71,834 research outputs found
Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation
There has been a large increase in the amount of work on hierarchical
low-rank approximation methods, where the interest is shared by multiple
communities that previously did not intersect. This objective of this article
is two-fold; to provide a thorough review of the recent advancements in this
field from both analytical and algebraic perspectives, and to present a
comparative benchmark of two highly optimized implementations of contrasting
methods for some simple yet representative test cases. We categorize the recent
advances in this field from the perspective of compute-memory tradeoff, which
has not been considered in much detail in this area. Benchmark tests reveal
that there is a large difference in the memory consumption and performance
between the different methods.Comment: 19 pages, 6 figure
A Multiscale Method for Model Order Reduction in PDE Parameter Estimation
Estimating parameters of Partial Differential Equations (PDEs) is of interest
in a number of applications such as geophysical and medical imaging. Parameter
estimation is commonly phrased as a PDE-constrained optimization problem that
can be solved iteratively using gradient-based optimization. A computational
bottleneck in such approaches is that the underlying PDEs needs to be solved
numerous times before the model is reconstructed with sufficient accuracy. One
way to reduce this computational burden is by using Model Order Reduction (MOR)
techniques such as the Multiscale Finite Volume Method (MSFV).
In this paper, we apply MSFV for solving high-dimensional parameter
estimation problems. Given a finite volume discretization of the PDE on a fine
mesh, the MSFV method reduces the problem size by computing a
parameter-dependent projection onto a nested coarse mesh. A novelty in our work
is the integration of MSFV into a PDE-constrained optimization framework, which
updates the reduced space in each iteration. We also present a computationally
tractable way of differentiating the MOR solution that acknowledges the change
of basis. As we demonstrate in our numerical experiments, our method leads to
computational savings particularly for large-scale parameter estimation
problems and can benefit from parallelization.Comment: 22 pages, 4 figures, 3 table
Solution of the Linearly Structured Partial Polynomial Inverse Eigenvalue Problem
In this paper, linearly structured partial polynomial inverse eigenvalue
problem is considered for the matrix polynomial of arbitrary degree
. Given a set of eigenpairs (), this problem
concerns with computing the matrices for
of specified linear structure such that the matrix
polynomial has
the given eigenpairs as its eigenvalues and eigenvectors. Many practical
applications give rise to the linearly structured structured matrix polynomial.
Therefore, construction of the linearly structured matrix polynomial is the
most important aspect of the polynomial inverse eigenvalue problem(PIEP). In
this paper, a necessary and sufficient condition for the existence of the
solution of this problem is derived. Additionally, we characterize the class of
all solutions to this problem by giving the explicit expressions of solutions.
The results presented in this paper address some important open problems in the
area of PIEP raised in De Teran, Dopico and Van Dooren [SIAM Journal on Matrix
Analysis and Applications, (), pp ]. An attractive
feature of our solution approach is that it does not impose any restriction on
the number of eigendata for computing the solution of PIEP. The proposed method
is validated with various numerical examples on a spring mass problem
Chapter 10: Algebraic Algorithms
Our Chapter in the upcoming Volume I: Computer Science and Software
Engineering of Computing Handbook (Third edition), Allen Tucker, Teo Gonzales
and Jorge L. Diaz-Herrera, editors, covers Algebraic Algorithms, both symbolic
and numerical, for matrix computations and root-finding for polynomials and
systems of polynomials equations. We cover part of these large subjects and
include basic bibliography for further study. To meet space limitation we cite
books, surveys, and comprehensive articles with pointers to further references,
rather than including all the original technical papers.Comment: 41.1 page
A literature survey of matrix methods for data science
Efficient numerical linear algebra is a core ingredient in many applications
across almost all scientific and industrial disciplines. With this survey we
want to illustrate that numerical linear algebra has played and is playing a
crucial role in enabling and improving data science computations with many new
developments being fueled by the availability of data and computing resources.
We highlight the role of various different factorizations and the power of
changing the representation of the data as well as discussing topics such as
randomized algorithms, functions of matrices, and high-dimensional problems. We
briefly touch upon the role of techniques from numerical linear algebra used
within deep learning
A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions
This paper introduces a new framework for constructing the Discrete Empirical
Interpolation Method DEIM projection operator. The interpolation node selection
procedure is formulated using the QR factorization with column pivoting, and it
enjoys a sharper error bound for the DEIM projection error. Furthermore, for a
subspace given as the range of an orthonormal , the DEIM
projection does not change if is replaced by with arbitrary
unitary matrix . In a large-scale setting, the new approach allows
modifications that use only randomly sampled rows of , but with the
potential of producing good approximations with corresponding probabilistic
error bounds. Another salient feature of the new framework is that robust and
efficient software implementation is easily developed, based on readily
available high performance linear algebra packages.Comment: 19 page
Direct Inversion of the 3D Pseudo-polar Fourier Transform
The pseudo-polar Fourier transform is a specialized non-equally spaced
Fourier transform, which evaluates the Fourier transform on a near-polar grid,
known as the pseudo-polar grid. The advantage of the pseudo-polar grid over
other non-uniform sampling geometries is that the transformation, which samples
the Fourier transform on the pseudo-polar grid, can be inverted using a fast
and stable algorithm. For other sampling geometries, even if the non-equally
spaced Fourier transform can be inverted, the only known algorithms are
iterative. The convergence speed of these algorithms as well as their accuracy
are difficult to control, as they depend both on the sampling geometry as well
as on the unknown reconstructed object. In this paper, we present a direct
inversion algorithm for the three-dimensional pseudo-polar Fourier transform.
The algorithm is based only on one-dimensional resampling operations, and is
shown to be significantly faster than existing iterative inversion algorithms
Literature survey on low rank approximation of matrices
Low rank approximation of matrices has been well studied in literature.
Singular value decomposition, QR decomposition with column pivoting, rank
revealing QR factorization (RRQR), Interpolative decomposition etc are
classical deterministic algorithms for low rank approximation. But these
techniques are very expensive operations are required for matrices). There are several randomized algorithms available in the
literature which are not so expensive as the classical techniques (but the
complexity is not linear in n). So, it is very expensive to construct the low
rank approximation of a matrix if the dimension of the matrix is very large.
There are alternative techniques like Cross/Skeleton approximation which gives
the low-rank approximation with linear complexity in n . In this article we
review low rank approximation techniques briefly and give extensive references
of many techniques
Fast Algorithms for the Multi-dimensional Jacobi Polynomial Transform
We use the well-known observation that the solutions of Jacobi's differential
equation can be represented via non-oscillatory phase and amplitude functions
to develop a fast algorithm for computing multi-dimensional Jacobi polynomial
transforms. More explicitly, it follows from this observation that the matrix
corresponding to the discrete Jacobi transform is the Hadamard product of a
numerically low-rank matrix and a multi-dimensional discrete Fourier transform
(DFT) matrix. The application of the Hadamard product can be carried out via
fast Fourier transforms (FFTs), resulting in a nearly optimal algorithm
to compute the multidimensional Jacobi polynomial transform
Preconditioning Parametrized Linear Systems
Preconditioners are generally essential for fast convergence in the iterative
solution of linear systems of equations. However, the computation of a good
preconditioner can be expensive. So, while solving a sequence of many linear
systems, it is advantageous to recycle preconditioners, that is, update a
previous preconditioner and reuse the updated version. In this paper, we
introduce a simple and effective method for doing this. Although our approach
can be used for matrices changing slowly in any way, we focus on the important
case of sequences of the type , where the right hand side
may or may not change. More general changes in matrices will be discussed in a
future paper. We update preconditioners by defining a map from a new matrix to
a previous matrix, for example the first matrix in the sequence, and combine
the preconditioner for this previous matrix with the map to define the new
preconditioner. This approach has several advantages. The update is entirely
independent from the original preconditioner, so it can be applied to any
preconditioner. The possibly high cost of an initial preconditioner can be
amortized over many linear solves. The cost of updating the preconditioner is
more or less constant and independent of the original preconditioner. There is
flexibility in balancing the quality of the map with the computational cost. In
the numerical experiments section we demonstrate good results for several
applications.Comment: V2 Model Reduction discussion. V3 Theoretical section replaced with
experimental analysis of sparsity patterns. Top opt application added. Rail
replaced with Flow. Only early shifts for THT. ILUTP, SAM implementations
more efficient; results updated. ILUTP m-file included. New citations added.
V4 New pattern added to top opt sparsity pattern analysis/results. ILUTP
m-file minor update
- …