arXiv subjects

Giovanni Samaey

Publications and source records attributed to Giovanni Samaey.

At least 73 records · Page 4Linked to original sources

arXiv2016

DDE-BIFTOOL Manual - Bifurcation analysis of delay differential equations

DDEBIFTOOL is a collection of Matlab routines for numerical bifurcation analysis of systems of delay differential equations with discrete constant and state-dependent delays. The package supports continuation and stability analysis of steady state solutions and periodic solutions. Further one can compute and continue several local and global bifurcations: fold and Hopf bifurcations of steady states; folds, period doublings and torus bifurcations of periodic orbits; and connecting orbits between equilibria. To analyse the stability of steady state solutions, approximations are computed to the rightmost, stability-determining roots of the characteristic equation which can subsequently be used as starting values in a Newton procedure. For periodic solutions, approximations to the Floquet multipliers are computed. The manual describes the structure of the package, its routines, and its data and method parameter structures.

math.DS↗

arXiv2016

A high-order relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws

We present a general, high-order, fully explicit relaxation scheme which can be applied to any system of nonlinear hyperbolic conservation laws in multiple dimensions. The scheme consists of two steps. In a first (relaxation) step, the nonlinear hyperbolic conservation law is approximated by a kinetic equation with stiff BGK source term. Then, this kinetic equation is integrated in time using a projective integration method. After taking a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time step restriction on the outer time step is similar to the CFL condition for the hyperbolic conservation law. Moreover, the number of inner time steps is also independent of the stiffness of the BGK source term. We discuss stability and consistency, and illustrate with numerical results (linear advection, Burgers' equation and the shallow water and Euler equations) in one and two spatial dimensions.

math.NA↗

arXiv2016

Telescopic projective integration for kinetic equations with multiple relaxation times

We study a general, high-order, fully explicit numerical method for simulating kinetic equations with a BGK-type collision model with multiple relaxation times. In that case, the problem is stiff and its spectrum consists of multiple separated eigenvalue clusters. Projective integration methods are explicit integration schemes that first take a few small (inner) steps with a simple, explicit method, after which the solution is extrapolated forward in time over a large (outer) time step. These are very efficient schemes, provided there are only two clusters of eigenvalues. Telescopic projective integration methods generalize the idea of projective integration methods by constructing a hierarchy of projective levels. Here, we show how telescopic projective integration methods can be used to efficiently integrate kinetic equations with multiple relaxation times. We show that the required number of projective levels depends on the number of clusters, which in turn depends on the stiffness of the BGK source term. The size of the outer level time step only depends on the slowest time scale present in the model and is independent of the stiffness of the problem. We discuss stability and illustrate the approach with simulations in one and two spatial dimensions.

math.NA↗

arXiv2015

Variance-reduced simulation of stochastic agent-based models for tumor growth

We investigate a hybrid PDE/Monte Carlo technique for the variance reduced simulation of an agent-based multiscale model for tumor growth. The variance reduction is achieved by combining a simulation of the stochastic agent-based model on the microscopic scale with a deterministic solution of a simplified (coarse) partial differential equation (PDE) on the macroscopic scale as a control variable. We show that this technique is able to significantly reduce the variance with only the (limited) additional computational cost associated with the deterministic solution of the coarse PDE. We illustrate the performance with numerical experiments in different regimes, both in the avascular and vascular stage of tumor growth.

q-bio.QM↗

arXiv2014

Noise reduction in coarse bifurcation analysis of stochastic agent-based models: an example of consumer lock-in

We investigate coarse equilibrium states of a fine-scale, stochastic agent-based model of consumer lock-in in a duopolistic market. In the model, agents decide on their next purchase based on a combination of their personal preference and their neighbours' opinions. For agents with independent identically-distributed parameters and all-to-all coupling, we derive an analytic approximate coarse evolution-map for the expected average purchase. We then study the emergence of coarse fronts when spatial segregation is present in the relative perceived quality of products. We develop a novel Newton-Krylov method that is able to compute accurately and efficiently coarse fixed points when the underlying fine-scale dynamics is stochastic. The main novelty of the algorithm is in the elimination of the noise that is generated when estimating Jacobian-vector products using time-integration of perturbed initial conditions. We present numerical results that demonstrate the convergence properties of the numerical method, and use the method to show that macroscopic fronts in this model destabilise at a coarse symmetry-breaking bifurcation.

nlin.PS↗

arXiv2014

A high-order asymptotic-preserving scheme for kinetic equations using projective integration

We investigate a high-order, fully explicit, asymptotic-preserving scheme for a kinetic equation with linear relaxation, both in the hydrodynamic and diffusive scalings in which a hyperbolic, resp. parabolic, limiting equation exists. The scheme first takes a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution and estimate the time derivative of the slow components. These estimated time derivatives are then used in an (outer) Runge-Kutta method of arbitrary order. We show that, with an appropriate choice of inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting macroscopic equation. Moreover, the number of inner time steps is also independent of the scaling parameter. We analyse stability and consistency, and illustrate with numerical results.

math.NA↗

arXiv2014

On the acceleration of spatially distributed agent-based computations: a patch dynamics scheme

In recent years, individual-based/agent-based modeling has been applied to study a wide range of applications, ranging from engineering problems to phenomena in sociology, economics and biology. Simulating such agent-based models over extended spatiotemporal domains can be prohibitively expensive due to stochasticity and the presence of multiple scales. Nevertheless, many agent-based problems exhibit smooth behavior in space and time on a macroscopic scale, suggesting that a useful coarse-grained continuum model could be obtained. For such problems, the equation-free framework [16-18] can significantly reduce the computational cost. Patch dynamics is an essential component of this framework. This scheme is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales; it uses appropriately initialized simulations of the fine-scale agent-based model in a number of small "patches", which cover only a fraction of the spatiotemporal domain. In this work, we construct a finite-volume-inspired conservative patch dynamics scheme and apply it to a financial market agent-based model based on the work of Omurtag and Sirovich [22]. We first apply our patch dynamics scheme to a continuum approximation of the agent-based model, to study its performance and analyze its accuracy. We then apply the scheme to the agent-based model itself. Our computational experiments indicate that here, typically, the patch dynamics-based simulation requires only 20% of the full agent-based simulation in space, and need occur over only 10% of the temporal domain.

math.NA↗

arXiv2013

A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations

We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale-in-time systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate slow macroscopic model), which is iteratively corrected using a fine-scale propagator (accurately simulating the full microscopic dynamics). This correction is done in parallel over many subintervals, thereby reducing the wall-clock time needed to obtain the solution, compared to the integration of the full microscopic model. We provide a numerical analysis of the algorithm for a prototypical example of a micro-macro model, namely singularly perturbed ordinary differential equations. We show that the computed solution converges to the full microscopic solution (when the parareal iterations proceed) only if special care is taken during the coupling of the microscopic and macroscopic levels of description. The convergence rate depends on the modeling error of the approximate macroscopic model. We illustrate these results with numerical experiments.

math.NA↗

arXiv2012

Towards an efficient multiscale modeling of low-dimensional reactive systems: study of numerical closure procedures

In this paper, we present a study on how to develop an efficient multiscale simulation strategy for the dynamics of chemically active systems on low-dimensional supports. Such reactions are encountered in a wide variety of situations, ranging from heterogeneous catalysis to electrochemical or (membrane) biological processes, to cite a few. We analyzed in this context different techniques within the framework of an important multiscale approach known as the equation free method (EFM), which "bridges the multiscale gap" by building microscopic configurations using macroscopic-level information only. We hereby considered two simple reactive processes on a one-dimensional lattice, the simplicity of which allowed for an in-depth understanding of the parameters controlling the efficiency of this approach. We demonstrate in particular that it is not enough to base the EFM on the time evolution of the average concentrations of particles on the lattice, but that the time evolution of clusters of particles has to be included as well. We also show how important it is for the accuracy of this method to carefully choose the procedure with which microscopic states are constructed, starting from the measured macroscopic quantities. As we also demonstrate that some errors cannot be corrected by increasing the number of observed macroscopic variables, this work points towards which procedures should be used in order to generate efficient and reliable multiscale simulations of these systems.

physics.comp-ph↗

arXiv2011

Individual-based models for bacterial chemotaxis in the diffusion asymptotics

We discuss velocity-jump models for chemotaxis of bacteria with an internal state that allows the velocity jump rate to depend on the memory of the chemoattractant concentration along their path of motion. Using probabilistic techniques, we provide a pathwise result that shows that the considered process converges to an advection-diffusion process in the (long-time) diffusion limit. We also (re-)prove using the same approach that the same limiting equation arises for a related, simpler process with direct sensing of the chemoattractant gradient. Additionally, we propose a time discretization technique that retains these diffusion limits exactly, i.e., without error that depends on the time discretization. In the companion paper \cite{variance}, these results are used to construct a coupling technique that allows numerical simulation of the process with internal state with asymptotic variance reduction, in the sense that the variance vanishes in the diffusion limit.

math.NA↗

arXiv2011

Simulating individual-based models of bacterial chemotaxis with asymptotic variance reduction

We discuss variance reduced simulations for an individual-based model of chemotaxis of bacteria with internal dynamics. The variance reduction is achieved via a coupling of this model with a simpler process in which the internal dynamics has been replaced by a direct gradient sensing of the chemoattractants concentrations. In the companion paper \cite{limits}, we have rigorously shown, using a pathwise probabilistic technique, that both processes converge towards the same advection-diffusion process in the diffusive asymptotics. In this work, a direct coupling is achieved between paths of individual bacteria simulated by both models, by using the same sets of random numbers in both simulations. This coupling is used to construct a hybrid scheme with reduced variance. We first compute a deterministic solution of the kinetic density description of the direct gradient sensing model; the deviations due to the presence of internal dynamics are then evaluated via the coupled individual-based simulations. We show that the resulting variance reduction is \emph{asymptotic}, in the sense that, in the diffusive asymptotics, the difference between the two processes has a variance which vanishes according to the small parameter.

math.NA↗

arXiv2011

A micro/macro algorithm to accelerate Monte Carlo simulation of stochastic differential equations

We present and analyze a micro/macro acceleration technique for the Monte Carlo simulation of stochastic differential equations (SDEs) in which there is a separation between the (fast) time-scale on which individual trajectories of the SDE need to be simulated and the (slow) time-scale on which we want to observe the (macroscopic) function of interest. The method performs short bursts of microscopic simulation using an ensemble of SDE realizations, after which the ensemble is restricted to a number of macroscopic state variables. The resulting macroscopic state is then extrapolated forward in time and the ensemble is projected onto the extrapolated macroscopic state. We provide a first analysis of its convergence in terms of extrapolation time step and number of macroscopic state variables. The effects of the different approximations on the resulting error are illustrated via numerical experiments.

math.NA↗

arXiv2010

Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit

We investigate a projective integration scheme for a kinetic equation in the limit of vanishing mean free path, in which the kinetic description approaches a diffusion phenomenon. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large time step on the diffusion time scale. We show that, with an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the diffusion equation, whereas the required number of inner steps does not depend on the mean free path. We also provide a consistency result. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the diffusion equation in the limit of vanishing mean free path. The analysis is illustrated with numerical results, and we present an application to the Su-Olson test.

math.AP↗

arXiv2010

A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells

We propose a numerical procedure to study closure approximations for FENE dumbbells in terms of chosen macroscopic state variables, enabling to test straightforwardly which macroscopic state variables should be included to build good closures. The method involves the reconstruction of a polymer distribution related to the conditional equilibrium of a microscopic Monte Carlo simulation, conditioned upon the desired macroscopic state. We describe the procedure in detail, give numerical results for several strategies to define the set of macroscopic state variables, and show that the resulting closures are related to those obtained by a so-called quasi-equilibrium approximation \cite{Ilg:2002p10825}.

math-ph↗

arXiv2007

Coarse-grained analysis of a lattice Boltzmann model for planar streamer fronts

We study the traveling wave solutions of a lattice Boltzmann model for the planar streamer fronts that appear in the transport of electrons through a gas in a strong electrical field. To mimic the physical properties of the impact ionization reaction, we introduce a reaction matrix containing reaction rates that depend on the electron velocities. Via a Chapman--Enskog expansion, one is able to find only a rough approximation for a macroscopic evolution law that describes the traveling wave solution. We propose to compute these solutions with the help of a coarse-grained time-stepper, which is an effective evolution law for the macroscopic fields that only uses appropriately initialized simulations of the lattice Boltzmann model over short time intervals. The traveling wave solution is found as a fixed point of the sequential application of the coarse-grained time-stepper and a shift-back operator. The fixed point is then computed with a Newton-Krylov Solver. We compare the resulting solutions with those of the approximate PDE model, and propose a method to find the minimal physical wave speed.

physics.comp-ph↗

arXiv2006

Coarse-grained computation of traveling waves of lattice Boltzmann models with Newton-Krylov solvers

For many complex dynamical systems, a separation of scales prevails between the (microscopic) level of description of the available model, and the (macroscopic) level at which one would like to observe and analyze the system. For this type of problems, an ``equation-free'' framework has recently been proposed. Using appropriately initialized microscopic simulations, one can build a coarse-grained time-stepper to approximate a time-stepper for the unavailable macroscopic model. Here, we show how one can use this coarse-grained time-stepper to compute coarse-grained traveling wave solutions of a lattice Boltzmann model. In a moving frame, emulated by performing a shift-back operation after the coarse-grained time-step, the traveling wave appears as a steady state, which is computed using an iterative method, such as Newton--GMRES. To accelerate convergence of the GMRES procedure, a macroscopic model-based preconditioner is used, which is derived from the lattice Boltzmann model using a Chapman--Enskog expansion. We illustrate the approach on a lattice Boltzmann model for the Fisher equation and on a model for ionization waves.

physics.comp-ph↗

arXiv2004

Patch dynamics with buffers for homogenization problems

An important class of problems exhibits smooth behaviour on macroscopic space and time scales, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, an "equation-free" framework has been proposed, of which patch dynamics is an essential component. Patch dynamics is designed to perform numerical simulations of an unavailable macroscopic equation on macroscopic time and length scales; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes (patches), which cover only a fraction of the space-time domain. To reduce the effect of the artificially introduced box boundaries, we use buffer regions to "shield" the boundary artefacts from the interior of the domain for short time intervals. We analyze the accuracy of this scheme for a diffusion homogenization problem with periodic heterogeneity, and propose a simple heuristic to determine a sufficient buffer size. The algorithm performance is illustrated through a set of numerical examples, which include a non-linear reaction-diffusion equation and the Kuramoto--Sivashinsky equation.

physics.comp-ph↗

arXiv2003

The gap-tooth scheme for homogenization problems

An important class of problems exhibits smooth behaviour in space and time on a macroscopic scale, while only a microscopic evolution law is known. For such time-dependent multi-scale problems, an ``equation-free framework'' has been proposed, of which the gap-tooth scheme is an essential component. The gap-tooth scheme is designed to approximate a time-stepper for an unavailable macroscopic equation in a macroscopic domain; it uses appropriately initialized simulations of the available microscopic model in a number of small boxes, which cover only a fraction of the domain. We analyze the convergence of this scheme for a parabolic homogenization problem with non-linear reaction. In this case, the microscopic model is a partial differential equation with rapidly oscillating coefficients, while the unknown macroscopic model is approximated by the homogenized equation. We show that our method approximates a finite difference scheme of arbitrary (even) order for the homogenized equation when we appropriately constrain the microscopic problem in the boxes. We illustrate this theoretical result with numerical tests on several model problems. We also demonstrate that it is possible to obtain a convergent scheme without constraining the microscopic code, by introducing buffer regions around the computational boxes.

physics.comp-ph↗