259 research outputs found
Local discontinuous Galerkin methods for fractional ordinary differential equations
This paper discusses the upwinded local discontinuous Galerkin methods for
the one-term/multi-term fractional ordinary differential equations (FODEs). The
natural upwind choice of the numerical fluxes for the initial value problem for
FODEs ensures stability of the methods. The solution can be computed element by
element with optimal order of convergence in the norm and
superconvergence of order at the downwind point of each
element. Here is the degree of the approximation polynomial used in an
element and () represents the order of the one-term
FODEs. A generalization of this includes problems with classic 'th-term
FODEs, yielding superconvergence order at downwind point as
. The underlying mechanism of the
superconvergence is discussed and the analysis confirmed through examples,
including a discussion of how to use the scheme as an efficient way to evaluate
the generalized Mittag-Leffler function and solutions to more generalized
FODE's.Comment: 17 pages, 7 figure
Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems
Reduced basis methods are popular for approximately solving large and complex
systems of differential equations. However, conventional reduced basis methods
do not generally preserve conservation laws and symmetries of the full order
model. Here, we present an approach for reduced model construction, that
preserves the symplectic symmetry of dissipative Hamiltonian systems. The
method constructs a closed reduced Hamiltonian system by coupling the full
model with a canonical heat bath. This allows the reduced system to be
integrated with a symplectic integrator, resulting in a correct dissipation of
energy, preservation of the total energy and, ultimately, in the stability of
the solution. Accuracy and stability of the method are illustrated through the
numerical simulation of the dissipative wave equation and a port-Hamiltonian
model of an electric circuit
Structure Preserving Model Reduction of Parametric Hamiltonian Systems
While reduced-order models (ROMs) have been popular for efficiently solving
large systems of differential equations, the stability of reduced models over
long-time integration is of present challenges. We present a greedy approach
for ROM generation of parametric Hamiltonian systems that captures the
symplectic structure of Hamiltonian systems to ensure stability of the reduced
model. Through the greedy selection of basis vectors, two new vectors are added
at each iteration to the linear vector space to increase the accuracy of the
reduced basis. We use the error in the Hamiltonian due to model reduction as an
error indicator to search the parameter space and identify the next best basis
vectors. Under natural assumptions on the set of all solutions of the
Hamiltonian system under variation of the parameters, we show that the greedy
algorithm converges with exponential rate. Moreover, we demonstrate that
combining the greedy basis with the discrete empirical interpolation method
also preserves the symplectic structure. This enables the reduction of the
computational cost for nonlinear Hamiltonian systems. The efficiency, accuracy,
and stability of this model reduction technique is illustrated through
simulations of the parametric wave equation and the parametric Schrodinger
equation
IMEX evolution of scalar fields on curved backgrounds
Inspiral of binary black holes occurs over a time-scale of many orbits, far
longer than the dynamical time-scale of the individual black holes. Explicit
evolutions of a binary system therefore require excessively many time steps to
capture interesting dynamics. We present a strategy to overcome the
Courant-Friedrichs-Lewy condition in such evolutions, one relying on modern
implicit-explicit ODE solvers and multidomain spectral methods for elliptic
equations. Our analysis considers the model problem of a forced scalar field
propagating on a generic curved background. Nevertheless, we encounter and
address a number of issues pertinent to the binary black hole problem in full
general relativity. Specializing to the Schwarzschild geometry in Kerr-Schild
coordinates, we document the results of several numerical experiments testing
our strategy.Comment: 28 pages, uses revtex4. Revised in response to referee's report. One
numerical experiment added which incorporates perturbed initial data and
adaptive time-steppin
Waves 2005 Conference
Held at Brown University, Providence, RI, June 20--24, 2005International audienc
Deep convolutional neural networks for estimating porous material parameters with ultrasound tomography
We study the feasibility of data based machine learning applied to ultrasound
tomography to estimate water-saturated porous material parameters. In this
work, the data to train the neural networks is simulated by solving wave
propagation in coupled poroviscoelastic-viscoelastic-acoustic media. As the
forward model, we consider a high-order discontinuous Galerkin method while
deep convolutional neural networks are used to solve the parameter estimation
problem. In the numerical experiment, we estimate the material porosity and
tortuosity while the remaining parameters which are of less interest are
successfully marginalized in the neural networks-based inversion. Computational
examples confirms the feasibility and accuracy of this approach
From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex
The electrostatic interpretation of the Jacobi-Gauss quadrature points is exploited to obtain interpolation points suitable for approximation of smooth functions defined on a simplex. Moreover, several new estimates, based on extensive numerical studies, for approximation along the line using Jacobi-Gauss-Lobatto quadrature points as the nodal sets are presented. The electrostatic analogy is extended to the two-dimensional case, with the emphasis being on nodal sets inside a triangle for which two very good matrices of nodal sets are presented. The matrices are evaluated by computing the Lebesgue constants and they share the property that the nodes along the edges of the simplex are the Gauss-Lobatto quadrature points of the Chebyshev and Legendre polynomials, respectively. This makes the resulting nodal sets particularly well suited for integration with conventional spectral methods and supplies a new nodal basis for h - p finite element methods
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