Håkon Hoel

Håkon Hoel, Kody J. H. Law, Raul Tempone

This work embeds a multilevel Monte Carlo sampling strategy into the Monte Carlo step of the ensemble Kalman filter (EnKF) in the setting of finite dimensional signal evolution and noisy discrete-time observations. The signal dynamics is assumed to be governed by a stochastic differential equation (SDE), and a hierarchy of time grids is introduced for multilevel numerical integration of that SDE. The resulting multilevel EnKF is proved to asymptotically outperform EnKF in terms of computational cost versus approximation accuracy. The theoretical results are illustrated numerically.

Håkon Hoel, Kenneth Hvistendahl Karlsen, Nils Henrik Risebro et al.

Finite volume methods are proposed for computing approximate pathwise entropy/kinetic solutions to conservation laws with a rough path dependent flux function. For a convex flux, it is demonstrated that rough path oscillations may lead to "cancellations" in the solution. Making use of this property, we show that for $α$-H{ö}lder continuous rough paths the convergence rate of the numerical methods can improve from $\mathcal{O}(\text{COST}^{-γ})$, for some $γ\in \left[α/(12-8α), α/(10-6α)\right]$, with $α\in (0, 1)$, to $\mathcal{O}(\text{COST}^{-\min(1/4,α/2)})$. Numerical examples support the theoretical results.

Christian Bayer, Håkon Hoel, Petr Plecháč et al.

Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-independent Schrödinger observables for a molecular system with an electron spectral gap, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding Hamilton-Jacobi equation, bypasses the usual separation of nuclei and electron wave functions, includes caustic states and gives a different perspective on the Born-Oppenheimer approximation, Schrödinger Hamiltonian systems and numerical simulation in molecular dynamics modeling at constant energy microcanonical ensembles.

Eric Joseph Hall, Håkon Hoel, Mattias Sandberg et al.

We derive computable error estimates for finite element approximations of linear elliptic partial differential equations (PDE) with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.

Håkon Hoel, Juho Häppölä, Raúl Tempone

A formal mean square error expansion (MSE) is derived for Euler--Maruyama numerical solutions of stochastic differential equations (SDE). The error expansion is used to construct a pathwise a posteriori adaptive time stepping Euler--Maruyama method for numerical solutions of SDE, and the resulting method is incorporated into a multilevel Monte Carlo (MLMC) method for weak approximations of SDE. This gives an efficient MSE adaptive MLMC method for handling a number of low-regularity approximation problems. In low-regularity numerical example problems, the developed adaptive MLMC method is shown to outperform the uniform time stepping MLMC method by orders of magnitude, producing output whose error with high probability is bounded by TOL>0 at the near-optimal MLMC cost rate O(TOL^{-2}log(TOL)^4).