2,733 research outputs found
Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices
We propose a Preconditioned Locally Harmonic Residual (PLHR) method for
computing several interior eigenpairs of a generalized Hermitian eigenvalue
problem, without traditional spectral transformations, matrix factorizations,
or inversions. PLHR is based on a short-term recurrence, easily extended to a
block form, computing eigenpairs simultaneously. PLHR can take advantage of
Hermitian positive definite preconditioning, e.g., based on an approximate
inverse of an absolute value of a shifted matrix, introduced in [SISC, 35
(2013), pp. A696-A718]. Our numerical experiments demonstrate that PLHR is
efficient and robust for certain classes of large-scale interior eigenvalue
problems, involving Laplacian and Hamiltonian operators, especially if memory
requirements are tight
Closer to the solutions: iterative linear solvers
The solution of dense linear systems received much attention after the second world war, and by the end of the sixties, most of the problems associated with it had been solved. For a long time, Wilkinson's \The Algebraic Eigenvalue Problem" [107], other than the title suggests, became also the standard textbook for the solution of linear systems. When it became clear that partial dierential equations could be solved numerically, to a level of accuracy that was of interest for application areas (such as reservoir engineering, and reactor diusion modeling), there was a strong need for the fast solution of the discretized systems, and iterative methods became popular for these problems
Dynamic Iterative Pursuit
For compressive sensing of dynamic sparse signals, we develop an iterative
pursuit algorithm. A dynamic sparse signal process is characterized by varying
sparsity patterns over time/space. For such signals, the developed algorithm is
able to incorporate sequential predictions, thereby providing better
compressive sensing recovery performance, but not at the cost of high
complexity. Through experimental evaluations, we observe that the new algorithm
exhibits a graceful degradation at deteriorating signal conditions while
capable of yielding substantial performance gains as conditions improve.Comment: 6 pages, 7 figures. Accepted for publication in IEEE Transactions on
Signal Processin
Computing and deflating eigenvalues while solving multiple right hand side linear systems in Quantum Chromodynamics
We present a new algorithm that computes eigenvalues and eigenvectors of a
Hermitian positive definite matrix while solving a linear system of equations
with Conjugate Gradient (CG). Traditionally, all the CG iteration vectors could
be saved and recombined through the eigenvectors of the tridiagonal projection
matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm
capitalizes on the iteration vectors produced by CG to update only a small
window of vectors that approximate the eigenvectors. While this window is
restarted in a locally optimal way, the CG algorithm for the linear system is
unaffected. Yet, in all our experiments, this small window converges to the
required eigenvectors at a rate identical to unrestarted Lanczos. After the
solution of the linear system, eigenvectors that have not accurately converged
can be improved in an incremental fashion by solving additional linear systems.
In this case, eigenvectors identified in earlier systems can be used to
deflate, and thus accelerate, the convergence of subsequent systems. We have
used this algorithm with excellent results in lattice QCD applications, where
hundreds of right hand sides may be needed. Specifically, about 70 eigenvectors
are obtained to full accuracy after solving 24 right hand sides. Deflating
these from the large number of subsequent right hand sides removes the dreaded
critical slowdown, where the conditioning of the matrix increases as the quark
mass reaches a critical value. Our experiments show almost a constant number of
iterations for our method, regardless of quark mass, and speedups of 8 over
original CG for light quark masses.Comment: 22 pages, 26 eps figure
Paved with Good Intentions: Analysis of a Randomized Block Kaczmarz Method
The block Kaczmarz method is an iterative scheme for solving overdetermined
least-squares problems. At each step, the algorithm projects the current
iterate onto the solution space of a subset of the constraints. This paper
describes a block Kaczmarz algorithm that uses a randomized control scheme to
choose the subset at each step. This algorithm is the first block Kaczmarz
method with an (expected) linear rate of convergence that can be expressed in
terms of the geometric properties of the matrix and its submatrices. The
analysis reveals that the algorithm is most effective when it is given a good
row paving of the matrix, a partition of the rows into well-conditioned blocks.
The operator theory literature provides detailed information about the
existence and construction of good row pavings. Together, these results yield
an efficient block Kaczmarz scheme that applies to many overdetermined
least-squares problem
A mathematical and computational review of Hartree-Fock SCF methods in Quantum Chemistry
We present here a review of the fundamental topics of Hartree-Fock theory in
Quantum Chemistry. From the molecular Hamiltonian, using and discussing the
Born-Oppenheimer approximation, we arrive to the Hartree and Hartree-Fock
equations for the electronic problem. Special emphasis is placed in the most
relevant mathematical aspects of the theoretical derivation of the final
equations, as well as in the results regarding the existence and uniqueness of
their solutions. All Hartree-Fock versions with different spin restrictions are
systematically extracted from the general case, thus providing a unifying
framework. Then, the discretization of the one-electron orbitals space is
reviewed and the Roothaan-Hall formalism introduced. This leads to a exposition
of the basic underlying concepts related to the construction and selection of
Gaussian basis sets, focusing in algorithmic efficiency issues. Finally, we
close the review with a section in which the most relevant modern developments
(specially those related to the design of linear-scaling methods) are commented
and linked to the issues discussed. The whole work is intentionally
introductory and rather self-contained, so that it may be useful for non
experts that aim to use quantum chemical methods in interdisciplinary
applications. Moreover, much material that is found scattered in the literature
has been put together here to facilitate comprehension and to serve as a handy
reference.Comment: 64 pages, 3 figures, tMPH2e.cls style file, doublesp, mathbbol and
subeqn package
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