Anders Szepessy

Xin Huang, Aku Kammonen, Anamika Pandey et al.

The machine learning random Fourier feature method for data in high dimension is computationally and theoretically attractive since the optimization is based on a convex standard least squares problem and independent sampling of Fourier frequencies. The challenge is to sample the Fourier frequencies well. This work proves convergence of a data adaptive method based on resampling the frequencies asymptotically optimally, as the number of nodes and amount of data tend to infinity. Numerical results based on resampling and adaptive random walk steps together with approximations of the least squares problem by conjugate gradient iterations confirm the analysis for regression and classification problems.

Thomas Björk, Anders Szepessy, Raul Tempone et al.

We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on \Ito stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

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.

Ashraful Kadir, Mattias Sandberg, Anders Szepessy

Ehrenfest and Car-Parrinello molecular dynamics are computational alternatives to approximate Born-Oppenheimer molecular dynamics without solving the electron eigenvalue problem at each time-step. A non-trivial issue is to choose the artificial electron mass parameter appearing in the Car-Parrinello method to achieve both good accuracy and high computational efficiency. In this paper, we propose an algorithm, motivated by the Landau-Zener probability, to systematically choose an artificial mass dynamically, which makes the Car-Parrinello and Ehrenfest molecular dynamics methods dependent only on the problem data. Numerical experiments for simple model problems show that the time-dependent adaptive artificial mass parameter improves the efficiency of the Car-Parrinello and Ehrenfest molecular dynamics.

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.

Christian Bayer, Hakon Hoel, Ashraful Kadir et al.

The difference of the values of observables for the time-independent Schroedinger equation, with matrix valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states and the electron/nuclei mass ratio. The paper first proves an error estimate (depending on the electron/nuclei mass ratio and the probability to be in excited states) for this difference of microcanonical observables, assuming that molecular dynamics space-time averages converge, with a rate related to the maximal Lyapunov exponent. The error estimate is uniform in the number of particles and the analysis does not assume a uniform lower bound on the spectral gap of the electron operator and consequently the probability to be in excited states can be large. A numerical method to determine the probability to be in excited states is then presented, based on Ehrenfest molecular dynamics and stability analysis of a perturbed eigenvalue problem.