2,543 research outputs found
New recurrence relationships between orthogonal polynomials which lead to new Lanczos-type algorithms
Lanczos methods for solving Ax = b consist in constructing a sequence of vectors (Xk),k = 1,... such that rk = b-AXk= Pk(A)r0, where Pk is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(εi) = (y, Air0). Let P(1)k be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to c(1) defined as c(1)(εi) = c(εi+1). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for Pk and one for P(1)k. We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all
Revision of TR-09-25: A Hybrid Variational/Ensemble Filter Approach to Data Assimilation
Two families of methods are widely used in data assimilation: the
four dimensional variational (4D-Var) approach, and the ensemble Kalman filter
(EnKF) approach. The two families have been developed largely through parallel
research efforts. Each method has its advantages and disadvantages. It is of
interest to develop hybrid data assimilation
algorithms that can combine the relative strengths of the two approaches.
This paper proposes a subspace approach to investigate the theoretical equivalence between the suboptimal
4D-Var method (where only a small number of optimization iterations are
performed) and the practical EnKF method (where only a small number of ensemble
members are used) in a linear Gaussian setting. The analysis motivates a new
hybrid algorithm: the optimization directions obtained from a short window
4D-Var run are used to construct the EnKF initial ensemble.
The proposed hybrid method is computationally less expensive than a full
4D-Var, as only short assimilation windows are considered. The hybrid method has the potential to
perform better than the regular EnKF due to its look-ahead property.
Numerical results
show that the proposed hybrid ensemble filter method performs better than the
regular EnKF method for both linear and nonlinear test problems
Complexity transitions in global algorithms for sparse linear systems over finite fields
We study the computational complexity of a very basic problem, namely that of
finding solutions to a very large set of random linear equations in a finite
Galois Field modulo q. Using tools from statistical mechanics we are able to
identify phase transitions in the structure of the solution space and to
connect them to changes in performance of a global algorithm, namely Gaussian
elimination. Crossing phase boundaries produces a dramatic increase in memory
and CPU requirements necessary to the algorithms. In turn, this causes the
saturation of the upper bounds for the running time. We illustrate the results
on the specific problem of integer factorization, which is of central interest
for deciphering messages encrypted with the RSA cryptosystem.Comment: 23 pages, 8 figure
Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods
In this paper, we develop a new tensor-product based preconditioner for
discontinuous Galerkin methods with polynomial degrees higher than those
typically employed. This preconditioner uses an automatic, purely algebraic
method to approximate the exact block Jacobi preconditioner by Kronecker
products of several small, one-dimensional matrices. Traditional matrix-based
preconditioners require storage and
computational work, where is the degree of basis polynomials used, and
is the spatial dimension. Our SVD-based tensor-product preconditioner requires
storage, work in two spatial
dimensions, and work in three spatial dimensions.
Combined with a matrix-free Newton-Krylov solver, these preconditioners allow
for the solution of DG systems in linear time in per degree of freedom in
2D, and reduce the computational complexity from to
in 3D. Numerical results are shown in 2D and 3D for the
advection and Euler equations, using polynomials of degree up to . For
many test cases, the preconditioner results in similar iteration counts when
compared with the exact block Jacobi preconditioner, and performance is
significantly improved for high polynomial degrees .Comment: 40 pages, 15 figure
Spectral methods in general relativistic astrophysics
We present spectral methods developed in our group to solve three-dimensional
partial differential equations. The emphasis is put on equations arising from
astrophysical problems in the framework of general relativity.Comment: 51 pages, elsart (Elsevier Preprint), 19 PostScript figures,
submitted to Journal of Computational & Applied Mathematic
NFFT meets Krylov methods: Fast matrix-vector products for the graph Laplacian of fully connected networks
The graph Laplacian is a standard tool in data science, machine learning, and
image processing. The corresponding matrix inherits the complex structure of
the underlying network and is in certain applications densely populated. This
makes computations, in particular matrix-vector products, with the graph
Laplacian a hard task. A typical application is the computation of a number of
its eigenvalues and eigenvectors. Standard methods become infeasible as the
number of nodes in the graph is too large. We propose the use of the fast
summation based on the nonequispaced fast Fourier transform (NFFT) to perform
the dense matrix-vector product with the graph Laplacian fast without ever
forming the whole matrix. The enormous flexibility of the NFFT algorithm allows
us to embed the accelerated multiplication into Lanczos-based eigenvalues
routines or iterative linear system solvers and even consider other than the
standard Gaussian kernels. We illustrate the feasibility of our approach on a
number of test problems from image segmentation to semi-supervised learning
based on graph-based PDEs. In particular, we compare our approach with the
Nystr\"om method. Moreover, we present and test an enhanced, hybrid version of
the Nystr\"om method, which internally uses the NFFT.Comment: 28 pages, 9 figure
The Anderson model of localization: a challenge for modern eigenvalue methods
We present a comparative study of the application of modern eigenvalue
algorithms to an eigenvalue problem arising in quantum physics, namely, the
computation of a few interior eigenvalues and their associated eigenvectors for
the large, sparse, real, symmetric, and indefinite matrices of the Anderson
model of localization. We compare the Lanczos algorithm in the 1987
implementation of Cullum and Willoughby with the implicitly restarted Arnoldi
method coupled with polynomial and several shift-and-invert convergence
accelerators as well as with a sparse hybrid tridiagonalization method. We
demonstrate that for our problem the Lanczos implementation is faster and more
memory efficient than the other approaches. This seemingly innocuous problem
presents a major challenge for all modern eigenvalue algorithms.Comment: 16 LaTeX pages with 3 figures include
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