2,782 research outputs found
Local error estimates for adaptive simulation of the Reaction-Diffusion Master Equation via operator splitting
The efficiency of exact simulation methods for the reaction-diffusion master
equation (RDME) is severely limited by the large number of diffusion events if
the mesh is fine or if diffusion constants are large. Furthermore, inherent
properties of exact kinetic-Monte Carlo simulation methods limit the efficiency
of parallel implementations. Several approximate and hybrid methods have
appeared that enable more efficient simulation of the RDME. A common feature to
most of them is that they rely on splitting the system into its reaction and
diffusion parts and updating them sequentially over a discrete timestep. This
use of operator splitting enables more efficient simulation but it comes at the
price of a temporal discretization error that depends on the size of the
timestep. So far, existing methods have not attempted to estimate or control
this error in a systematic manner. This makes the solvers hard to use for
practitioners since they must guess an appropriate timestep. It also makes the
solvers potentially less efficient than if the timesteps are adapted to control
the error. Here, we derive estimates of the local error and propose a strategy
to adaptively select the timestep when the RDME is simulated via a first order
operator splitting. While the strategy is general and applicable to a wide
range of approximate and hybrid methods, we exemplify it here by extending a
previously published approximate method, the Diffusive Finite-State Projection
(DFSP) method, to incorporate temporal adaptivity
Adaptive finite element method assisted by stochastic simulation of chemical systems
Stochastic models of chemical systems are often analysed by solving the corresponding\ud
Fokker-Planck equation which is a drift-diffusion partial differential equation for the probability\ud
distribution function. Efficient numerical solution of the Fokker-Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with non-negligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the probability density
3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries
Recent advances in electron microscopy have enabled the imaging of single
cells in 3D at nanometer length scale resolutions. An uncharted frontier for in
silico biology is the ability to simulate cellular processes using these
observed geometries. Enabling such simulations requires watertight meshing of
electron micrograph images into 3D volume meshes, which can then form the basis
of computer simulations of such processes using numerical techniques such as
the Finite Element Method. In this paper, we describe the use of our recently
rewritten mesh processing software, GAMer 2, to bridge the gap between poorly
conditioned meshes generated from segmented micrographs and boundary marked
tetrahedral meshes which are compatible with simulation. We demonstrate the
application of a workflow using GAMer 2 to a series of electron micrographs of
neuronal dendrite morphology explored at three different length scales and show
that the resulting meshes are suitable for finite element simulations. This
work is an important step towards making physical simulations of biological
processes in realistic geometries routine. Innovations in algorithms to
reconstruct and simulate cellular length scale phenomena based on emerging
structural data will enable realistic physical models and advance discovery at
the interface of geometry and cellular processes. We posit that a new frontier
at the intersection of computational technologies and single cell biology is
now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies
available upon reques
Strong convergence for split-step methods in stochastic jump kinetics
Mesoscopic models in the reaction-diffusion framework have gained recognition
as a viable approach to describing chemical processes in cell biology. The
resulting computational problem is a continuous-time Markov chain on a discrete
and typically very large state space. Due to the many temporal and spatial
scales involved many different types of computationally more effective
multiscale models have been proposed, typically coupling different types of
descriptions within the Markov chain framework.
In this work we look at the strong convergence properties of the basic first
order Strang, or Lie-Trotter, split-step method, which is formed by decoupling
the dynamics in finite time-steps. Thanks to its simplicity and flexibility,
this approach has been tried in many different combinations.
We develop explicit sufficient conditions for path-wise well-posedness and
convergence of the method, including error estimates, and we illustrate our
findings with numerical examples. In doing so, we also suggest a certain
partition of unity representation for the split-step method, which in turn
implies a concrete simulation algorithm under which trajectories may be
compared in a path-wise sense
Efficient Reactive Brownian Dynamics
We develop a Split Reactive Brownian Dynamics (SRBD) algorithm for particle
simulations of reaction-diffusion systems based on the Doi or volume reactivity
model, in which pairs of particles react with a specified Poisson rate if they
are closer than a chosen reactive distance. In our Doi model, we ensure that
the microscopic reaction rules for various association and disassociation
reactions are consistent with detailed balance (time reversibility) at
thermodynamic equilibrium. The SRBD algorithm uses Strang splitting in time to
separate reaction and diffusion, and solves both the diffusion-only and
reaction-only subproblems exactly, even at high packing densities. To
efficiently process reactions without uncontrolled approximations, SRBD employs
an event-driven algorithm that processes reactions in a time-ordered sequence
over the duration of the time step. A grid of cells with size larger than all
of the reactive distances is used to schedule and process the reactions, but
unlike traditional grid-based methods such as Reaction-Diffusion Master
Equation (RDME) algorithms, the results of SRBD are statistically independent
of the size of the grid used to accelerate the processing of reactions. We use
the SRBD algorithm to compute the effective macroscopic reaction rate for both
reaction- and diffusion-limited irreversible association in three dimensions.
We also study long-time tails in the time correlation functions for reversible
association at thermodynamic equilibrium. Finally, we compare different
particle and continuum methods on a model exhibiting a Turing-like instability
and pattern formation. We find that for models in which particles diffuse off
lattice, such as the Doi model, reactions lead to a spurious enhancement of the
effective diffusion coefficients.Comment: To appear in J. Chem. Phy
A comparison of Eulerian and Lagrangian transport and non-linear reaction algorithms
When laboratory-measured chemical reaction rates are used in simulations at the field-scale, the models typically overpredict the apparent reaction rates. The discrepancy is primarily due to poorer mixing of chemically distinct waters at the larger scale. As a result, realistic field-scale predictions require accurate simulation of the degree of mixing between fluids. The Lagrangian particle-tracking (PT) method is a now-standard way to simulate the transport of conservative or sorbing solutes. The method’s main advantage is the absence of numerical dispersion (and its artificial mixing) when simulating advection. New algorithms allow particles of different species to interact in nonlinear (e.g., bimolecular) reactions. Therefore, the PT methods hold a promise of more accurate field-scale simulation of reactive transport because they eliminate the masking effects of spurious mixing due to advection errors inherent in grid-based methods. A hypothetical field-scale reaction scenario is constructed and run in PT and Eulerian (finite-volume/finite-difference) simulators. Grid-based advection schemes considered here include 1st- to 3rd-order spatially accurate total-variation-diminishing flux-limiting schemes, both of which are widely used in current transport/reaction codes. A homogeneous velocity field in which the Courant number is everywhere unity, so that the chosen Eulerian methods incur no error when simulating advection, shows that both the Eulerian and PT methods can achieve convergence in the L1 (integrated concentration) norm, but neither shows stricter pointwise convergence. In this specific case with a constant dispersion coefficient and bimolecular reaction A+B¿P, the correct total amount of product is 0.221MA0, where MA0 is the original mass of reactant A. When the Courant number drops, the grid-based simulations can show remarkable errors due to spurious over- and under-mixing. In a heterogeneous velocity field (keeping the same constant and isotropic dispersion), the PT simulations show an increased reaction total from 0.221MA0 to 0.372MA0 due to fluid deformation, while the 1st-order Eulerian simulations using ˜ 106 cells (with a classical grid Peclet number ¿x/aL of 10) have total product of 0.53MA0, or approximately twice as much additional reaction due to advection error. The 3rd-order TVD algorithm fares better, with total product of 0.394MA0, or about 1.14 times the increased reaction total. A very strict requirement on grid Peclet numbers for Eulerian simulations will be required for realistic reactions because of their nonlinear nature. We analytically estimate the magnitude of the effect for the end-member cases of very fast and very slow reactions and show that in either case, the mass produced is proportional to View the MathML source where Pe is the Peclet number. Therefore, extra mass is produced according to View the MathML source where the dispersion includes any numerical dispersion error. We test two PT methods, one that kills particles upon reaction and another that decrements a particle’s mass. For the bimolecular reaction studied here, the computational demands of the particle-killing methods are much smaller than, and the particle-number-preserving algorithm are on par with, the fastest Eulerian methods.Peer ReviewedPostprint (author's final draft
Explicit-in-Time Variational Formulations for Goal-Oriented Adaptivity
Goal-Oriented Adaptivity (GOA) is a powerful tool to accurately approximate physically relevant features of the solution of Partial Differential Equations (PDEs). It delivers optimal grids to solve challenging engineering problems. In time dependent problems, GOA requires to represent the error in the Quantity of Interest (QoI) as an integral over the whole space-time domain in order to reduce it via adaptive refinements. A full space-time variational formulation of the problem allows the aforementioned error representation. Thus, variational spacetime formulations for PDEs have been of great interest in the last decades, among other things, because they allow to develop mesh-adaptive algorithms. Since it is known that implicit time marching schemes have variational structure, they are often employed for GOA in time-domain problems. When coming to explicit-intime methods, these were introduced for Ordinary
In this dissertation, we prove that the explicit Runge-Kutta (RK) methods can be expressed as discontinuous-in-time Petrov-Galerkin (dPG) methods for the linear advection-diffusion equation. We systematically build trial and test functions that, after exact integration in time, lead to one, two, and general stage explicit RK methods. This approach enables us to reproduce the existing time domain goal-oriented adaptive algorithms using explicit methods in time. Here, we employ the lowest order dPG formulation that we propose to recover the Forward Euler method and we derive an appropriate error representation. Then, we propose an explicit-in-time goal-oriented adaptive algorithm that performs local refinements in space. In terms of time domain adaptivity, we impose the Courant-Friedrichs-Lewy (CFL) condition to ensure the stability of the method. We provide some numerical results in one-dimensional (1D)+time for the diffusion and advection-diffusion equations to show the performance of the proposed algorithm.
On the other hand, time-domain adaptive algorithms involve solving a dual problem that runs backwards in time. This process is, in general, computationally expensive in terms of memory storage. In this work, we dene a pseudo-dual problem that runs forwards in time. We also describe a forward-in-time adaptive algorithm that works for some specific problems. Although it is not possible to dene a general dual problem running forwards in time that provides information about future states, we provide numerical evidence via one-dimensional problems in space to illustrate the efficiency of our algorithm as well as its limitations. As a complementary method, we propose a hybrid algorithm that employs the classical backward-in-time dual problem once and then performs the adaptive process forwards in time. We also generalize a novel error representation for goal-oriented adaptivity using (unconventional) pseudo-dual problems in the context of frequency-domain wave-propagation problems to the time-dependent wave equation. We show via 1D+time numerical results that the upper bounds for the new error representation are sharper than the classical ones. Therefore, this new error representation can be used to design more efficient goal-oriented adaptive methodologies.
Finally, as classical Galerkin methods may lead to instabilities in advection-dominated-diffusion problems and therefore, inappropriate refinements, we propose a novel stabilized discretization method, which we call Isogeometric Residual Minimization (iGRM) with direction splitting. This method combines the benefits resulting from Isogeometric Analysis (IGA), residual minimization, and Alternating Direction Implicit (ADI) methods. We employ second order ADI time integrator schemes, B-spline basis functions in space and, at each time step, we solve a stabilized mixed method based on residual minimization. We show that the resulting system of linear equations has a Kronecker product structure, which results in a linear computational cost of the direct solver, even using implicit time integration schemes together with the stabilized mixed formulation. We test our method in 2D and 3D+time advection-diffusion problems. The derivation of a time-domain goal-oriented strategy based on iGRM will be considered in future works
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