5,109 research outputs found
Efficient Explicit Time Stepping of High Order Discontinuous Galerkin Schemes for Waves
This work presents algorithms for the efficient implementation of
discontinuous Galerkin methods with explicit time stepping for acoustic wave
propagation on unstructured meshes of quadrilaterals or hexahedra. A crucial
step towards efficiency is to evaluate operators in a matrix-free way with
sum-factorization kernels. The method allows for general curved geometries and
variable coefficients. Temporal discretization is carried out by low-storage
explicit Runge-Kutta schemes and the arbitrary derivative (ADER) method. For
ADER, we propose a flexible basis change approach that combines cheap face
integrals with cell evaluation using collocated nodes and quadrature points.
Additionally, a degree reduction for the optimized cell evaluation is presented
to decrease the computational cost when evaluating higher order spatial
derivatives as required in ADER time stepping. We analyze and compare the
performance of state-of-the-art Runge-Kutta schemes and ADER time stepping with
the proposed optimizations. ADER involves fewer operations and additionally
reaches higher throughput by higher arithmetic intensities and hence decreases
the required computational time significantly. Comparison of Runge-Kutta and
ADER at their respective CFL stability limit renders ADER especially beneficial
for higher orders when the Butcher barrier implies an overproportional amount
of stages. Moreover, vector updates in explicit Runge--Kutta schemes are shown
to take a substantial amount of the computational time due to their memory
intensity
An implicit algorithm for validated enclosures of the solutions to variational equations for ODEs
We propose a new algorithm for computing validated bounds for the solutions
to the first order variational equations associated to ODEs. These validated
solutions are the kernel of numerics computer-assisted proofs in dynamical
systems literature. The method uses a high-order Taylor method as a predictor
step and an implicit method based on the Hermite-Obreshkov interpolation as a
corrector step. The proposed algorithm is an improvement of the -Lohner
algorithm proposed by Zgliczy\'nski and it provides sharper bounds.
As an application of the algorithm, we give a computer-assisted proof of the
existence of an attractor set in the R\"ossler system, and we show that the
attractor contains an invariant and uniformly hyperbolic subset on which the
dynamics is chaotic, that is, conjugated to subshift of finite type with
positive topological entropy.Comment: 33 pages, 11 figure
Symmetric tensor decomposition
We present an algorithm for decomposing a symmetric tensor, of dimension n
and order d as a sum of rank-1 symmetric tensors, extending the algorithm of
Sylvester devised in 1886 for binary forms. We recall the correspondence
between the decomposition of a homogeneous polynomial in n variables of total
degree d as a sum of powers of linear forms (Waring's problem), incidence
properties on secant varieties of the Veronese Variety and the representation
of linear forms as a linear combination of evaluations at distinct points. Then
we reformulate Sylvester's approach from the dual point of view. Exploiting
this duality, we propose necessary and sufficient conditions for the existence
of such a decomposition of a given rank, using the properties of Hankel (and
quasi-Hankel) matrices, derived from multivariate polynomials and normal form
computations. This leads to the resolution of polynomial equations of small
degree in non-generic cases. We propose a new algorithm for symmetric tensor
decomposition, based on this characterization and on linear algebra
computations with these Hankel matrices. The impact of this contribution is
two-fold. First it permits an efficient computation of the decomposition of any
tensor of sub-generic rank, as opposed to widely used iterative algorithms with
unproved global convergence (e.g. Alternate Least Squares or gradient
descents). Second, it gives tools for understanding uniqueness conditions, and
for detecting the rank
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