Espen R. Jakobsen

Simone Cifani, Espen R. Jakobsen, Kenneth H. Karlsen

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (Lévy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments.

Simone Cifani, Espen R. Jakobsen

We study a class of degenerate convection diffusion equations with a fractional nonlinear diffusion term. These equations are natural generalizations of anomalous diffusion equations, fractional conservations laws, local convection diffusion equations, and some fractional Porous medium equations. In this paper we define weak entropy solutions for this class of equations and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable functions. Then we introduce a monotone conservative numerical scheme and prove convergence toward an Entropy solution in the class of bounded integrable functions of bounded variation. We then extend the well-posedness results to non-local terms based on general Lévy type operators, and establish some connections to fully non-linear HJB equations. Finally, we present some numerical experiments to give the reader an idea about the qualitative behavior of solutions of these equations.

Simone Cifani, Espen R. Jakobsen

First we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations. Secondly we develop a new Kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods - even when the principal derivatives have any fractional order between 1 and 2! The class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general Levy diffusion. Special cases are conservation laws, fractional conservation laws, certain fractional porous medium equations, and new strongly degenerate equations.

Simone Cifani, Espen R. Jakobsen

We introduce and analyze a spectral vanishing viscosity approximation of periodic fractional conservation laws. The fractional part of these equations can be a fractional Laplacian or other non-local operators that are generators of pure jump Lévy processes. To accommodate for shock solutions, we first extend to the periodic setting the Kružkov-Alibaud entropy formulation and prove well-posedness. Then we introduce the numerical method, which is a non-linear Fourier Galerkin method with an additional spectral viscosity term. This type of approximation was first introduced by Tadmor for pure conservation laws. We prove that this {\em non-monotone} method converges to the entropy solution of the problem, that it retains the spectral accuracy of the Fourier method, and that it diagonalizes the fractional term reducing dramatically the computational cost induced by this term. We also derive a robust $L^1$-error estimate, and provide numerical experiments for the fractional Burgers' equation.

Fabio Camilli, Espen R. Jakobsen

We construct a finite element like scheme for fully non-linear integro-partial differential equations arising in optimal control of jump-processes. Special cases of these equations include optimal portfolio and option pricing equations in Finance. The schemes are monotone and robust. We prove that they converge in very general situations, including degenerate equations, multiple dimensions, relatively low regularity of the data, and for most (if not all) types of jump-models used in Finance. In all cases we provide (probably optimal) error bounds. These bounds apply when grids are unstructured and integral terms are very singular, two features that are new or highly unusual in this setting.

S. Knutsen Furset, Geir-Arne Fuglstad, Espen R. Jakobsen

The Matérn covariance model is ubiquitous in spatial modelling, but there is no default choice for spatio-temporal modelling. In this paper, we consider the recently proposed ``diffusion-based'' extension of the spatial Matérn covariance model to a spatio-temporal non-separable covariance model that allows fractional smoothnesses in space and in time. The model is described in terms of a space-time fractional stochastic partial differential equation, but currently proposed computational approaches have strong restrictions on the possible smoothnesses in time. We propose a discretization method based on rational approximations in time to handle arbitrary smoothnesses, which leads to a vector autoregressive moving average process (VARMA). We prove that the covariance function of the approximation converges pointwise, determine explicit convergence rates as a function of spatial and temporal resolutions and the accuracy of the rational approximation, and conduct numerical verification to demonstrate small pointwise error for low orders of the VARMA process. Through a simulation study, we demonstrate that the parameters can be estimated back and that correctly specifying the temporal smoothness is especially important for forecasting. The approach is illustrated for three months of daily mean temperatures in mainland France.

Félix del Teso, Jørgen Endal, Espen R. Jakobsen

We develop unified and easy to use framework to study robust fully discrete numerical methods for nonlinear degenerate diffusion equations $$ \partial_t u-\mathfrak{L}[φ(u)]=f(x,t) \qquad\text{in}\qquad \mathbb{R}^N\times(0,T), $$ where $\mathfrak{L}$ is a general symmetric Lévy type diffusion operator. Included are both local and nonlocal problems with e.g. $\mathfrak{L}=Δ$ or $\mathfrak{L}=-(-Δ)^{\frac\alpha2}$, $α\in(0,2)$, and porous medium, fast diffusion, and Stefan type nonlinearities $φ$. By robust methods we mean that they converge even for nonsmooth solutions and under very weak assumptions on the data. We show that they are $L^p$-stable for $p\in[1,\infty]$, compact, and convergent in $C([0,T];L_{\text{loc}}^p(\mathbb{R}^N))$ for $p\in[1,\infty)$. The first part of this project is given in \cite{DTEnJa18a} and contains the unified and easy to use theoretical framework. This paper is devoted to schemes and testing. We study many different problems and many different concrete discretizations, proving that the results of \cite{DTEnJa18a} apply and testing the schemes numerically. Our examples include fractional diffusions of different orders, and Stefan problems, porous medium, and fast diffusion nonlinearities. Most of the convergence results and many schemes are completely new for nonlocal versions of the equation, including results on high order methods, the powers of the discrete Laplacian method, and discretizations of fast diffusions. Some of the results and schemes are new even for linear and local problems.

Félix del Teso, Jørgen Endal, Espen R. Jakobsen

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem problem for the anomalous diffusion equation $\partial_tu-\mathcal{L}^μ[φ(u)]=0$. Here $\mathcal{L}^μ$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function $φ:\mathbb{R} \to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain $L^1$-contraction and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations.

Kristian Debrabant, Espen R. Jakobsen

For linear and fully non-linear diffusion equations of Bellman-Isaacs type, we introduce a class of approximation schemes based on differencing and interpolation. As opposed to classical numerical methods, these schemes work for general diffusions with coefficient matrices that may be non-diagonal dominant and arbitrarily degenerate. In general such schemes have to have a wide stencil. Besides providing a unifying framework for several known first order accurate schemes, our class of schemes includes new first and higher order versions. The methods are easy to implement and more efficient than some other known schemes. We prove consistency and stability of the methods, and for the monotone first order methods, we prove convergence in the general case and robust error estimates in the convex case. The methods are extensively tested.

Simone Cifani, Espen R. Jakobsen, Kenneth H. Karlsen

We propose, analyze, and demonstrate a discontinuous Galerkin method for fractal conservation laws. Various stability estimates are established along with error estimates for regular solutions of linear equations. Moreover, in the nonlinear case and whenever piecewise constant elements are utilized, we prove a rate of convergence toward the unique entropy solution. We present numerical results for different types of solutions of linear and nonlinear fractal conservation laws.

Guy Barles, Espen R. Jakobsen

We obtain non-symmetric upper and lower bounds on the rate of convergence of general monotone approximation/numerical schemes for parabolic Hamilton Jacobi Bellman Equations by introducing a new notion of consistency. We apply our general results to various schemes including finite difference schemes, splitting methods and the classical approximation by piecewise constant controls.

Espen R. Jakobsen

Recently, Krylov, Barles, and Jakobsen developed the theory for estimating errors of monotone approximation schemes for the Bellman equation (a convex Isaacs equation). In this paper we consider an extension of this theory to a class of non-convex multidimensional Isaacs equations. This is the first result of this kind for non-convex multidimensional fully non-linear problems. To get the error bound, a key intermediate step is to introduce a penalization approximation. We conclude by (i) providing new error bounds for penalization approximations extending earlier results by e.g. Benssousan and Lions, and (ii) obtaining error bounds for approximation schemes for the penalization equation using very precise a priori bounds and a slight generalization of the recent theory of Krylov, Barles, and Jakobsen.