that as long as selected coefficients or terms are accurate to within that percentage of their estimated values the same basis is optimal in particular if the objective function coefficients are accurate to within the maximum tolerance percentage of their specified values then the same solution is optimal

analysis of the domain mapping method for elliptic diffusion problems on random domains in this article we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains in particular based on the decay of the karhunen loeve expansion of the domain perturbation field we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension for the implementation of a related approximation scheme like quasi monte carlo quadrature stochastic collocation etc we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field this simplifies the implementation and yields a non intrusive approach having this machinery at hand we can easily transfer it to stochastic interface problems the theoretical findings are complemented by numerical examples for both stochastic interface problems and boundary value problems on random domains

a multivariate faa di bruno formula with applications a multivariate faa di bruno formula for computing arbitrary partial derivatives of a function composition is presented it is shown by way of a general identity how such derivatives can also be expressed in the form of an infinite series applications to stochastic processes and multivariate cumulants are then delineated

parameterized outer estimation of ae solution sets to parametric interval linear systems abstract we consider linear algebraic equations where the elements of the matrix and of the right hand side vector are linear functions of interval parameters and their parametric ae solution sets which are defined by applying universal and existential quantifiers to the interval parameters usually interval methods find numerical interval vector that contains an ae solution set in this work we propose a method that generates an outer estimate of a parametric ae solution set in form of a linear parametric interval function called parameterized outer solution p solution parameterized outer solution is proposed for the parametric united solution set in kolev 2014 and takes precedence over the classical interval solution enclosure when the latter is part of other problems involving the same parameters the method we present generalizes the method from kolev 2016 for parametric ae solution sets it is also a parameterized analogue of a method from popova and hladik 2013 and produces the same interval enclosure as the method from the last reference

construction of the largest sensitivity region for general linear programs this paper develops an alternative approach to post optimality analysis for general linear program models which provides the largest sensitivity region of any single or simultaneous change of right hand side of constraints and the coefficients of decision variables in the objective function the goal is a theoretical unification of various types of sensitivity analyses as well as advancement in the practical implementation of post optimality analysis we extend our proposed approach in the construction of sensitivity region to maintain the degenerate vertex for models with degenerate optimal solution as well as maintaining the multiple solutions for the models with non unique optimal solutions as a by product the paper resolves the paradoxical situation known as the more for less less for more situations the proposed method is based on the given optimal solution s therefore it is easy to understand easy to implement and provides useful information to the decision makers the methodology and their computational algorithms are presented and discussed in the context of some illustrative numerical examples