23,575 research outputs found
An iterative semi-implicit scheme with robust damping
An efficient, iterative semi-implicit (SI) numerical method for the time
integration of stiff wave systems is presented. Physics-based assumptions are
used to derive a convergent iterative formulation of the SI scheme which
enables the monitoring and control of the error introduced by the SI operator.
This iteration essentially turns a semi-implicit method into a fully implicit
method. Accuracy, rather than stability, determines the timestep. The scheme is
second-order accurate and shown to be equivalent to a simple preconditioning
method. We show how the diffusion operators can be handled so as to yield the
property of robust damping, i.e., dissipating the solution at all values of the
parameter \mathcal D\dt, where is a diffusion operator and \dt
the timestep. The overall scheme remains second-order accurate even if the
advection and diffusion operators do not commute. In the limit of no physical
dissipation, and for a linear test wave problem, the method is shown to be
symplectic. The method is tested on the problem of Kinetic Alfv\'en wave
mediated magnetic reconnection. A Fourier (pseudo-spectral) representation is
used. A 2-field gyrofluid model is used and an efficacious k-space SI operator
for this problem is demonstrated. CPU speed-up factors over a CFL-limited
explicit algorithm ranging from to several hundreds are obtained,
while accurately capturing the results of an explicit integration. Possible
extension of these results to a real-space (grid) discretization is discussed.Comment: Submitted to the Journal of Computational Physics. Clarifications and
caveats in response to referees, numerical demonstration of convergence rate,
generalized symplectic proo
Spectral Methods for Numerical Relativity. The Initial Data Problem
Numerical relativity has traditionally been pursued via finite differencing.
Here we explore pseudospectral collocation (PSC) as an alternative to finite
differencing, focusing particularly on the solution of the Hamiltonian
constraint (an elliptic partial differential equation) for a black hole
spacetime with angular momentum and for a black hole spacetime superposed with
gravitational radiation. In PSC, an approximate solution, generally expressed
as a sum over a set of orthogonal basis functions (e.g., Chebyshev
polynomials), is substituted into the exact system of equations and the
residual minimized. For systems with analytic solutions the approximate
solutions converge upon the exact solution exponentially as the number of basis
functions is increased. Consequently, PSC has a high computational efficiency:
for solutions of even modest accuracy we find that PSC is substantially more
efficient, as measured by either execution time or memory required, than finite
differencing; furthermore, these savings increase rapidly with increasing
accuracy. The solution provided by PSC is an analytic function given
everywhere; consequently, no interpolation operators need to be defined to
determine the function values at intermediate points and no special
arrangements need to be made to evaluate the solution or its derivatives on the
boundaries. Since the practice of numerical relativity by finite differencing
has been, and continues to be, hampered by both high computational resource
demands and the difficulty of formulating acceptable finite difference
alternatives to the analytic boundary conditions, PSC should be further pursued
as an alternative way of formulating the computational problem of finding
numerical solutions to the field equations of general relativity.Comment: 15 pages, 5 figures, revtex, submitted to PR
Composing Scalable Nonlinear Algebraic Solvers
Most efficient linear solvers use composable algorithmic components, with the
most common model being the combination of a Krylov accelerator and one or more
preconditioners. A similar set of concepts may be used for nonlinear algebraic
systems, where nonlinear composition of different nonlinear solvers may
significantly improve the time to solution. We describe the basic concepts of
nonlinear composition and preconditioning and present a number of solvers
applicable to nonlinear partial differential equations. We have developed a
software framework in order to easily explore the possible combinations of
solvers. We show that the performance gains from using composed solvers can be
substantial compared with gains from standard Newton-Krylov methods.Comment: 29 pages, 14 figures, 13 table
A Robust Solution Procedure for Hyperelastic Solids with Large Boundary Deformation
Compressible Mooney-Rivlin theory has been used to model hyperelastic solids,
such as rubber and porous polymers, and more recently for the modeling of soft
tissues for biomedical tissues, undergoing large elastic deformations. We
propose a solution procedure for Lagrangian finite element discretization of a
static nonlinear compressible Mooney-Rivlin hyperelastic solid. We consider the
case in which the boundary condition is a large prescribed deformation, so that
mesh tangling becomes an obstacle for straightforward algorithms. Our solution
procedure involves a largely geometric procedure to untangle the mesh: solution
of a sequence of linear systems to obtain initial guesses for interior nodal
positions for which no element is inverted. After the mesh is untangled, we
take Newton iterations to converge to a mechanical equilibrium. The Newton
iterations are safeguarded by a line search similar to one used in
optimization. Our computational results indicate that the algorithm is up to 70
times faster than a straightforward Newton continuation procedure and is also
more robust (i.e., able to tolerate much larger deformations). For a few
extremely large deformations, the deformed mesh could only be computed through
the use of an expensive Newton continuation method while using a tight
convergence tolerance and taking very small steps.Comment: Revision of earlier version of paper. Submitted for publication in
Engineering with Computers on 9 September 2010. Accepted for publication on
20 May 2011. Published online 11 June 2011. The final publication is
available at http://www.springerlink.co
Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation: Extended Materials
We study the seismic inverse problem for the recovery of subsurface
properties in acoustic media. In order to reduce the ill-posedness of the
problem, the heterogeneous wave speed parameter to be recovered is represented
using a limited number of coefficients associated with a basis of eigenvectors
of a diffusion equation, following the regularization by discretization
approach. We compare several choices for the diffusion coefficient in the
partial differential equations, which are extracted from the field of image
processing. We first investigate their efficiency for image decomposition
(accuracy of the representation with respect to the number of variables and
denoising). Next, we implement the method in the quantitative reconstruction
procedure for seismic imaging, following the Full Waveform Inversion method,
where the difficulty resides in that the basis is defined from an initial model
where none of the actual structures is known. In particular, we demonstrate
that the method is efficient for the challenging reconstruction of media with
salt-domes. We employ the method in two and three-dimensional experiments and
show that the eigenvector representation compensates for the lack of low
frequency information, it eventually serves us to extract guidelines for the
implementation of the method.Comment: 45 pages, 37 figure
Seismic Ray Impedance Inversion
This thesis investigates a prestack seismic inversion scheme implemented in the ray
parameter domain. Conventionally, most prestack seismic inversion methods are
performed in the incidence angle domain. However, inversion using the concept of
ray impedance, as it honours ray path variation following the elastic parameter
variation according to Snell’s law, shows the capacity to discriminate different
lithologies if compared to conventional elastic impedance inversion.
The procedure starts with data transformation into the ray-parameter domain and then
implements the ray impedance inversion along constant ray-parameter profiles. With
different constant-ray-parameter profiles, mixed-phase wavelets are initially estimated
based on the high-order statistics of the data and further refined after a proper well-to-seismic
tie. With the estimated wavelets ready, a Cauchy inversion method is used to
invert for seismic reflectivity sequences, aiming at recovering seismic reflectivity
sequences for blocky impedance inversion. The impedance inversion from reflectivity
sequences adopts a standard generalised linear inversion scheme, whose results are
utilised to identify rock properties and facilitate quantitative interpretation. It has also
been demonstrated that we can further invert elastic parameters from ray impedance
values, without eliminating an extra density term or introducing a Gardner’s relation
to absorb this term.
Ray impedance inversion is extended to P-S converted waves by introducing the
definition of converted-wave ray impedance. This quantity shows some advantages in
connecting prestack converted wave data with well logs, if compared with the shearwave
elastic impedance derived from the Aki and Richards approximation to the
Zoeppritz equations. An analysis of P-P and P-S wave data under the framework of
ray impedance is conducted through a real multicomponent dataset, which can reduce
the uncertainty in lithology identification.Inversion is the key method in generating those examples throughout the entire thesis
as we believe it can render robust solutions to geophysical problems. Apart from the
reflectivity sequence, ray impedance and elastic parameter inversion mentioned above,
inversion methods are also adopted in transforming the prestack data from the offset
domain to the ray-parameter domain, mixed-phase wavelet estimation, as well as the
registration of P-P and P-S waves for the joint analysis.
The ray impedance inversion methods are successfully applied to different types of
datasets. In each individual step to achieving the ray impedance inversion, advantages,
disadvantages as well as limitations of the algorithms adopted are detailed. As a
conclusion, the ray impedance related analyses demonstrated in this thesis are highly
competent compared with the classical elastic impedance methods and the author
would like to recommend it for a wider application
Solution of the inverse scattering problem by T-matrix completion. II. Simulations
This is Part II of the paper series on data-compatible T-matrix completion
(DCTMC), which is a method for solving nonlinear inverse problems. Part I of
the series contains theory and here we present simulations for inverse
scattering of scalar waves. The underlying mathematical model is the scalar
wave equation and the object function that is reconstructed is the medium
susceptibility. The simulations are relevant to ultrasound tomographic imaging
and seismic tomography. It is shown that DCTMC is a viable method for solving
strongly nonlinear inverse problems with large data sets. It provides not only
the overall shape of the object but the quantitative contrast, which can
correspond, for instance, to the variable speed of sound in the imaged medium.Comment: This is Part II of a paper series. Part I contains theory and is
available at arXiv:1401.3319 [math-ph]. Accepted in this form to Phys. Rev.
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