Jason Frank

• Ph.D. 2000 Delft University of Technology
• Professor at Utrecht University

This study focuses on the motion of passive tracers induced by the joint action of tidal and residual currents in shallow seas with an irregular bottom topography. Interest in this problem has rapidly increased in recent years, because of the detection of large-scale pollution of marine waters by plastics. Early simplified models considered advecti...

Li et al. (2018) have proposed a regularization of the forward-backward sweep iteration for solving the Pontryagin maximum principle in optimal control problems. The authors prove the global convergence of the iteration in the continuous time case. In this article we show that their proof can be extended to the case of numerical discretization by s...

Li, Chen, Tai & E. (J. Machine Learning Research, 2018) have proposed a regularization of the forward-backward sweep iteration for solving the Pontryagin maximum principle in optimal control problems. The authors prove the global convergence of the iteration in the continuous time case. In this article we show that their proof can be extended to th...

Solar radiation management (SRM) has been proposed as a means to reduce global warming in spite of high greenhouse-gas concentrations and to lower the chance of warming-induced tipping points. However, SRM may cause economic damages and its feasibility is still uncertain. To investigate the trade-off between these (economic) gains and damages, we i...

Models incorporating delay have been frequently used to understand climate variability phenomena, but often the delay is introduced through an ad-hoc physical reasoning, such as the propagation time of waves. In this paper, the Mori-Zwanzig formalism is introduced as a way to systematically derive delay models from systems of partial differential e...

Solar Radiation Management (SRM) has been proposed as a means to reduce global warming in spite of high greenhouse gas concentrations and lower the chance of warming-induced tipping points. However, SRM may cause economic damages, and its feasibility is still uncertain. To investigate the trade-off between these gains and damages, we incorporate SR...

In this paper we propose a new sequential data assimilation method for non-linear ordinary differential equations with compact state space. The method is designed so that the Lyapunov exponents of the corresponding estimation error dynamics are negative, i.e. the estimation error decays exponentially fast. The latter is shown to be the case for gen...

A minimal requirement for simulating multi-scale systems is to reproduce the statistical behavior of the slow variables. In particular, a good numerical method should accurately aproximate the probability density function of the continuous-time slow variables. In this note we use results from homogenization and from backward error analysis to quant...

In this article we develop algorithms for data assimilation based upon a computational time dependent stable/unstable splitting. Our particular method is based upon shadowing refinement and synchronization techniques and is motivated by work on Assimilation in the Unstable Subspace (AUS) and Pseudo-orbit Data Assimilation (PDA). The algorithm utili...

In this article we develop algorithms for data assimilation based upon a computational time dependent stable/unstable splitting. Our particular method is based upon shadowing refinement and synchronization techniques and is motivated by work on Assimilation in the Unstable Subspace (AUS) and Pseudo-orbit Data Assimilation (PDA). The algorithm utili...

The paper presents a new state estimation algorithm for a bilinear equation representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS) equations on a torus in R2. This state equation is subject to uncertain but bounded noise in the input (Kolmogorov forcing) and initial conditions, and its output is incomplete and contains boun...

Models used in simulation may give accurate short-term trajectories but distort long-term (statistical) properties. In this work, we augment a given approximate model with a control law (a ‘thermostat’) that gently perturbs the dynamical system to target a thermodynamic state consistent with a set of prescribed (possibly evolving) observations. As...

Dynamical systems methodology is a mature complementary approach to forward simulation which can be used to investigate many aspects of climate dynamics. With this paper, a review is given on the methods to analyse deterministic and stochastic climate models and show that these are not restricted to low-dimensional toy models, but that they can be...

Point vortex models are frequently encountered in conceptual studies in geophysical fluid dynamics, but also in practical applications, for instance, in aeronautics. In spherical geometry, the motion of vortex centres is governed by a dynamical system with a known Poisson structure. We construct Poisson integration methods for these dynamics by spl...

A key challenge for an efficient splitting technique is defining the importance function. If the rare event set consists of multiple separated subsets this challenge becomes bigger since the most likely path to the rare event set may be very different from the most likely path to an intermediate level. We propose to mitigate this problem of path de...

In this paper we propose a state estimation method for linear parabolic partial differential equations (PDE) that accounts for errors in the model, truncation, and observations. It is based on an extension of the Galerkin projection method. The extended method models projection coefficients, representing the state of the PDE in some basis, by means...

We explore the direct modification of the pseudo-spectral truncation of 2D, incompressible fluid dynamics to maintain a prescribed kinetic energy spectrum. The method provides a means of simulating fluid states with defined spectral properties, for the purpose of matching simulation statistics to given information, arising from observations, theore...

The paper presents symplectic Möbius integrators for Riccati equations. All proposed methods preserve symmetry, positivity and quadratic invariants for the Riccati equations, and non-stationary Lyapunov functions. In addition, an efficient and numerically stable discretization procedure based on reinitialization for the associated linear Hamiltonia...

Many physical problems arising in biological or material sciences involve solving partial differential equations in deformable interfaces or complex domains. For instance, the surfactant (an amphiphilic molecular) which usually favors the presence in ...

As intermittent renewable energy penetrates electrical power grids more and more, assessing grid reliability is of increasing concern for grid operators. Monte Carlo simulation is a robust and popular technique to estimate indices for grid reliability, but the involved computational intensity may be too high for typical reliability analyses. We sho...

We study a Hamiltonian toy model for a Lagrangian fluid parcel in the semi-geostrophic limit which exhibits slow and fast dynamics. We first reinject unresolved fast dynamics into the deterministic equation through a stochastic parametrization that respects the conservation of the energy of the deterministic system. In a second step we use stochast...

Thermal bath coupling mechanisms as utilized in molecular dynamics are applied to partial differential equation models. Working from a semi-discrete (Fourier mode) formulation for the Burgers–Hopf or Korteweg–de Vries equation, we introduce auxiliary variables and stochastic perturbations in order to drive the system to sample a target ensemble whi...

In this paper we solve two initial value problems for two-dimensional internal gravity waves. The waves are contained in a uniformly stratified, square-shaped domain whose sidewalls are tilted with respect to the direction of gravity. We consider several disturbances of the initial stream function field and solve both for its free evolution and for...

This paper presents a probabilistic power flow model subject to connection temperature constraints. Renewable power generation is included and modelled stochastically in order to reflect its intermittent nature. In contrast to conventional models that enforce connection current constraints, short-term current overloading is allowed. Temperature con...

We develop a hydrostatic Hamiltonian particle-mesh (HPM) method for efficient long-term numerical integration of the atmosphere. In the HPM method, the hydrostatic approximation is interpreted as a holonomic constraint for the vertical position of particles. This can be viewed as defining a set of vertically buoyant horizontal meshes, with the alti...

A broad array of canonical sampling methods are available for molecular simulation based on stochastic-dynamical perturbation of Newtonian dynamics, including Langevin dynamics, Stochastic Velo- city Rescaling, and methods that combine Nosé-Hoover dynamics with stochastic perturbation. In this article we discuss several stochastic-dynamical thermos...

In this note we study the asymptotic limit of large variance in a stochastically perturbed thermostat model, the Nosé-Hoover-Langevin device. We show that in this limit, the model reduces to a Langevin equation with one-dimensional Wiener process, and that the perturbation is in the direction of the conjugate momentum vector. Numerical experiments...

Although our climate is ultimately driven by (nonuniform) solar heating, many aspects of the flow can be understood qualitatively from forcing-free and frictionless dynamics. In the limit of zero forcing and dissipation, our weather system falls under the realm of Hamiltonian fluid dynamics and the flow conserves potential vorticity (PV), energy an...

Although our climate is ultimately driven by (nonuniform) solar heating, many aspects of the flow can be understood qualitatively from forcing-free and frictionless dynamics. In the limit of zero forcing and dissipation, our weather system falls under the realm of Hamiltonian fluid dynamics and the flow conserves potential vorticity (PV), energy an...

We conduct long-time simulations with a Hamiltonian particle-mesh method for ideal fluid flow, to determine the statistical mean vorticity field of the discretization. Lagrangian and Eulerian statistical models are proposed for the discrete dynamics, and these are compared against numerical experiments. The observed results are in excellent agreeme...

Based on the thermodynamic concept of a reservoir, we investigate a computational model for interaction with unresolved degrees of freedom (a thermal bath). We assume that a finite restricted system can be modelled by a generalized canonical ensemble, described by a density which is a smooth function of the energy of the restricted system. A thermo...

The results of statistical analysis of simulation data obtained from long time integrations of geophysical fluid models greatly depend on the conservation properties of the numerical discretization used. This is illustrated for quasi-geostrophic flow with topographic forcing, for which a well established statistical mechanics exists. Statistical me...

Although Runge-Kutta and partitioned Runge-Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multi-Hamiltonian PDEs, they do not always lead to well-defined numerical methods. We consider the case study of the nonlinear Schrodinger equation in detail, for which the previously known multisymplecti...

We describe the remapped particle-mesh method, a new mass-conserving method for solving the density equation which is suitable for combining with semi-Lagrangian methods for compressible flow applied to numerical weather prediction. In addition to the conservation property, the remapped particle-mesh method is computationally efficient and at least...

The parallel implementation of GCR is addressed, with particular focus on communication costs associated with orthogonalization processes. This consideration brings up questions concerning the use of Householder reflections with GCR. To precondition the GCR method a block Gauss-Jacobi method is used. Approximate solvers are used to obtain a solutio...

In this paper we discuss the conservation of wave action under numerical discretization by variational and multisymplectic methods. Both the general wave action conservation defined with respect to a smooth, periodic, one-parameter ensemble of flow realizations and the specific wave action based on an approximated and averaged Lagrangian are addres...

Multisymplectic methods have recently been proposed as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. Their excellent long time behavior for a variety of Hamiltonian wave equations has been demonstrated in a number of numerical studies. A theoretical investigation and justification of multisymplectic methods is still la...

We consider energy-conserving semi-discretizations of linear wave equations on nonuniform grids. Specifically we study explicit and implicit skew-adjoint finite difference methods, based on the assumption of an underlying smooth mapping from a uniform grid, applied to the first and second order wave equations. Our interest is in internal reflection...

In this note we show that multisymplectic Runge-Kutta box schemes, of which the Gauss-Legendre methods are the most important, preserve a discrete conservation law of wave action. The result follows by loop integration over an ensemble of flow realizations, and the local energy-momentum conservation law for continuous variables in semi-discretizati...

A key aspect of the recently proposed Hamiltonian particle-mesh (HPM) method is its time-staggered discretization combined with a regularization of the continuous governing equations. In this article, the time discretization aspect of the HPM method is analysed for the linearized, rotating, shallow-water equations with orography, and the combined e...

This paper addresses nonphysical reflections encountered in the discretization of wave equations on nonuniform grids. Such nonphysical solutions are commonly attributed to spurious modes in the numerical dispersion relation. We provide an example of a discretization in which a (nonspurious) physical mode is spuriously energized at a grid nonuniform...

To complete the description of the HPM method, we need to find an inexpensive smoothing operator that averages out fluctuations over the sphere on some length scale Λ. Following Merilees' pseudospectral code (Merilees, 1973), we compute derivatives by employing one-dimensional fast Fourier transforms (FFTs) along the longitudinal and the latitudina...

The Landau–Lifshitz equation (LLE) governing the flow of magnetic spin in a ferromagnetic material is a PDE with a noncanonical Hamiltonian structure. In this paper we derive a number of new formulations of the LLE as a partial differential equation on a multisymplectic structure. Using this form we show that the standard central spatial discretiza...

We develop a particle-mesh method for two-layer shallow-water equations subject to the rigid-lid approximation. The method is based on the recently proposed Hamiltonian particle-mesh (HPM) method and the interpretation of the rigid-lid approximation as a set of holonomic constraints. The suggested spatial discretization leads to a constrained Hamil...

We propose a particle-mesh method for rotating shallow-water equations. The spatially truncated equations are Hamiltonian and satisfy Kelvin circulation theorem. The generation of non-smooth components in the layer depth is avoided by applying a smoothing operator similar to that in the context of α-Euler models.

Virtual environments have shown great promise as a research tool in science and engineering. In this paper we study a classical problem in mathematics: that of approximating globally optimal Fekete point configurations. We found a highly interactive virtual environment, combined with a time-critical computation, can provide valuable insight into th...

A new particle-mesh method is proposed for the rotating shallow-water equations. The spatially truncated equations are Hamiltonian and satisfy a Kelvin circulation theorem. The generation of non-smooth components in the layer-depth is avoided by applying a smoothing operator similar to what has recently been discussed in the context of α-Euler mode...

Kelvin's circulation theorem and its implications for potential vorticity (PV) conservation are among the most fundamental concepts in ideal fluid dynamics. In this note, we discuss the numerical treatment of these concepts with the Smoothed Particle Hydrodynamics (SPH) and related methods. We show that SPH satisfies an exact circulation theorem in...

In this paper we outline a new particle-mesh method for rapidly rotating shallow water flows based on a set of regularized equations of motion. The time-stepping method uses an operator splitting of the equations into an Eulerian gravity wave part and a Lagrangian advection part. An essential ingredient is the advection of absolute vorticity by mea...

Efficient parallel algorithms are required to simulate incompressible turbulent flows in complex two- and three-dimensional domains. The incompressible Navier–Stokes equations are discretized in general coordinates on a structured grid. For a flow on a general domain we use an unstructured decomposition of the domain into subdomains of simple shape...

A block preconditioner is considered in a parallel computing environment. This preconditioner has good parallel properties, however, the convergence deteriorates when the number of blocks increases. Two different techniques are studied to accelerate the convergence: overlapping at the interfaces and using a coarse grid correction. It appears that t...

In this article we introduce new bounds on the effective condition number of deflated and preconditioned-deflated symmetric positive definite linear systems. For the case of a subdomain deflation such as that of Nicolaides [SIAM J. Numer. Anal., 24 (1987), pp. 355--365], these theorems can provide direction in choosing a proper decomposition into s...

We generalize the extended backward differentiation formulas (EBDFs) introduced by Cash and by Psihoyios and Cash so that the system matrix in the modified Newton process can be block-diagonalized, enabling an efficient parallel implementation. The purpose of this paper is to justify the use of diagonalizable EBDFs on parallel computers and to offe...

A Deated iccg method is applied to problems with extreme contrasts in the coef- cients, such as those encountered in reservoir simulation, where the permeabilities of various rocks dier orders of magnitude. The coecient matrix, which is symmetric and positive denite, has a very large condition number. We prove that the the number of small eigenvalu...

Virtual environments have shown great promise as a research tool in science and engineering. In this paper, we study a classical problem in mathematics: that of approximating globally optimal Fekete point configurations. We found that a highly interactive virtual environment, combined with a time-critical computation, can provide valuable insight i...

A block-preconditioner is considered in a parallel computing environment. This preconditioner has good parallel properties, however the convergence deteriorates when the number of blocks increases. Two different techniques are studied to accelerate the convergence: overlapping at the interfaces and using a coarse grid correction. It appears that th...

The extended backward differentiation formulas (EBDFs) and their modified form (MEBDF) were proposed by Cash in the 1980s for solving initial-value problems (IVPs) for stiff systems of ordinary differential equations (ODEs). In a recent performance evaluation of various IVP solvers, including a variable-step-variable-order implementation of the MEB...

Practical, structure-preserving methods for integrating classical Heisenberg spin systems are discussed. Two new integrators are derived and compared, including (1) a symmetric energy and spin-length preserving integrator based on a Red-Black splitting of the spin sites combined with a staggered timestepping scheme and (2) a (Lie-Poisson) symplecti...

In many applications, large systems of ordinary differential equations (ODEs) have to be solved numerically that have both stiff and nonstiff parts. A popular approach in such cases is to integrate the stiff parts implicitly and the nonstiff parts explicitly. In this paper we study a class of implicit-explicit (IMEX) linear multistep methods intend...

Solution of large linear systems encountered in computational fluid dynamics often naturally leads to some form of domain decomposition, especially when it is desired to use parallel machines. It has been proposed to use approximate solvers to obtain fast but rough solutions on the separate subdomains. In this paper a number of approximate solvers...

Johannes Gerardus (Jan) Verwer overleed op 16 februari 2011, geheel onverwachts, slechts vier weken na zijn pensionering bij het Centrum Wiskunde & Informatica (CWI). Jan Verwer was een vooraanstaand onderzoeker op het gebied van de numerieke wiskunde voor gewone en partiële differentiaalvergelijkingen, en een succesvol en natuurlijk leider in de w...

Thesis (M.S.)--University of Kansas, Aerospace Engineering, 1994. Includes bibliographical references.