7,556 research outputs found
Fast Multipole Preconditioners for Sparse Matrices Arising from Elliptic Equations
Among optimal hierarchical algorithms for the computational solution of
elliptic problems, the Fast Multipole Method (FMM) stands out for its
adaptability to emerging architectures, having high arithmetic intensity,
tunable accuracy, and relaxable global synchronization requirements. We
demonstrate that, beyond its traditional use as a solver in problems for which
explicit free-space kernel representations are available, the FMM has
applicability as a preconditioner in finite domain elliptic boundary value
problems, by equipping it with boundary integral capability for satisfying
conditions at finite boundaries and by wrapping it in a Krylov method for
extensibility to more general operators. Here, we do not discuss the well
developed applications of FMM to implement matrix-vector multiplications within
Krylov solvers of boundary element methods. Instead, we propose using FMM for
the volume-to-volume contribution of inhomogeneous Poisson-like problems, where
the boundary integral is a small part of the overall computation. Our method
may be used to precondition sparse matrices arising from finite
difference/element discretizations, and can handle a broader range of
scientific applications. Compared with multigrid methods, it is capable of
comparable algebraic convergence rates down to the truncation error of the
discretized PDE, and it offers potentially superior multicore and distributed
memory scalability properties on commodity architecture supercomputers.
Compared with other methods exploiting the low rank character of off-diagonal
blocks of the dense resolvent operator, FMM-preconditioned Krylov iteration may
reduce the amount of communication because it is matrix-free and exploits the
tree structure of FMM. We describe our tests in reproducible detail with freely
available codes and outline directions for further extensibility.Comment: 17 pages, 9 figure
A multigrid scheme for 3D Monge-Amp\`ere equations
The elliptic Monge-Amp\`ere equation is a fully nonlinear partial
differential equation which has been the focus of increasing attention from the
scientific computing community. Fast three dimensional solvers are needed, for
example in medical image registration but are not yet available. We build fast
solvers for smooth solutions in three dimensions using a nonlinear
full-approximation storage multigrid method. Starting from a second-order
accurate centered finite difference approximation, we present a nonlinear
Gauss-Seidel iterative method which has a mechanism for selecting the convex
solution of the equation. The iterative method is used as an effective
smoother, combined with the full-approximation storage multigrid method.
Numerical experiments are provided to validate the accuracy of the finite
difference scheme and illustrate the computational efficiency of the proposed
multigrid solver.Comment: 18 pages, 1 figure, 7 tables, 41 references. Accepted by
International Journal of Computer Mathematics (published online: 21 Nov 2016
A fast and memory-efficient spectral Galerkin scheme for distributed elliptic optimal control problems
Many scientific and engineering challenges can be formulated as optimization
problems which are constrained by partial differential equations (PDEs). These
include inverse problems, control problems, and design problems. As a major
challenge, the associated optimization procedures are inherently large-scale.
To ensure computational tractability, the design of efficient and robust
iterative methods becomes imperative. To meet this challenge, this paper
introduces a fast and memory-efficient preconditioned iterative scheme for a
class of distributed optimal control problems governed by
convection-diffusion-reaction (CDR) equations. As an alternative to low-order
discretizations and Schur-complement block preconditioners, the scheme combines
a high-order spectral Galerkin method with an efficient preconditioner tailored
specifically for the CDR application. The preconditioner is matrix-free and can
be applied within linear complexity where the proportionality constant is
small. Numerical results demonstrate that the preconditioner is ideal in the
sense that appropriate Krylov subspace methods converge within a low number of
iterations, independently of the problem size and the Tikhonov regularization
parameter
High-performance modeling acoustic and elastic waves using the Parallel Dichotomy Algorithm
A high-performance parallel algorithm is proposed for modeling the
propagation of acoustic and elastic waves in inhomogeneous media. An initial
boundary-value problem is replaced by a series of boundary-value problems for a
constant elliptic operator and different right-hand sides via the integral
Laguerre transform. It is proposed to solve difference equations by the
conjugate gradient method for acoustic equations and by the GMRES method
for modeling elastic waves. A preconditioning operator was the Laplace operator
that is inverted using the variable separation method. The novelty of the
proposed algorithm is using the Dichotomy Algorithm (Terekhov, 2010), which was
designed for solving a series of tridiagonal systems of linear equations, in
the context of the preconditioning operator inversion. Via considering
analytical solutions, it is shown that modeling wave processes for long
instants of time requires high-resolution meshes. The proposed parallel
fine-mesh algorithm enabled to solve real application seismic problems in
acceptable time and with high accuracy. By solving model problems, it is
demonstrated that the considered parallel algorithm possesses high performance
and efficiency over a wide range of the number of processors (from 2 to 8192).Comment: The formula (2) has been correcte
An Efficient Coarse Grid Projection Method for Quasigeostrophic Models of Large-Scale Ocean Circulation
This paper puts forth a coarse grid projection (CGP) multiscale method to
accelerate computations of quasigeostrophic (QG) models for large scale ocean
circulation. These models require solving an elliptic sub-problem at each time
step, which takes the bulk of the computational time. The method we propose
here is a modular approach that facilitates data transfer with simple
interpolations and uses black-box solvers for solving the elliptic sub-problem
and potential vorticity equations in the QG flow solvers. After solving the
elliptic sub-problem on a coarsened grid, an interpolation scheme is used to
obtain the fine data for subsequent time stepping on the full grid. The
potential vorticity field is then updated on the fine grid with savings in
computational time due to the reduced number of grid points for the elliptic
solver. The method is applied to both single layer barotropic and two-layer
stratified QG ocean models for mid-latitude oceanic basins in the beta plane,
which are standard prototypes of more realistic ocean dynamics. The method is
found to accelerate these computations while retaining the same level of
accuracy in the fine-resolution field. A linear acceleration rate is obtained
for all the cases we consider due to the efficient linear-cost fast Fourier
transform based elliptic solver used. We expect the speed-up of the CGP method
to increase dramatically for versions of the method that use other, suboptimal,
elliptic solvers, which are generally quadratic cost. It is also demonstrated
that numerical oscillations due to lower grid resolutions, in which the Munk
scales are not resolved adequately, are effectively eliminated with CGP method.Comment: International Journal for Multiscale Computational Engineering, 2013.
arXiv admin note: substantial text overlap with arXiv:1212.0140,
arXiv:1212.0922, arXiv:1104.273
The WR-HSS iteration method for a system of linear differential equations and its applications to the unsteady discrete elliptic problem
We consider the numerical method for non-self-adjoint positive definite
linear differential equations, and its application to the unsteady discrete
elliptic problem, which is derived from spatial discretization of the unsteady
elliptic problem with Dirichlet boundary condition. Based on the idea of the
alternating direction implicit (ADI) iteration technique and the
Hermitian/skew-Hermitian splitting (HSS), we establish a waveform relaxation
(WR) iteration method for solving the non-self-adjoint positive definite linear
differential equations, called the WR-HSS method. We analyze the convergence
property of the WR-HSS method, and prove that the WR-HSS method is
unconditionally convergent to the solution of the system of linear differential
equations. In addition, we derive the upper bound of the contraction factor of
the WR-HSS method in each iteration which is only dependent on the Hermitian
part of the corresponding non-self-adjoint positive definite linear
differential operator. Finally, the applications of the WR-HSS method to the
unsteady discrete elliptic problem demonstrate its effectiveness and the
correctness of the theoretical results.Comment: 30 pages, 5 figures, 13 table
A multidomain spectral method for solving elliptic equations
We present a new solver for coupled nonlinear elliptic partial differential
equations (PDEs). The solver is based on pseudo-spectral collocation with
domain decomposition and can handle one- to three-dimensional problems. It has
three distinct features. First, the combined problem of solving the PDE,
satisfying the boundary conditions, and matching between different subdomains
is cast into one set of equations readily accessible to standard linear and
nonlinear solvers. Second, touching as well as overlapping subdomains are
supported; both rectangular blocks with Chebyshev basis functions as well as
spherical shells with an expansion in spherical harmonics are implemented.
Third, the code is very flexible: The domain decomposition as well as the
distribution of collocation points in each domain can be chosen at run time,
and the solver is easily adaptable to new PDEs. The code has been used to solve
the equations of the initial value problem of general relativity and should be
useful in many other problems. We compare the new method to finite difference
codes and find it superior in both runtime and accuracy, at least for the
smooth problems considered here.Comment: 31 pages, 8 figure
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
Matrix-equation-based strategies for convection-diffusion equations
We are interested in the numerical solution of nonsymmetric linear systems
arising from the discretization of convection-diffusion partial differential
equations with separable coefficients and dominant convection. Preconditioners
based on the matrix equation formulation of the problem are proposed, which
naturally approximate the original discretized problem. For certain types of
convection coefficients, we show that the explicit solution of the matrix
equation can effectively replace the linear system solution. Numerical
experiments with data stemming from two and three dimensional problems are
reported, illustrating the potential of the proposed methodology
An extrapolation cascadic multigrid method combined with a fourth order compact scheme for 3D poisson equation
In this paper, we develop an EXCMG method to solve the three-dimensional
Poisson equation on rectangular domains by using the compact finite difference
(FD) method with unequal meshsizes in different coordinate directions. The
resulting linear system from compact FD discretization is solved by the
conjugate gradient (CG) method with a relative residual stopping criterion. By
combining the Richardson extrapolation and tri-quartic Lagrange interpolation
for the numerical solutions from two-level of grids (current and previous
grids), we are able to produce an extremely accurate approximation of the
actual numerical solution on the next finer grid, which can greatly reduce the
number of relaxation sweeps needed. Additionally, a simple method based on the
midpoint extrapolation formula is used for the fourth-order FD solutions on
two-level of grids to achieve sixth-order accuracy on the entire fine grid
cheaply and directly. The gradient of the numerical solution can also be easily
obtained through solving a series of tridiagonal linear systems resulting from
the fourth-order compact FD discretizations. Numerical results show that our
EXCMG method is much more efficient than the classical V-cycle and W-cycle
multigrid methods. Moreover, only few CG iterations are required on the finest
grid to achieve full fourth-order accuracy in both the -norm and
-norm for the solution and its gradient when the exact solution
belongs to . Finally, numerical result shows that our EXCMG method is
still effective when the exact solution has a lower regularity, which widens
the scope of applicability of our EXCMG method.Comment: Accepted for publication in Journal of Scientific Computing. arXiv
admin note: text overlap with arXiv:1506.0298
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