6,403 research outputs found
Accurate macroscale modelling of spatial dynamics in multiple dimensions
Developments in dynamical systems theory provides new support for the
macroscale modelling of pdes and other microscale systems such as Lattice
Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically
resolving subgrid microscale dynamics the dynamical systems approach constructs
accurate closures of macroscale discretisations of the microscale system. Here
we specifically explore reaction-diffusion problems in two spatial dimensions
as a prototype of generic systems in multiple dimensions. Our approach unifies
into one the modelling of systems by a type of finite elements, and the
`equation free' macroscale modelling of microscale simulators efficiently
executing only on small patches of the spatial domain. Centre manifold theory
ensures that a closed model exist on the macroscale grid, is emergent, and is
systematically approximated. Dividing space either into overlapping finite
elements or into spatially separated small patches, the specially crafted
inter-element/patch coupling also ensures that the constructed discretisations
are consistent with the microscale system/PDE to as high an order as desired.
Computer algebra handles the considerable algebraic details as seen in the
specific application to the Ginzburg--Landau PDE. However, higher order models
in multiple dimensions require a mixed numerical and algebraic approach that is
also developed. The modelling here may be straightforwardly adapted to a wide
class of reaction-diffusion PDEs and lattice equations in multiple space
dimensions. When applied to patches of microscopic simulations our coupling
conditions promise efficient macroscale simulation.Comment: some figures with 3D interaction when viewed in Acrobat Reader. arXiv
admin note: substantial text overlap with arXiv:0904.085
Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization
We introduce a new variational method for the numerical homogenization of
divergence form elliptic, parabolic and hyperbolic equations with arbitrary
rough () coefficients. Our method does not rely on concepts of
ergodicity or scale-separation but on compactness properties of the solution
space and a new variational approach to homogenization. The approximation space
is generated by an interpolation basis (over scattered points forming a mesh of
resolution ) minimizing the norm of the source terms; its
(pre-)computation involves minimizing quadratic (cell)
problems on (super-)localized sub-domains of size .
The resulting localized linear systems remain sparse and banded. The resulting
interpolation basis functions are biharmonic for , and polyharmonic
for , for the operator -\diiv(a\nabla \cdot) and can be seen as a
generalization of polyharmonic splines to differential operators with arbitrary
rough coefficients. The accuracy of the method ( in energy norm
and independent from aspect ratios of the mesh formed by the scattered points)
is established via the introduction of a new class of higher-order Poincar\'{e}
inequalities. The method bypasses (pre-)computations on the full domain and
naturally generalizes to time dependent problems, it also provides a natural
solution to the inverse problem of recovering the solution of a divergence form
elliptic equation from a finite number of point measurements.Comment: ESAIM: Mathematical Modelling and Numerical Analysis. Special issue
(2013
Explicit bounds on eigenfunctions and spectral functions on manifolds hyperbolic near a point
We derive explicit bounds for the remainder term in the local Weyl law for
locally hyperbolic manifolds, we also give the estimates of the derivative of
this remainder. We use these to obtain explicit bounds for the C^k-norms of the
L^2-normalised eigenfunctions in the case spectrum of the Laplacian is
discrete, e.g. for closed Riemannian manifolds. We also derive bounds for the
local heat trace. Our estimates are purely local and therefore also hold for
any manifold at points near which the metric is locally hyperbolic.Comment: 23 pages, 2 figure
Compressive Wave Computation
This paper considers large-scale simulations of wave propagation phenomena.
We argue that it is possible to accurately compute a wavefield by decomposing
it onto a largely incomplete set of eigenfunctions of the Helmholtz operator,
chosen at random, and that this provides a natural way of parallelizing wave
simulations for memory-intensive applications.
This paper shows that L1-Helmholtz recovery makes sense for wave computation,
and identifies a regime in which it is provably effective: the one-dimensional
wave equation with coefficients of small bounded variation. Under suitable
assumptions we show that the number of eigenfunctions needed to evolve a sparse
wavefield defined on N points, accurately with very high probability, is
bounded by C log(N) log(log(N)), where C is related to the desired accuracy and
can be made to grow at a much slower rate than N when the solution is sparse.
The PDE estimates that underlie this result are new to the authors' knowledge
and may be of independent mathematical interest; they include an L1 estimate
for the wave equation, an estimate of extension of eigenfunctions, and a bound
for eigenvalue gaps in Sturm-Liouville problems.
Numerical examples are presented in one spatial dimension and show that as
few as 10 percents of all eigenfunctions can suffice for accurate results.
Finally, we argue that the compressive viewpoint suggests a competitive
parallel algorithm for an adjoint-state inversion method in reflection
seismology.Comment: 45 pages, 4 figure
Sparse Discrete Ordinates Method in Radiative Transfer
The stationary monochromatic radiative transfer equation (RTE) is a partial differential transport equation stated on a five-dimensional phase space, the Cartesian product of physical and angular domain. We solve the RTE with a Galerkin FEM in physical space and collocation in angle, corresponding to a discrete ordinates method (DOM). To reduce the complexity of the problem and to avoid the "curse of dimension", we adapt the sparse grid combination technique to the solution space of the RTE and show that we obtain a sparse DOM which uses essentially only as many degrees of freedom as required for a purely spatial transport problem. For smooth solutions, the convergence rates deteriorate only by a logarithmic factor. We compare the sparse DOM to the standard full DOM and a sparse tensor product approach developed earlier with Galerkin FEM in physical space and a spectral method in angle. Numerical experiments confirm our finding
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