31,023 research outputs found
Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations
Computational costs of numerically solving multidimensional partial
differential equations (PDEs) increase significantly when the spatial
dimensions of the PDEs are high, due to large number of spatial grid points.
For multidimensional reaction-diffusion equations, stiffness of the system
provides additional challenges for achieving efficient numerical simulations.
In this paper, we propose a class of Krylov implicit integration factor (IIF)
discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion
equations on high spatial dimensions. The key ingredient of spatial DG
discretization is the multiwavelet bases on nested sparse grids, which can
significantly reduce the numbers of degrees of freedom. To deal with the
stiffness of the DG spatial operator in discretizing reaction-diffusion
equations, we apply the efficient IIF time discretization methods, which are a
class of exponential integrators. Krylov subspace approximations are used to
evaluate the large size matrix exponentials resulting from IIF schemes for
solving PDEs on high spatial dimensions. Stability and error analysis for the
semi-discrete scheme are performed. Numerical examples of both scalar equations
and systems in two and three spatial dimensions are provided to demonstrate the
accuracy and efficiency of the methods. The stiffness of the reaction-diffusion
equations is resolved well and large time step size computations are obtained
Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
Kolmogorov -widths and low-rank approximations are studied for families of
elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay
of the -widths can be controlled by that of the error achieved by best
-term approximations using polynomials in the parametric variable. However,
we prove that in certain relevant instances where the diffusion coefficients
are piecewise constant over a partition of the physical domain, the -widths
exhibit significantly faster decay. This, in turn, yields a theoretical
justification of the fast convergence of reduced basis or POD methods when
treating such parametric PDEs. Our results are confirmed by numerical
experiments, which also reveal the influence of the partition geometry on the
decay of the -widths.Comment: 27 pages, 6 figure
Spatially Distributed Stochastic Systems: equation-free and equation-assisted preconditioned computation
Spatially distributed problems are often approximately modelled in terms of
partial differential equations (PDEs) for appropriate coarse-grained quantities
(e.g. concentrations). The derivation of accurate such PDEs starting from finer
scale, atomistic models, and using suitable averaging, is often a challenging
task; approximate PDEs are typically obtained through mathematical closure
procedures (e.g. mean-field approximations). In this paper, we show how such
approximate macroscopic PDEs can be exploited in constructing preconditioners
to accelerate stochastic simulations for spatially distributed particle-based
process models. We illustrate how such preconditioning can improve the
convergence of equation-free coarse-grained methods based on coarse
timesteppers. Our model problem is a stochastic reaction-diffusion model
capable of exhibiting Turing instabilities.Comment: 8 pages, 6 figures, submitted to Journal of Chemical Physic
Ergodic BSDEs and related PDEs with Neumann boundary conditions under weak dissipative assumptions
We study a class of ergodic BSDEs related to PDEs with Neumann boundary
conditions. The randomness of the drift is given by a forward process under
weakly dissipative assumptions with an invertible and bounded diffusion matrix.
Furthermore, this forward process is reflected in a convex subset of not
necessary bounded. We study the link of such EBSDEs with PDEs and we apply our
results to an ergodic optimal control problem
A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces
In this paper we present a high-order kernel method for numerically solving
diffusion and reaction-diffusion partial differential equations (PDEs) on
smooth, closed surfaces embedded in . For two-dimensional
surfaces embedded in , these types of problems have received
growing interest in biology, chemistry, and computer graphics to model such
things as diffusion of chemicals on biological cells or membranes, pattern
formations in biology, nonlinear chemical oscillators in excitable media, and
texture mappings. Our kernel method is based on radial basis functions (RBFs)
and uses a semi-discrete approach (or the method-of-lines) in which the surface
derivative operators that appear in the PDEs are approximated using
collocation. The method only requires nodes at "scattered" locations on the
surface and the corresponding normal vectors to the surface. Additionally, it
does not rely on any surface-based metrics and avoids any intrinsic coordinate
systems, and thus does not suffer from any coordinate distortions or
singularities. We provide error estimates for the kernel-based approximate
surface derivative operators and numerically study the accuracy and stability
of the method. Applications to different non-linear systems of PDEs that arise
in biology and chemistry are also presented
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