the dynamics of maps tangent to the identity and with nonvanishing index let f be a germ of a holomorphic self map of c 2 at the origin o tangent to the identity and with o as a nondicritical isolated fixed point a parabolic curve for f is a holomorphic f invariant curve with o on the boundary attracted by o under the action of f it has been shown by m abate 2001 that if the characteristic direction v p t o c 2 has residual index not belonging to q then there exist parabolic curves for f tangent to v in this paper we prove using a different method that the conclusion still holds just assuming that the residual index is not vanishing at least when f is regular along v

analytical and formal classifications of quasi homogeneous foliations in c2 0 we prove a result of classification for germs of formal and convergent quasi homogeneous foliations in c 2 with fixed separatrix basically we prove that the analytical and formal class of such a foliation depend respectively only on the analytical and formal class of its representation of projective holonomy

embeddings of submanifolds and normal bundles this paper studies the embeddings of a complex submanifold s inside a complex manifold m in particular we are interested in comparing the embedding of s in m with the embedding of s as the zero section in the total space of the normal bundle ns of s in m we explicitly describe some cohomological classes allowing to measure the difference between the two embeddings in the spirit of the work by grauert griffiths and camacho movasati and sad we are also able to explain the geometrical meaning of the separate vanishing of these classes our results hold for any codimension but even for curves in a surface we generalize previous results due to laufert and camacho movasati and sad

keeping track of limit cycles this article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields it introduces the notion of limit periodic sets and shows that the hilbert s 16th problem can be reduced to the existence of a uniform bound on its cyclicity it also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the noetherian property of the bautin ideal

analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients we consider the problem of numerically approximating statistical moments of the solution of a time dependent linear parabolic partial differential equation pde whose coefficients and or forcing terms are spatially correlated random fields the stochastic coefficients of the pde are approximated by truncated karhunen loeve expansions driven by a finite number of uncorrelated random variables after approximating the stochastic coefficients the original stochastic pde turns into a new deterministic parametric pde of the same type the dimension of the parameter set being equal to the number of random variables introduced after proving that the solution of the parametric pde problem is analytic with respect to the parameters we consider global polynomial approximations based on tensor product total degree or sparse polynomial spaces and constructed by either a stochastic galerkin or a stochastic collocation approach we derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional monte carlo method for this class of problems copyright 2009 john wiley sons ltd

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

error analysis of a stochastic collocation method for parabolic partial differential equations with random input data a stochastic collocation method for solving linear parabolic partial differential equations with random coefficients forcing terms and initial conditions is analyzed the input data are assumed to depend on a finite number of random variables unlike previous analyses a wider range of situations are considered including input data that depend nonlinearly on the random variables and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate the exponential decay of the interpolation error in the probability space for both finite element semidiscrete spatial discretizations and for finite element crank nicolson fully discrete space time discretizations ingredients in the convergence analysis include the proof of the analyticity with respect to the probabilistic parameters of the semidiscrete and fully discrete approximate solutions a numerical example is provided to illustrate the analyses

residual indices of holomorphic maps relative to singular curves of fixed points on surfaces let m be a two dimensional complex manifold and let f m to m be a holomorphic map that fixes pointwise a possibly singular compact reduced and globally irreducible curve c subset m we give a notion of degeneracy of f at a point of c it turns out that f is non degenerate at one point if and only if it is non degenerate at every point of c when f is non degenerate on c we define a residual index for f at each point of c then we prove that the sum of the indices is equal to the self intersection number of c

higher order pathwise numerical approximations of spdes with additive noise in this article the pathwise numerical approximation of semilinear parabolic stochastic partial differential equations spdes driven by additive noise is considered a new numerical scheme for the time and space discretization of such spdes is proposed and this scheme is shown to converge for such spdes faster than standard numerical schemes such as the linear implicit euler scheme or the linear implicit crank nicholson scheme the suggested scheme takes advantage of the smoothing effect of the dominant linear operator and of two linear functionals of the noise process of the spde the abstract setting under which the scheme is analyzed includes spdes driven by fractional brownian motions too

semi implicit integration factor methods on sparse grids for high dimensional systems numerical methods for partial differential equations in high dimensional spaces are often limited by the curse of dimensionality though the sparse grid technique based on a one dimensional hierarchical basis through tensor products is popular for handling challenges such as those associated with spatial discretization the stability conditions on time step size due to temporal discretization such as those associated with high order derivatives in space and stiff reactions remain here we incorporate the sparse grids with the implicit integration factor method iif that is advantageous in terms of stability conditions for systems containing stiff reactions and diffusions we combine iif in which the reaction is treated implicitly and the diffusion is treated explicitly and exactly with various sparse grid techniques based on the finite element and finite difference methods and a multi level combination approach the overall method is found to be efficient in terms of both storage and computational time for solving a wide range of pdes in high dimensions in particular the iif with the sparse grid combination technique is flexible and effective in solving systems that may include cross derivatives and non constant diffusion coefficients extensive numerical simulations in both linear and nonlinear systems in high dimensions along with applications of diffusive logistic equations and fokker planck equations demonstrate the accuracy efficiency and robustness of the new methods indicating potential broad applications of the sparse grid based integration factor method

holder classes with boundary conditions as interpolation spaces on etudie la caracterisation de certains espaces d interpolation reels entre le domaine d un operateur differentiel elliptique avec des conditions aux limites generales et l espace de banach de fonctions continues dans lequel le domaine est plonge

germs of holomorphic vector fields in c3 without a separatrix oblatum 28 xii 1988 26 iii 1990 10 x 1990 13 vi 1991 30 ix 1991 supported by universite de toulouse conacyt cnrs and caicyt 87 336 fulltext preview

topological invariants and equidesingularization for holomorphic vector fields on etudie l invariance du nombre de milnor et ses applications on demontre un theoreme d equidesingularisation

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

nonlinear filtering revisited a spectral approach the objective of this paper is to develop an approach to nonlinear filtering based on the cameron martin version of wiener chaos expansion this approach gives rise to a new numerical scheme for nonlinear filtering the main feature of this algorithm is that it allows one to separate the computations involving the observations from those dealing only with the system parameters and to shift the latter off line

darboux integrating factors inverse problems abstract we discuss planar polynomial vector fields with prescribed darboux integrating factors in a nondegenerate affine geometric setting we establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems and show that the space of vector fields with a given integrating factor modulo a subspace of explicitly known standard vector fields has finite dimension for several classes of examples we determine this space explicitly

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

weak approximation of stochastic partial differential equations the nonlinear case we study the error of the euler scheme applied to a stochastic partial dierential equation we prove that as it is often the case the weak order of convergence is twice the strong order a key ingredient in our proof is malliavin calculus which enables us to get rid of the irregular terms of the error we apply our method to the case a semilinear stochastic heat equation driven by a space time white noise

solving parabolic stochastic partial differential equations via averaging over characteristics the method of characteristics the averaging over the characteristic formula and the weak sense numerical integration of ordinary stochastic differential equations together with the monte carlo technique are used to propose numerical methods for linear stochastic partial differential equations spdes their orders of convergence in the mean square sense and in the sense of almost sure convergence are obtained a variance reduction technique for the monte carlo procedures is considered layer methods for linear and semilinear spdes are constructed and the corresponding convergence theorems are proved the approach developed is supported by numerical experiments

on maximal regularity for the cauchy dirichlet parabolic problem with fractional time derivative abstract we prove two maximal regularity results in spaces of continuous and holder continuous functions for a linear cauchy dirichlet problem with a fractional time derivative d t this derivative is intended in the sense of caputo and is taken in 0 2 in case 1 we obtain maximal regularity results for parabolic problems already known in mathematical literature

extendable cohomologies for complex analytic varieties we introduce a cohomology called extendable cohomology for abstract complex singular varieties based on suitable differential forms beside a study of the general properties of such a cohomology we show that given a complex vector bundle one can compute its topological chern classes using the extendable chern classes defined via a chern weil type theory we also prove that the localizations of the extendable chern classes represent the localizations of the respective topological chern classes thus obtaining an abstract residue theorem for compact singular complex analytic varieties as an application of our theory we prove a camacho sad type index theorem for holomorphic foliations of singular complex varieties

invariant curves and topological invariants for real plane analytic vector fields abstract let z be a germ of a singular real analytic vector field at o r 2 we give conditions on the multiplicity and the milnor number of z which imply that the foliation defined by z has a characteristic orbit or an analytic invariant curve with the hypothesis that z is a real generalized curve then it is proved that for a non dicritical real generalized curve the multiplicity mod 2 is invariant under bilipshitz homeomorphisms preserving foliations

nonlinear perturbations of a class of holomorphic semigroups of growth order α by comparison theorems for volterra equations abstract this paper studies nonlinear perturbation of a holomorphic semigroup of growth order in a banach space x the generator a of a holomorphic semigroup is assumed to be almost sectorial and it is also assumed that the part a 0 in the closure x 0 of the domain d a is sectorial a nonlinear perturbing operator b is assumed to be locally continuous from a subset of a real interpolation space between x 0 and d a 0 into x existence and uniqueness of mild solutions to the associated cauchy problems are proved by using comparison theorems for integral equations of volterra type the result obtained is applied to show the global well posedness of drift diffusion systems with no flux boundary conditions in the two dimensional case

index theorems for holomorphic self maps let m be a complex manifold and s subset m a possibly singular subvariety of m let f colon m to m be a holomorphic map such that f restricted to s is the identity we show that one can associate to f a holomorphic section x f of a sheaf related to the embedding of s in m and that such a section reads the dynamical behavior of f along s in particular we prove that under generic hypotheses the canonical section x f induces a holomorphic action in the sense of bott on the normal bundle of the regular part of s in m and this allows to obtain for holomorphic self maps with non isolated fixed points index theorems similar to camacho sad baum bott and variation index theorems for holomorphic foliations finally we apply our index theorems to obtain information about topology and dynamics of holomorphic self maps of surfaces with a compact curve of fixed points

weak order for the discretization of the stochastic heat equation in this paper we study the approximation of the distribution of x t hilbert valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as dx t ax t dt q 1 2 d w t quad x 0 x in h quad t in 0 t driven by a gaussian space time noise whose covariance operator q is given we assume that a alpha is a finite trace operator for some alpha 0 and that q is bounded from h into d a beta for some beta geq 0 it is not required to be nuclear or to commute with a the discretization is achieved thanks to finite element methods in space parameter h 0 and implicit euler schemes in time parameter delta t t n we define a discrete solution x n h and for suitable functions varphi defined on h we show that e varphi x n h e varphi x t o h 2 gamma delta t gamma noindent where gamma 1 alpha beta let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations

a stochastic collocation method for the second order wave equation with a discontinuous random speed in this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions the speed is piecewise smooth in the physical space and depends on a finite number of random variables the numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space this approach leads to the solution of uncoupled deterministic problems as in the monte carlo method we consider both full and sparse tensor product spaces of orthogonal polynomials we provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data in particular we show that unlike in elliptic and parabolic problems the solution to hyperbolic problems is not in general analytic with respect to the random variables therefore the rate of convergence may only be algebraic an exponential fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data we present numerical examples which confirm the analysis and show that the collocation method is a valid alternative to the more traditional monte carlo method for this class of problems

a mild ito formula for spdes this article introduces a new somehow mild ito type formula for the solution process of a stochastic partial differential equation of evolutionary type

weighted admissibility and wellposedness of linear systems in banach spaces we study linear control systems in infinite dimensional banach spaces governed by analytic semigroups for p in 1 infty and alpha in mathbb r we introduce the notion of l p admissibility of type alpha for unbounded observation and control operators generalizing earlier work by le merdy j london math soc 2 67 2003 pp 715 738 and haak and le merdy houston j math 31 2005 pp 1153 1167 we give conditions under which l p admissibility of type alpha is characterized by boundedness conditions which are similar to those in the well known weiss conjecture we also study l p wellposedness of type alpha for the full system here we use recent ideas due to pruss and simonett arch math basel 82 2004 pp 415 431 our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take lebesgue and besov spaces as state space this extends the considerations in c i byrnes e j dynam control systems 8 2002 pp 341 370 to non hilbertian settings and to p neq2

numerical methods for stochastic systems preserving symplectic structure stochastic hamiltonian systems with multiplicative noise phase flows of which preserve symplectic structure are considered to construct symplectic methods for such systems sufficiently general fully implicit schemes i e schemes with implicitness both in deterministic and stochastic terms are needed a new class of fully implicit methods for stochastic systems is proposed increments of wiener processes in these fully implicit schemes are substituted by some truncated random variables a number of symplectic integrators is constructed special attention is paid to systems with separable hamiltonians some results of numerical experiments are presented they demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones

an inverse problem for a parabolic integrodifferential model in the theory of combustion abstract in this paper we investigate an abstract inverse problem that can be applied to the evolution equation u t t x u t x 0 t k t s u s x d s u t t x d x f u t x u t x given suitable initial boundary conditions here f is a given function and in the case f u t x u t x e u t x the evolution equation has applications in the theory of combustion since we identify the convolution memory kernel k and the temperature u we associate an additional measurement on the temperature of type x u t x d x g t where and g are given functions the novelty with respect to the existing literature is the presence of the term u t t x d x in the evolution equation that is motivated by a model in the theory of combustion we prove a local in time existence theorem and a global in time uniqueness result

high order integrator for sampling the invariant distribution of a class of parabolic stochastic pdes with additive space time noise we introduce a time integrator to sample with high order of accuracy the invariant distribution for a class of semilinear spdes driven by an additive space time noise combined with a postprocessor the new method is a modification with negligible overhead of the standard linearized implicit euler maruyama method we first provide an analysis of the integrator when applied for sdes finite dimension where we prove that the method has order 2 for the approximation of the invariant distribution instead of 1 we then perform a stability analysis of the integrator in the semilinear spde context and we prove in a linear case that a higher order of convergence is achieved numerical experiments including the semilinear heat equation driven by space time white noise confirm the theoretical findings and illustrate the efficiency of the approach

meromorphic vector fields with single valued solutions on complex surfaces abstract we study ordinary differential equations in the complex domain given by meromorphic vector fields on compact complex kahler surfaces we prove that if such an equation has a maximal single valued solution in particular an entire one whose image is not contained in a proper analytic subset then up to a bimeromorphic transformation either the vector field is holomorphic or it preserves a fibration

cubature on wiener space it is well known that there is a mathematical equivalence between solving parabolic partial differential equations pdes and the integration of certain functionals on wiener space monte carlo simulation of stochastic differential equations sdes is a naive approach based on this underlying principle in finite dimensions it is well known that cubature can be a very effective approach to integration we discuss the appropriate extension of this idea to wiener space in the process we develop high order numerical schemes valid for high dimensional sdes and semi elliptic pdes

complete algebraic vector fields on affine surfaces part ii in this work we study the singularities at infinity of algebraic vector fields in dimension 2 these singularities will be classified under a mild assumption the general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in part ii our main results are local and therefore can be carried over more general surfaces whereas we deal with complete vector fields the results also apply to complete ones thanks to a theorem of forstneric 7

weak convergence analysis of the linear implicit euler method for semilinear stochastic partial differential equations with additive noise abstract in this paper we analyze the weak error of a semi discretization in time by the linear implicit euler method for semilinear stochastic partial differential equations spdes with additive noise the main result reveals how the weak order depends on the regularity of noise and that the order of weak convergence is twice that of strong convergence in particular the linear implicit euler method for spdes driven by trace class noise achieves an almost optimal order 1 for arbitrarily small 0

weak approximations a malliavin calculus approach we introduce a variation of the proof for weak approximations that is suitable for studying the densities of stochastic processes which are evaluations of the flow generated by a stochastic differential equation on a random variable that may be anticipating our main assumption is that the process and the initial random variable have to be smooth in the malliavin sense furthermore if the inverse of the malliavin covariance matrix associated with the process under consideration is sufficiently integrable then approximations for densities and distributions can also be achieved we apply these ideas to the case of stochastic differential equations with boundary conditions and the composition of two diffusions

galerkin finite element methods for stochastic parabolic partial differential equations we study the finite element method for stochastic parabolic partial differential equations driven by nuclear or space time white noise in the multidimensional case the discretization with respect to space is done by piecewise linear finite elements and in time we apply the backward euler method the noise is approximated by using the generalized l2 projection operator optimal strong convergence error estimates in the l2 and dot h 1 norms with respect to the spatial variable are obtained the proof is based on appropriate nonsmooth data error estimates for the corresponding deterministic parabolic problem the computational analysis and numerical example are given

local levi flat hypersurfaces invariants by a codimension one holomorphic foliation in this paper we study codimension one holomorphic foliations leaving invariant real analytic hypersurfaces in particular we prove that a germ of real analytic levi flat hypersurface with sufficiently small singular set is given by the zeroes of the imaginary part of a holomorphic function

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

sparse high order fem for elliptic spdes abstract we describe the analysis and the implementation of two finite element fe algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients they are based on separation of deterministic and stochastic parts of the input data by a karhunen loeve expansion truncated after m terms with a change of measure we convert the problem to a sequence of m dimensional parametric deterministic problems two sparse high order polynomial approximations of the random solution s joint pdf s parametrized in the input data s karhunen loeve expansion coordinates are analyzed a sparse stochastic galerkin fem sparse sgfem and a sparse stochastic collocation fem sparse scfem a priori and a posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution s joint pdf s to the stochastic regularity of the input data scfem and sgfem yield deterministic approximations of the random solutions joint pdf s that converge spectrally in the number of deterministic problems to be solved numerical examples with random inputs of small correlation length in diffusion problems are presented high order gpc approximations of solutions with stochastic parameter spaces of dimension up to m 80 are computed on workstations

holomorphic dynamics near germs of singular curves let m be a two dimensional complex manifold p m and open image in new window a germ of holomorphic foliation of m at p letopen image in new window be a germ of an irreducible possibly singular curve at p in m which is a separatrix for open image in new window we prove that if the camacho sad suwa index ind open image in new window then there exists another separatrix for open image in new window at p a similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points

invariant curves and analytic integrability of a planar vector field abstract we give an expression of the irreducible invariant curves at the singular point for analytically integrable systems we provide an expression of its primitive first integral this fact allows us to obtain necessary conditions of analytic integrability at degenerate singular points

on the poincare problem abstract we derive some restrictions on the possible degrees of algebraic invariant curves and on the possible form of algebraic integrating factors for plane polynomial vector fields whose stationary points at infinity satisfy a certain genericity condition the method is elementary and we show by example that it also yields strong results for certain non generic vector fields

adaptive weak approximation of stochastic differential equations adaptive time stepping methods based on the monte carlo euler method for weak approximation of ito stochastic differential equations are developed the main result is new expansions of the computational error with computable leading order term in a posteriori form based on stochastic flows and discrete dual backward problems the expansions lead to efficient and accurate computation of error estimates adaptive algorithms for either stochastic time steps or deterministic time steps are described numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps stochastic time steps use brownian bridges and require more work for a given number of time steps deterministic time steps may yield more time steps but require less work for example in the limit of vanishing error tolerance the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps c 2001 john wiley sons inc

the law of the euler scheme for stochastic differential equations i convergence rate of the distribution function we study the approximation problem ofef x t byef x t n where x t is the solution of a stochastic differential equation x t n is defined by the euler discretization scheme with stept n andf is a given function for smoothf s talay and tubaro have shown that the erroref x t f x t n can be expanded in powers of 1 n which permits to construct romberg extrapolation precedures to accelerate the convergence rate here we prove that the expansion exists also whenf is only supposed measurable and bounded under an additional nondegeneracy condition of hormander type for the infinitesimal generator of x t to obtain this result we use the stochastic variations calculus in the second part of this work we will consider the density of the law ofx t n and compare it to the density of the law ofx t

a recursive sparse grid collocation method for differential equations with white noise we consider a sparse grid collocation method in conjunction with a time discretiza tion of the differential equations for computing expectations of functionals of solutions to differential equations perturbed by time dependent white noise we first analyze the error of smolyak s sparse grid collocation used to evaluate expectations of functionals of solutions to stochastic differential equations discretized by the euler scheme we show theoretically and numerically that this algo rithm can have satisfactory accuracy for small noise magnitude or small integration time however it does not converge either with decrease of the euler scheme s time step size or with increase of smolyak s sparse grid level subsequently we use this method as a building block for proposing a new algorithm by combining sparse grid collocation with a recursive procedure this approach allows us to numerically integrate linear stochastic partial differential equations over longer times which is illustrated in numerical tests on a stochastic advection diffusion equation

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

complex interpolation and fourier multipliers for the spacesb p q s andf p q s of besov hardy sobolev type the case 0 p 0 q

convergence of numerical schemes for the solution of parabolic stochastic partial differential equations we consider the numerical solution of the stochastic partial differential equation u t 2 u x 2 u w x t where w is space time white noise using finite differences for this equation gyongy has obtained an estimate of the rate of convergence for a simple scheme based on integrals of w over a rectangular grid we investigate the extent to which this order of convergence can be improved and find that better approximations are possible for the case of additive noise u 1 if we wish to estimate space averages of the solution rather than pointwise estimates or if we are permitted to generate other functionals of the noise but for multiplicative noise u u we show that no such improvements are possible

maximal lp lq regularity of the linearized initial boundary value problem for motion of compressible viscous fluids we discuss the initial boundary value problem for the linearized system of equations which describe motion of compressible viscous barotropic fluids in a bounded domain with the navier boundary condition this problem has uniquely a solution in the anisotropic sobolev space wq1 w p1 wp q2 1 n for any 1 p 1 q globally in time moreover exponentially weighted lp lq estimates for solutions globally in time can be established we prove the above properties by resolvent estimates for the linearized operator of the above system the theory of analytic semigroups on banach spaces and the operator valued fourier multiplier theorem on umd spaces

wiener chaos solutions of linear stochastic evolution equations a new method is described for constructing a generalized solution of a stochastic evolution equation existence uniqueness regularity and a probabilistic representation of this wiener chaos solution are established for a large class of equations as an application of the general theory new results are obtained for several types of the passive scalar equation

analyticity of solutions for a generalized euler equation abstract we consider the so called lake and great lake equations which are shallow water equations that describe the long time motion of an inviscid incompressible fluid contained in a shallow basin with a slowly spatially varying bottom a free upper surface and vertical side walls under the influence of gravity and in the limit of small characteristic velocities and very small surface amplitude if these equations are posed on a space periodic domain and the initial data are real analytic the solution remains real analytic for all times the proof is based on a characterization of gevrey classes in terms of decay of fourier coefficients in particular our result recovers known results for the euler equations in two and three spatial dimensions we believe the proof is new

numerical analysis of semilinear stochastic evolution equations in banach spaces the solution of stochastic evolution equations generally relies on numerical computation here usually the main idea is to discretize the spde spatially obtaining a system of sdes that can be solved by e g the euler scheme in this paper we investigate the discretization error of semilinear stochastic evolution equations in lp spaces resp banach spaces the space discretization may be done by galerkin approximation for the time discretization we consider the implicit euler the explicit euler scheme and the crank nicholson scheme in the last section we give some examples i e we consider an spdes driven by nuclear wiener noise approximated by wavelets and delay equation approximated by finite differences

on the differentiability of solutions of stochastic evolution equations with respect to their initial values abstract in this article we study the differentiability of solutions of parabolic semilinear stochastic evolution equations sees with respect to their initial values we prove that if the nonlinear drift coefficients and the nonlinear diffusion coefficients of the considered sees are n times continuously frechet differentiable then the solutions of the considered sees are also n times continuously frechet differentiable with respect to their initial values moreover a key contribution of this work is to establish suitable enhanced regularity properties of the derivative processes of the considered see in the sense that the dominating linear operator appearing in the see smoothes the higher order derivative processes

wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics in this paper we propose a numerical method based on wiener chaos expansion and apply it to solve the stochastic burgers and navier stokes equations driven by brownian motion the main advantage of the wiener chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner the separation principle effectively reduces a stochastic equation to its associated propagator a system of deterministic equations for the coefficients of the wiener chaos expansion simple formulas for statistical moments of the stochastic solution are presented these formulas only involve the solutions of the propagator we demonstrate that for short time solutions the numerical methods based on the wiener chaos expansion are more efficient and accurate than those based on the monte carlo simulations

a multistage wiener chaos expansion method for stochastic advection diffusion reaction equations using wiener chaos expansion wce we develop numerical algorithms for solving second order linear parabolic stochastic partial differential equations spdes we propose a deterministic wce based algorithm for computing moments of the spde solutions without any use of the monte carlo technique we also compare the proposed deterministic algorithm with two other numerical methods based on the monte carlo technique and demonstrate that the new method is more efficient for highly accurate solutions numerical tests verify that the scheme is of mean square order o frac delta n 2 sqrt n 1 for diffusion and for diffusion reaction spdes with constant or variable coefficients where delta is the time step and n is the wiener chaos order

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

weak approximation of the stochastic wave equation we investigate the accuracy of approximation of e f u t where u t email protected 0 is the solution of the stochastic wave equation driven by the space time white noise and f is an r valued function defined on the hilbert space l 2 r the approximation is done by the leap frog scheme we show that under certain conditions on f the approximation by the leap frog scheme is of order two

liouvillian integration and bernoulli foliations analytic foliations in the 2 dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable it is shown that such a condition leads to the existence of a liouville type first integral and under generic extra conditions it is proven that these foliations can be defined by bernoulli equations

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

on invariance of plurigenera for foliations on surfaces let x be a smooth algebraic surface a foliation f on x is roughly speaking a subline bundle t f of the tangent bundle of x the dual of t f is called the canonical bundle of the foliation k f in the last few years birational methods have been successfully used in order to study foliations more precisely geometric properties of the foliation are translated into properties of the canonical bundle of the foliation one of the most important invariants describing the properties of a line bundle l is its kodaira dimension kappa l which measures the growth of the global sections of l and its tensor powers the kodaira dimension of a foliation f is defined as the kodaira dimension of its canonical bundle kappa k f in their fundamental works brunella and mcquillan give a classfication of foliations on surfaces on the model of enriques kodaira classification of surfaces the next step is the study of the behaviour of families of foliations brunella proves that for a family of foliations x t f t of dimension one on surfaces satisfying certain hypotheses of regularity the kodaira dimension of the foliation does not depend on t by analogy with siu s invariance of plurigenera it is natural to ask whether for a family of foliations x t f t the dimensions of global sections of the canonical bundle and its powers depend on t in this talk we will discuss to which extent an invariance of plurigenera for foliations is true and under which hypotheses on the family of foliations it holds

a semi discrete scheme for the stochastic nonlinear schrodinger equation we study the convergence of a semi discretized version of a numerical scheme for a stochastic nonlinear schrodinger equation the nonlinear term is a power law and the noise is multiplicative with a stratonovich product our scheme is implicit in the deterministic part of the equation as is usual for conservative equations we also use an implicit discretization of the noise which is better suited to stratonovich products we consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate however in the semi discrete case no ito formula is available and we have to use a discrete form of this tool also in the course of the proof we need to introduce a cut off of the diffusion coefficient which allows to treat the nonlinearity then we prove convergence by a compactness argument due to the presence of noise and to the implicit discretization of the noise this is rather complicated and technical we finally obtain convergence of the discrete solutions in various topologies

a free boundary problem modeling electrostatic mems i linear bending effects the dynamical and stationary behaviors of a fourth order evolution equation with clamped boundary conditions and a singular nonlocal reaction term which is coupled to an elliptic free boundary problem in a non smooth domain are investigated the equation arises in the modeling of microelectromechanical systems and includes two positive parameters lambda and varepsilon related to the applied voltage and the aspect ratio of the device respectively local and global well posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as a criterion for global existence excluding the occurrence of finite time singularities which are not physically relevant existence of a stable steady state is shown for sufficiently small lambda non existence of steady states is also established when varepsilon is small enough and lambda is large enough depending on varepsilon

quasi monte carlo finite element methods for a class of elliptic partial differential equations with random coefficients in this paper quasi monte carlo qmc methods are applied to a class of elliptic partial differential equations pdes with random coefficients where the random coefficient is parametrized by a co

weak error analysis for semilinear stochastic volterra equations with additive noise we prove a weak error estimate for the approximation in space and time of a semilinear stochastic volterra integro differential equation driven by additive space time gaussian noise we treat this equation in an abstract framework in which parabolic stochastic partial differential equations are also included as a special case the approximation in space is performed by a standard finite element method and in time by an implicit euler method combined with a convolution quadrature the weak rate of convergence is proved to be twice the strong rate as expected our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics the proof does not rely on a kolmogorov equation instead it is based on a duality argument from malliavin calculus