Haut PDF Expansion of the global error for numerical schemes solving stochastic differential equations

Expansion of the global error for numerical schemes solving stochastic differential equations

Expansion of the global error for numerical schemes solving stochastic differential equations

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Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations

Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations

ity criterium. More precisely, it consists in dening reduced approximation spaces V M and S M such that they verify simultaneously equations (69) and (71). One then shows that reduced basis are solution of an invariant subspace problem. This problem can be assimilated to an eigenproblem whose dominant eigenspace leads to the researched reduced basis functions. The GSD method has been initially introduced for solving a particular class of linear elliptic stochastic partial dierential equations [89]. In this context, the method appears as a natural extension of Hilbert Karhunen-Loève decom- position (see appendix B). Dedicated algorithms, inspired from classical algorithms for solving eigenproblems, have been proposed for the construction of reduced basis func- tions. The main advantage of these algorithms is that they only require the resolution of a few deterministic problems, with a well mastered mathematical structure, and of a few stochastic algebraic equations. Computational costs are then drastically reduced. Moreover, stochastic equations and deterministic problems being uncoupled, the GSD method allows for recovering a part of non intrusivity for Galerkin spectral approaches. In [90], the method has been used for solving a nonlinear stochastic elliptic problem for which a classical global nonlinear solver led to the resolution of successive linear stochastic problems. Each linear stochastic problem were solved by GSD algorithms proposed in [90], with a re-use and an enrichment of the reduced basis of deterministic functions at each iteration of the nonlinear solver. The GSD method has been extended to a wider class of linear problems in [91], where it has been also proposed some new ecient algorithms for building the generalized decomposition. More recently, a natural extension to the non-linear context has been proposed in [94].
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On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations

On strong $L^2$ convergence of time numerical schemes for the stochastic 2D Navier-Stokes equations

Numerical schemes and algorithms have been introduced to best approximate and con- struct solutions for PDEs. A similar approach has started to emerge for stochastic models and in particular SPDEs and has known a strong interest by the probability commu- nity. Many algorithms based on either finite difference, finite element or spectral Galerkin methods (for the space discretization), and on either Euler schemes, Crank-Nicolson or Runge-Kutta schemes (for the temporal discretization) have been introduced for both the linear and nonlinear cases. Their rates of convergence have been widely investigated. The literature on numerical analysis for SPDEs is now very extensive. When the models are either linear, have global Lipschitz properties or more generally some monotonicity prop- erty, then there is extensive literature, see [1, 2]. Moreover, in this case the convergence is proven to be in mean square. When nonlinearities are involved that are not of Lipschitz or monotone type, then a rate of convergence in mean square is difficult to obtain. Indeed, because of the stochastic perturbation, there is no way of using the Gronwall lemma after taking the expectation of the error bound because it involves a nonlinear term that is usually in a quadratic form. One way of getting around it is to localize the nonlinear term in order to get a linear inequality and then use the Gronwall lemma. This gives rise to a rate of convergence in probability, that was first introduced by J. Printems [16].
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An eXtended Stochastic Finite Element Method for solving stochastic partial differential equations on random domains

An eXtended Stochastic Finite Element Method for solving stochastic partial differential equations on random domains

Fig. 15. Example 1. Integration precision: convergence of global error indicator ε k with respect to k for isotropic and anisotropic procedures. 7.1.3 Quality of the X-SFEM solution Some examples of post-processing will now illustrate the quality of the solution obtained by X-SFEM. The solution being explicit in terms of basic random variables, post-processing can be performed at a very low cost. The following results have been obtained with the mesh presented on figure 13 and with a stochastic integration based on a stochastic partition of order k = 2. Figure 16 presents the response surfaces of the horizontal displacement for two points: x 1 = (1, 0.5) which is surely inside the random domain and x2 = (1, 0.95) which is possibly inside or outside the random domain. For both points, we observe that the approximation matches very well the exact physical solution. For point x2, we observe that the non-physical part of the solution is the unique possible prolongation (in the approximation space) of the physical part of the solution.
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An efficient spectral method for the numerical solution to stochastic differential equations

An efficient spectral method for the numerical solution to stochastic differential equations

is directly sought as a polynomial function of a Gaussian process (G t ) t∈[0,T ] and the coefficients of the expansion are obtained by solving ordinary differen- tial equations (ODE). The macroscopic quantities (mean, variance,...) can be computed directly from the coefficients of the decomposition. Thus, two major drawbacks of competing methods can be avoided: the curse of dimensionality for Polynomial chaos expansions and the resort to Monte-Carlo simulations, with slow convergence, for Euler-like or exact sampling techniques. Assuming that the ODEs can be solved with high degree of accuracy, then the error of our method mainly comes from the interpolation error.
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Numerical Methods and Deep Learning for Stochastic Control Problems and Partial Differential Equations

Numerical Methods and Deep Learning for Stochastic Control Problems and Partial Differential Equations

In this paper, we combine different ideas from the mathematics (numerical probability) and the computer science (reinforcement learning) communities to propose and compare several algo- rithms based on dynamic programming (DP), and deep neural networks (DNN) for the approxima- tion/learning of (i) the optimal policy, and then of (ii) the value function. Notice that this differs from the classical approach in DP recalled above, where we first approximate the Q-optimal state/control value function, and then approximate the optimal control. Our learning of the optimal policy is achieved in the spirit of [ HE16 ] by DNN, but sequentially in time though DP instead of a global learning over the whole period 0, . . . , N − 1. Once we get an approximation of the optimal policy, and recalling the martingale property ( 5.1.3 ), we approximate the value function by Monte-Carlo (MC) regression based on simulations of the forward process with the approximated optimal control. In particular, we avoid the issue of a priori endogenous simulation of the controlled process in the classical Q-approach. The MC regressions for the approximation of the optimal policy and/or value function, are performed according to different features leading to algorithmic variants: Performance iteration (PI) or hybrid iteration (HI), and regress now or regress later/quantization in the spirit of [ LS01 ] or [ GY04 ]. Numerical results on several applications are devoted to a companion paper [ BHLP18 ]. The theoretical contribution of the current paper is to provide a detailed convergence analysis of our three proposed algorithms: Theorem 5.4.1 for the NNContPI Algo based on control learning by performance iteration with DNN, Theorem 5.4.2 for the Hybrid-Now Algo based on control learning by DNN and then value function learning by regress-now method, and Theorem 5.4.3 for the Hybrid-LaterQ Algo based on control learning by DNN and then value function learning by regress later method combined with quantization. We rely mainly on arguments from statistical learning and non parametric regression as developed notably in the book [ GKKW02 ], for giving estimates of approximated control and value function in terms of the universal approximation error of the neural networks, and of the statistical error in the estimation of network functions.
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Solving Partial Differential Equations with Chaotic Asynchronous Schemes in Multi- Interaction Systems

Solving Partial Differential Equations with Chaotic Asynchronous Schemes in Multi- Interaction Systems

If another phenomenon acts on the same constituent, the corresponding interaction can only act before or after the previous one. If this last phenomenon is modeled by PDEs with its own numerical scheme, the resulting numerical resolution of both phenomena within asynchronous iterations can be viewed as a sort of stochastic splitting method applied to both schemes. Another way to solve this problem could be to write a new single scheme valid for the PDE system obtained by both phenomena; this is not very accurate in terms of structural coupling, thus moves away from the MIS principles for complex system modeling. Anyway, if one of the phenomena involved in the complex sys- tem and acting on such a constituent is not modeled by PDEs, its intrication with PDE-based models won’t be possible for time intervals smaller than the time step
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High weak order discretization schemes for stochastic differential equation

High weak order discretization schemes for stochastic differential equation

Our initial motivation for the study of the error ε n (f ) comes from the second point of view (numerical probabilities) but all the results of this paper are significant from both perspectives. Let us mention that the difficulty of the analysis and the interest of the result depend on the regularity of the test function f. It turns out that if f is a smooth function, then the analysis of the error is rather simple, using a Taylor type expansion in short time first, and a concatenation argument after. However, the study is much more subtle if f is simply a bounded and measurable test function - this is the so called convergence in total variation distance. A lot of work has been done in this direction in the case of the CLT. In particular, Bhattacharya and Rao [15] obtained the convergence when f (x) = 1 A (x) where A is a measurable set that belongs to a large class (including convex sets). From that point, one would hope to get such results for every measurable set A and consequently for every measurable and bounded test function f. Eventually, the seminal result of Prokhorov [62] clarified this point: He proved that the convergence in total variation in the CLT may not be obtained without some regularity assumptions on the law of Z k . Essentially, one has to assume that the law of Z k has an absolute
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Accelerated finite elements schemes for parabolic stochastic partial differential equations

Accelerated finite elements schemes for parabolic stochastic partial differential equations

1. Introduction We are interested in finite elements approximations for Cauchy problems for stochastic parabolic PDEs of the form of equation (2.1) below. Such kind of equations arise in various fields of sciences and engineering, for example in nonlinear filtering of partially observed diffusion processes. Therefore these equations have been intensively studied in the litera- ture, and theories for their solvability and numerical methods for approximations of their solutions have been developed. Since the computational effort to get reasonably accurate numerical solutions grow rapidly with the dimension d of the state space, it is important to investigate the possibility of accelerating the convergence of spatial discretisations by Richardson extrapolation. About a century ago Lewis Fry Richardson had the idea in [18] that the speed of convergence of numerical approximations, which depend on some parameter h converging to zero, can be increased if one takes appropriate linear combina- tions of approximations corresponding to different parameters. This method to accelerate the convergence, called Richardson extrapolation, works when the approximations admit a power series expansion in h at h = 0 with a remainder term, which can be estimated by a higher power of h. In such cases, taking appropriate mixtures of approximations with different parameters, one can eliminate all other terms but the zero order term and the remainder in the expansion. In this way, the order of accuracy of the mixtures is the exponent k + 1 of the power h k+1 , that estimates the remainder. For various numerical methods applied to solving deterministic partial differential equations (PDEs) it has been proved that such expansions exist and that Richardson extrapolations can spectacularly increase the speed of convergence of the methods, see, e.g., [16], [17] and [20]. Richard- son’s idea has also been applied to numerical solutions of stochastic equations. It was shown first in [21] that by Richardson extrapolation one can accelerate the weak conver- gence of Euler approximations of stochastic differential equations. Further results in this direction can be found in [14], [15] and the references therein. For stochastic PDEs the first result on accelerated finite difference schemes appears in [7], where it is shown that
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Weak error analysis for time and particle discretizations of some stochastic differential equations non linear in the sense of McKean

Weak error analysis for time and particle discretizations of some stochastic differential equations non linear in the sense of McKean

Title: Weak error analysis for time and particle discretizations of some stochastic differential equations non linear in the sense of McKean Summary: This thesis is dedicated to the theoretical and numerical study of the weak error for time and particle discretizations of some Stochastic Differential Equations non linear in the sense of McK- ean. In the first part, we address the weak error analysis for the time discretization of standard SDEs. More specifically, we study the convergence in total variation of the Euler-Maruyama scheme applied to d-dimensional SDEs with additive noise and a measurable drift coefficient. We prove weak convergence with order 1/2 when assuming boundedness on the drift coefficient. By adding more regularity to the drift, namely the drift has a spatial divergence in the sense of distributions with ρ-th power integrable with respect to the Lebesgue measure in space uniformly in time for some ρ ≥ d, the order of convergence at the terminal time improves to 1 up to some logarithmic factor. In dimension d = 1, this result is pre- served when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. In the second part of the thesis, we analyze the weak error for both time and particle discretizations of two classes of nonlinear SDEs in the sense of McKean. The first class consists in multi-dimensional SDEs with regular drift and diffusion coefficients in which the dependence in law intervenes through moments. The second class consists in one-dimensional SDEs with a constant diffusion coefficient and a singular drift coefficient where the dependence in law intervenes through the cumulative distribution function. We approximate the SDEs by the Euler-Maruyama schemes of the associated particle systems and obtain for both classes a weak order of convergence equal to 1 in time and particles. We also prove, for the second class, a trajectorial propagation of chaos result with optimal order 1/2 in particles as well as a strong order of convergence equal to 1 in time and 1/2 in particles. All our theoretical results are illustrated by numerical experiments.
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A review on estimation of stochastic differential equations for pharmacokinetic/pharmacodynamic models

A review on estimation of stochastic differential equations for pharmacokinetic/pharmacodynamic models

In a population context, there exist four estimation methods: Hermite expansion of the marginal likelihood, the combination of FOCE with the extended Kalman filtering and the use of a stochastic version of the EM algorithm and Bayesian approach. Note that Hermite expansion can not be directly extended when the diffusion is observed with noise whereas Kalman filter, Extended Kalman filter and FOCE methods are strictly designed for models with observation noise. Methods relying on the EM algorithm can be applied to models with or without noise, provided the complete likelihood belongs to the exponential family. Bayesian approach is finally the most adaptable method since it can be applied to any model (one or several trajectories, with or without noise observation) as soon as the transition density is explicit. If not, a Euler approximation has to be performed and the introduction of set of auxiliary latent data
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The Law of the Euler Scheme for Stochastic Differential Equations : II. Convergence Rate of the Density

The Law of the Euler Scheme for Stochastic Differential Equations : II. Convergence Rate of the Density

Unite´ de recherche INRIA Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE` S NANCY Unite´ de recherche INRIA Rennes, Ir[r]

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Obliquely Reflected Backward Stochastic Differential Equations

Obliquely Reflected Backward Stochastic Differential Equations

Abstract In this paper, we study existence and uniqueness to multidimensional Reflected Backward Stochastic Differential Equations in an open convex domain, allowing for oblique directions of reflection. In a Markovian framework, combining a priori estimates for penalised equations and compactness arguments, we obtain existence results under quite weak assumptions on the driver of the BSDEs and the direction of reflection, which is allowed to depend on both Y and Z. In a non Markovian framework, we obtain existence and uniqueness result for direction of reflection depending on time and Y . We make use in this case of stability estimates that require some smoothness conditions on the domain and the direction of reflection.
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Mean Field Forward-Backward Stochastic Differential Equations

Mean Field Forward-Backward Stochastic Differential Equations

The purpose of this note is to provide an existence result for the solution of Forward Backward Stochastic Differential Equations (FBSDEs) of the McKean-Vlasov type. Following the wave of interest created by the pathbreaking work of Lasry and Lions on mean field games [13, 14, 15], simple forms of Backward Stochastic Differential Equations (BSDEs) of McKean Vlasov type have been introduced and called of mean field type. Fully coupled FBSDEs are typically more involved and more difficult to solve than BSDEs. FBSDEs of mean field type occur naturally in the probabilistic analysis of mean field games and the optimal control of dynamics of the McKean Vlasov type as considered in [4, 3]. See also [1, 5, 17] for the particular case of Linear Quadratic (LQ) models. Detailed explanations on how these FBSDEs occur in these contexts and the particular models which were solved are given in Section 3 below.
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Image Denoising using Stochastic Differential Equations

Image Denoising using Stochastic Differential Equations

Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vi[r]

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Stochastic spikes and strong noise limits of stochastic differential equations

Stochastic spikes and strong noise limits of stochastic differential equations

m b+1 −x b+1 , and conditionally on M y→x the time is takes to do the down-crossing scales like λ −2 and in particular its law converges to that of the zero random variable. We shall not give the details because this is intuitive given the second computation. Let us note however that making this rigorous would not be difficult: we could describe the down-crossing from y to x with maximum m as the concatenation of two independent processes, the first is X started at y and conditioned to reach m before x, and the second is X stated at m and conditioned to reach x before touching m again. Both pieces are described as diffusions via Girsanov’s theorem, the additional drift term due to conditioning being explicit in terms of P [x,m] (y). For both pieces the trajectories
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Second order discretization schemes of stochastic differential systems for the computation of the invariant law

Second order discretization schemes of stochastic differential systems for the computation of the invariant law

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignemen[r]

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Higher-Order Numerical Schemes and Operator Splitting for Solving 3D Paraxial Wave Equations in Heterogeneous Media

Higher-Order Numerical Schemes and Operator Splitting for Solving 3D Paraxial Wave Equations in Heterogeneous Media

Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Ir[r]

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Residual equilibrium schemes for time dependent partial differential equations

Residual equilibrium schemes for time dependent partial differential equations

ically with a wide class of PDEs which admit steady states. The method is based on an underlying numerical discretization and is capable to preserve the steady states by keeping the accuracy of the underlying method. Therefore, the approach can be used also in conjunction with spectral techniques for which the direct construction of a steady state preserving approach is often impossible. Thanks to its structural simplicity it can be applied to very different PDEs, examples ranging from linear and nonlinear diffusion equations to Boltzmann equations and hyperbolic balance laws have been presented and show the effectiveness of the residual equilibrium approach. In principle the methods admits several improvement when applied to a specific kind of PDE, for example in order to preserve some peculiar property of the differential
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An Extension of Massera’s Theorem for N-Dimensional Stochastic Differential Equations

An Extension of Massera’s Theorem for N-Dimensional Stochastic Differential Equations

Additional information is available at the end of the chapter http://dx.doi.org/10.5772/intechopen.73183 Abstract In this chapter, we consider a periodic SDE in the dimension n ≥ 2, and we study the existence of periodic solutions for this type of equations using the Massera principle. On the other hand, we prove an analogous result of the Massera’s theorem for the SDE considered. Keywords: stochastic differential equations, periodic solution, Markov process, Massera theorem
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