6,927 research outputs found
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
Block Circulant and Toeplitz Structures in the Linearized HartreeāFock Equation on Finite Lattices: Tensor Approach
This paper introduces and analyses the new grid-based tensor approach to
approximate solution of the elliptic eigenvalue problem for the 3D
lattice-structured systems. We consider the linearized Hartree-Fock equation
over a spatial lattice for both periodic and
non-periodic problem setting, discretized in the localized Gaussian-type
orbitals basis. In the periodic case, the Galerkin system matrix obeys a
three-level block-circulant structure that allows the FFT-based
diagonalization, while for the finite extended systems in a box (Dirichlet
boundary conditions) we arrive at the perturbed block-Toeplitz representation
providing fast matrix-vector multiplication and low storage size. The proposed
grid-based tensor techniques manifest the twofold benefits: (a) the entries of
the Fock matrix are computed by 1D operations using low-rank tensors
represented on a 3D grid, (b) in the periodic case the low-rank tensor
structure in the diagonal blocks of the Fock matrix in the Fourier space
reduces the conventional 3D FFT to the product of 1D FFTs. Lattice type systems
in a box with Dirichlet boundary conditions are treated numerically by our
previous tensor solver for single molecules, which makes possible calculations
on rather large lattices due to reduced numerical
cost for 3D problems. The numerical simulations for both box-type and periodic
lattice chain in a 3D rectangular "tube" with up to
several hundred confirm the theoretical complexity bounds for the
block-structured eigenvalue solvers in the limit of large .Comment: 30 pages, 12 figures. arXiv admin note: substantial text overlap with
arXiv:1408.383
Approximation of the critical buckling factor for composite panels
This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented
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
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