Jørgen Endal

Matej Benko, Iwona Chlebicka, Jørgen Endal et al.

We present a significant advancement in the field of Langevin Monte Carlo (LMC) methods by introducing the Inexact Proximal Langevin Algorithm (IPLA). This novel algorithm broadens the scope of problems that LMC can effectively address while maintaining controlled computational costs. IPLA extends LMC's applicability to potentials that are convex, strongly convex in the tails, and exhibit polynomial growth, beyond the conventional $L$-smoothness assumption. Moreover, we extend LMC's applicability to super-quadratic potentials and offer improved convergence rates over existing algorithms. Additionally, we provide bounds on all moments of the Markov chain generated by IPLA, enhancing its analytical robustness.

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.