34,463 research outputs found
Solving rank structured Sylvester and Lyapunov equations
We consider the problem of efficiently solving Sylvester and Lyapunov
equations of medium and large scale, in case of rank-structured data, i.e.,
when the coefficient matrices and the right-hand side have low-rank
off-diagonal blocks. This comprises problems with banded data, recently studied
by Haber and Verhaegen in "Sparse solution of the Lyapunov equation for
large-scale interconnected systems", Automatica, 2016, and by Palitta and
Simoncini in "Numerical methods for large-scale Lyapunov equations with
symmetric banded data", SISC, 2018, which often arise in the discretization of
elliptic PDEs.
We show that, under suitable assumptions, the quasiseparable structure is
guaranteed to be numerically present in the solution, and explicit novel
estimates of the numerical rank of the off-diagonal blocks are provided.
Efficient solution schemes that rely on the technology of hierarchical
matrices are described, and several numerical experiments confirm the
applicability and efficiency of the approaches. We develop a MATLAB toolbox
that allows easy replication of the experiments and a ready-to-use interface
for the solvers. The performances of the different approaches are compared, and
we show that the new methods described are efficient on several classes of
relevant problems
Selecting Algorithms for Black Box Matrices: Checking for Matrix Properties That Can Simplify Computations
Processes to automate the selection of appropriate algorithms for various
matrix computations are described. In particular, processes to check for, and
certify, various matrix properties of black box matrices are presented. These
include sparsity patterns and structural properties that allow "superfast"
algorithms to be used in place of black-box algorithms. Matrix properties that
hold generically, and allow the use of matrix preconditioning to be reduced or
eliminated, can also be checked for and certified - notably including in the
small-field case, where this presently has the greatest impact on the
efficiency of the computation.Comment: Department of Computer Science Technical Report 2016-1085-0
Lanczos eigensolution method for high-performance computers
The theory, computational analysis, and applications are presented of a Lanczos algorithm on high performance computers. The computationally intensive steps of the algorithm are identified as: the matrix factorization, the forward/backward equation solution, and the matrix vector multiples. These computational steps are optimized to exploit the vector and parallel capabilities of high performance computers. The savings in computational time from applying optimization techniques such as: variable band and sparse data storage and access, loop unrolling, use of local memory, and compiler directives are presented. Two large scale structural analysis applications are described: the buckling of a composite blade stiffened panel with a cutout, and the vibration analysis of a high speed civil transport. The sequential computational time for the panel problem executed on a CONVEX computer of 181.6 seconds was decreased to 14.1 seconds with the optimized vector algorithm. The best computational time of 23 seconds for the transport problem with 17,000 degs of freedom was on the the Cray-YMP using an average of 3.63 processors
Hyperspectral image compression : adapting SPIHT and EZW to Anisotropic 3-D Wavelet Coding
Hyperspectral images present some specific characteristics that should be used by an efficient compression system. In compression, wavelets have shown a good adaptability to a wide range of data, while being of reasonable complexity. Some wavelet-based compression algorithms have been successfully used for some hyperspectral space missions. This paper focuses on the optimization of a full wavelet compression system for hyperspectral images. Each step of the compression algorithm is studied and optimized. First, an algorithm to find the optimal 3-D wavelet decomposition in a rate-distortion sense is defined. Then, it is shown that a specific fixed decomposition has almost the same performance, while being more useful in terms of complexity issues. It is shown that this decomposition significantly improves the classical isotropic decomposition. One of the most useful properties of this fixed decomposition is that it allows the use of zero tree algorithms. Various tree structures, creating a relationship between coefficients, are compared. Two efficient compression methods based on zerotree coding (EZW and SPIHT) are adapted on this near-optimal decomposition with the best tree structure found. Performances are compared with the adaptation of JPEG 2000 for hyperspectral images on six different areas presenting different statistical properties
Fast Hessenberg reduction of some rank structured matrices
We develop two fast algorithms for Hessenberg reduction of a structured
matrix where is a real or unitary diagonal
matrix and . The proposed algorithm for the
real case exploits a two--stage approach by first reducing the matrix to a
generalized Hessenberg form and then completing the reduction by annihilation
of the unwanted sub-diagonals. It is shown that the novel method requires
arithmetic operations and it is significantly faster than other
reduction algorithms for rank structured matrices. The method is then extended
to the unitary plus low rank case by using a block analogue of the CMV form of
unitary matrices. It is shown that a block Lanczos-type procedure for the block
tridiagonalization of induces a structured reduction on in a block
staircase CMV--type shape. Then, we present a numerically stable method for
performing this reduction using unitary transformations and we show how to
generalize the sub-diagonal elimination to this shape, while still being able
to provide a condensed representation for the reduced matrix. In this way the
complexity still remains linear in and, moreover, the resulting algorithm
can be adapted to deal efficiently with block companion matrices.Comment: 25 page
Comparison of five methods of computing the Dirichlet-Neumann operator for the water wave problem
We compare the effectiveness of solving Dirichlet-Neumann problems via the
Craig-Sulem (CS) expansion, the Ablowitz-Fokas-Musslimani (AFM) implicit
formulation, the dual AFM formulation (AFM*), a boundary integral collocation
method (BIM), and the transformed field expansion (TFE) method. The first three
methods involve highly ill-conditioned intermediate calculations that we show
can be overcome using multiple-precision arithmetic. The latter two methods
avoid catastrophic cancellation of digits in intermediate results, and are much
better suited to numerical computation.
For the Craig-Sulem expansion, we explore the cancellation of terms at each
order (up to 150th) for three types of wave profiles, namely band-limited,
real-analytic, or smooth. For the AFM and AFM* methods, we present an example
in which representing the Dirichlet or Neumann data as a series using the AFM
basis functions is impossible, causing the methods to fail. The example
involves band-limited wave profiles of arbitrarily small amplitude, with
analytic Dirichlet data. We then show how to regularize the AFM and AFM*
methods by over-sampling the basis functions and using the singular value
decomposition or QR-factorization to orthogonalize them. Two additional
examples are used to compare all five methods in the context of water waves,
namely a large-amplitude standing wave in deep water, and a pair of interacting
traveling waves in finite depth.Comment: 31 pages, 18 figures. (change from version 1: corrected error in
table on page 12
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