1,566 research outputs found
Gradient-type subspace iteration methods for the symmetric eigenvalue problem
This paper explores variants of the subspace iteration algorithm for
computing approximate invariant subspaces. The standard subspace iteration
approach is revisited and new variants that exploit gradient-type techniques
combined with a Grassmann manifold viewpoint are developed. A gradient method
as well as a conjugate gradient technique are described.
Convergence of the gradient-based algorithm is analyzed and a few numerical
experiments are reported, indicating that the proposed algorithms are sometimes
superior to a standard Chebyshev-based subspace iteration when compared in
terms of number of matrix vector products, but do not require estimating
optimal parameters. An important contribution of this paper to achieve this
good performance is the accurate and efficient implementation of an exact line
search. In addition, new convergence proofs are presented for the
non-accelerated gradient method that includes a locally exponential convergence
if started in a neighbourhood of the dominant
subspace with spectral gap .Comment: 29 page
Criterion for polynomial solutions to a class of linear differential equation of second order
We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where
\lambda_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential
equation, has a polynomial solution of degree n >0, then \delta_n=\lambda_n
s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}=
\lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1}\hbox{and}\quad
s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1},\quad n=1,2,.... Conversely (ii) if
\lambda_n\lambda_{n-1}\ne 0 and \delta_n=0, then the differential equation has
a polynomial solution of degree at most n. We show that the classical
differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first
and second kind), Gegenbauer, and the Hypergeometric type, etc, obey this
criterion. Further, we find the polynomial solutions for the generalized
Hermite, Laguerre, Legendre and Chebyshev differential equations.Comment: 12 page
Parallel algorithm with spectral convergence for nonlinear integro-differential equations
We discuss a numerical algorithm for solving nonlinear integro-differential
equations, and illustrate our findings for the particular case of Volterra type
equations. The algorithm combines a perturbation approach meant to render a
linearized version of the problem and a spectral method where unknown functions
are expanded in terms of Chebyshev polynomials (El-gendi's method). This
approach is shown to be suitable for the calculation of two-point Green
functions required in next to leading order studies of time-dependent quantum
field theory.Comment: 15 pages, 9 figure
Optimization via Chebyshev Polynomials
This paper presents for the first time a robust exact line-search method
based on a full pseudospectral (PS) numerical scheme employing orthogonal
polynomials. The proposed method takes on an adaptive search procedure and
combines the superior accuracy of Chebyshev PS approximations with the
high-order approximations obtained through Chebyshev PS differentiation
matrices (CPSDMs). In addition, the method exhibits quadratic convergence rate
by enforcing an adaptive Newton search iterative scheme. A rigorous error
analysis of the proposed method is presented along with a detailed set of
pseudocodes for the established computational algorithms. Several numerical
experiments are conducted on one- and multi-dimensional optimization test
problems to illustrate the advantages of the proposed strategy.Comment: 26 pages, 6 figures, 2 table
All-at-once preconditioning in PDE-constrained optimization
The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound constraints for the control are introduced. Numerical results will illustrate the competitiveness of our techniques
Numerical Approximations Using Chebyshev Polynomial Expansions
We present numerical solutions for differential equations by expanding the
unknown function in terms of Chebyshev polynomials and solving a system of
linear equations directly for the values of the function at the extrema (or
zeros) of the Chebyshev polynomial of order N (El-gendi's method). The
solutions are exact at these points, apart from round-off computer errors and
the convergence of other numerical methods used in connection to solving the
linear system of equations. Applications to initial value problems in
time-dependent quantum field theory, and second order boundary value problems
in fluid dynamics are presented.Comment: minor wording changes, some typos have been eliminate
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