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Publication# Stabilized Numerical Methods for Stochastic Differential Equations driven by Diffusion and Jump-Diffusion Processes

Abstract

Stochastic models that account for sudden, unforeseeable events play a crucial role in many different fields such as finance, economics, biology, chemistry, physics and so on. That kind of stochastic problems can be modeled by stochastic differential equations driven by jump-diffusion processes. In addition, there are situations, where a stochastic model is based on stochastic differential equations with multiple scales. Such stochastic problems are called stiff and lead for classical explicit integrators such as the Euler-Maruyama method to time stepsize restrictions due to stability issues. This opens the door for stabilized explicit numerical methods to efficiently tackle such situations. In this thesis we introduce first a stabilized multilevel Monte Carlo method for stiff stochastic differential equations. Using S-ROCK methods we show that this approach is very efficient for stochastic problems with multiple scales, but also for nonstiff problems with a significant noise part. Further, we present an improved version of the stabilized multilevel Monte Carlo method by considering S-ROCK methods with a higher weak order of convergence. Then we extend the S-ROCK methods to jump-diffusion processes. We study in detail the strong order of convergence of the newly introduced methods and we discuss the corresponding mean square stability domains. In the next part we present the multilevel Monte Carlo method for jump-diffusion processes. We state and prove a theorem that indicates the computational cost required to achieve a certain mean square accuracy. In the numerical section we compare the multilevel Monte Carlo approach to two variance reduction techniques, the antithetic and the control variates. We also show how the S-ROCK method for jump-diffusion processes, introduced in this thesis, can be used to create a stabilized multilevel Monte Carlo method for jump-diffusions that handles stiffness and considers the inclusion of jumps at the same time. Finally, we propose in this thesis a variable time stepping algorithm that uses S-ROCK methods to approximate weak solutions of stiff stochastic differential equations. A rigorous analytical study is carried out to derive a computable leading term of the time discretization error and an adaptive algorithm is suggested that adapts the time grid and adjusts the number of stages of the S-ROCK method simultaneously.

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The quantification of uncertainties can be particularly challenging for problems requiring long-time integration as the structure of the random solution might considerably change over time. In this respect, dynamical low-rank approximation (DLRA) is very appealing. It can be seen as a reduced basis method, thus solvable at a relatively low computational cost, in which the solution is expanded as a linear combination of a few deterministic functions with random coefficients. The distinctive feature of the DLRA is that both the deterministic functions and random coefficients are computed on the fly and are free to evolve in time, thus adjusting at each time to the current structure of the random solution. This is achieved by suitably projecting the dynamics onto the tangent space of a manifold consisting of all random functions with a fixed rank. In this thesis, we aim at further analysing and applying the DLR methods to time-dependent problems.Our first work considers the DLRA of random parabolic equations and proposes a class of fully discrete numerical schemes.Similarly to the continuous DLRA, our schemes are shown to satisfy a discrete variational formulation.By exploiting this property, we establish the stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a ``parabolic'' type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions. The discrete variational formulation is further applied in our second work, which derives a-priori and a-posteriori error estimates for the discrete DLRA of a random parabolic equation obtained by the three newly-proposed schemes. Under the assumption that the right-hand side of the dynamical system lies in the tangent space up to a small remainder, we show that the solution converges with standard convergence rates w.r.t. the time, spatial, and stochastic discretization parameters, with constants independent of singular values.We follow by presenting a residual-based a-posteriori error estimation for a heat equation with a random forcing term and a random diffusion coefficient which is assumed to depend affinely on a finite number of independent random variables. The a-posteriori error estimate consists of four parts: the finite element method error, the time discretization error, the stochastic collocation error, and the rank truncation error. These estimators are then used to drive an adaptive choice of FE mesh, collocation points, time steps, and time-varying rank.The last part of the thesis examines the idea of applying the DLR method in data assimilation problems, in particular the filtering problem. We propose two new filtering algorithms. They both rely on complementing the DLRA with a Gaussian component. More precisely, the DLR portion captures the non-Gaussian features in an evolving low-dimensional subspace through interacting particles, whereas each particle carries a Gaussian distribution on the whole ambient space. We study the effectiveness of these algorithms on a filtering problem for the Lorenz-96 system.

Mathematical models involving multiple scales are essential for the description of physical systems. In particular, these models are important for the simulation of time-dependent phenomena, such as the heat flow, where the Laplacian contains mixed and indistinguishable fast and slow modes. Stationary problems can also exhibit a multiscale nature. For example, elliptic equations governed by a diffusion coefficient with strong discontinuities have solutions characterized by regions with a high gradient. Simulating such models is very demanding, as the computational cost of standard numerical methods is usually ruled by the fastest dynamics or the smallest scale.
In the first part of this thesis, we develop multirate integration methods for deterministic and stochastic time-dependent problems with disparate time-scales. The cost of traditional schemes for such problems is prohibitive due to step size restrictions in the explicit case or solutions to large nonlinear systems in the implicit case. Existing multirate methods are either implicit or make use of interpolations, which trigger instabilities, or are based on a scale separation assumption, which is not satisfied by parabolic problems. Here we introduce a new framework based on modified equations which allows for the development of a whole new class of interpolation-free explicit multirate numerical methods, which do not need any scale separation, are stable and accurate. For deterministic problems, our methodology is based on the replacement of the original right-hand side by an averaged force, whose stiffness is reduced due to a fast but cheap auxiliary problem. Integrating the modified equation and the auxiliary problems by explicit schemes is generally cheaper than integrating the original problem. We thus introduce a multirate method based on stabilized explicit schemes and prove its efficiency, stability and accuracy. Numerical experiments show that standard schemes and our multirate approach provide essentially the same solutions; hence, the bottleneck caused by the stiffness of a few degrees of freedom is overcome without sacrificing accuracy. We also generalize the same framework to stochastic differential equations, where we need to introduce a damped diffusion term for which the resulting modified equation inherits the mean-square stability properties of the original problem. An interpolation-free stabilized explicit multirate method for stochastic equations is then derived.
In the second part of this thesis, we consider elliptic problems with high gradients and develop a local adaptive discontinuous Galerkin scheme. Local methods for such problems already exist in literature; however, they are usually based on iterations and have several downsides. In particular, their a priori error analysis is based on rather strong and nonphysical assumptions and they lack a rigorous a posteriori error analysis. The scheme that we propose is based on a coarse solution on the full domain which is subsequently improved by solving local elliptic problems only once on subdomains with artificial boundary conditions. The a priori error analysis is performed under minimal regularity assumptions due to the gradient discretization framework. Furthermore, we derive a posteriori error estimators based on conforming fluxes and potential reconstructions which can be used to identify the local subdomains on the fly, are free of undetermined constants and robust in singularly perturbed regimes.

Mathematical and numerical aspects of viscoelastic flows are investigated here. Two simplified mathematical models are considered. They are motivated by a splitting algorithm for solving viscoelastic flows with free surfaces. The first model is a simplified Oldroyd-B model. Existence on a fixed time interval is proved in several Banach spaces provided the data are small enough. Short time existence is also proved for arbitrarily large data in Hölder spaces for the time variable. These results are based on the maximal regularity property of the Stokes operator and on the analycity behavior of the corresponding semi-group. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates, using an implicit function theorem framework. Then, the extension of these results to a stochastic simplified Hookean dumbbells model is discussed. Because of the presence of the Brownian motion, existence in a fixed time interval, provided the data are small enough, is proved only in some of the Banach spaces considered previously. The dumbbells' elongation is split in two parts, one satisfying a standart stochastic differential equation, the other satisfying a partial differential equation with a stochastic source term. A finite element discretization in space is also proposed. Existence of the numerical solution is proved for small data, as well as a priori error estimates. A numerical algorithm for solving viscoelastic flows with free surfaces is also described. This algorithm is based on a splitting method in time and two different meshes are used for the space discretization. Convergence of the numerical model is checked for the pure extensional flow and the filling of a pipe. Then, numerical results are reported for the stretching of a filament and for jet buckling.