Dra. Marlliny Monsalve
Reaserch interests:
My research interests include numerical linear algebra, scientific
computing, matrix theory, perturbation analysis, and their application in
science and engineering. Currently I am developing numerical methods to solve nonlinear matrix
problems.
Publications:
• On two numerical methods for the solution of
largescale algebraic Riccati equations. IMA J. Numer. Anal., 34 pp. 904–920. (2014) (with V. Simoncini and D. Szyld).
Abstract (click to view)
The inexact NewtonKleinman method is an iterative scheme for numerically solving
large scale algebraic Riccati equations. At each iteration, the approximate solution of a Lyapunov
linear equation is required. Specifically designed projection of the Riccati equation onto an iteratively
generated approximation space provides a possible alternative. Our numerical experiments with
enriched approximation spaces seem to indicate that this latter approach is superior to Newtontype
strategies on realistic problems, thus giving experimental ground for recent developments in this
direction. As part of an explanation of why this is so, we derive several matrix relations between the
iterates produced by the same projection approach applied to both the (quadratic) Riccati equation
and its linear counterpart, the Lyapunov equation.
• Newton's method and secant methods: A longstanding relationship
from vectors to matrices. Portugaliae Mathematica, 68(4) pp. 431–475. (2011) (with M. Raydan).
Abstract (click to view)
Nonlinear matrix equations arise in different scientific topics,
such as applied statistics, control theory, and financial mathematics, among others. As in many other
scientific areas, Newton's method has played an important role when solving these matrix problems.
Under standard assumptions, the specialized Newton's methods that have been developed for specific problems exhibit
local and qquadratic convergence and require a suitable initial guess. They also
require, as usual, a significant amount of computational work per iteration, that in this case involve
several matrix factorizations per iterations. As expected, whenever a Newton's method can be developed, a secant method
can also be developed. Indeed, more recently, secant methods for solving specific nonlinear matrix problems have been
developed opening a new line of research. As in previous scenarios, these specialized secant methods exhibit
local and qsuperlinear convergence, also require a suitable initial guess, and avoid the use of derivatives in the formulation of the
schemes. In this review we start by recalling the presence of Newton's method and the secant methods, and also their classical relationship, in different and sometimes unexpected scenarios
for vector problems. Then we present and describe the state of the art in the use of Newton's method and also the secant method in the space of matrices. A second objective is to present
a unified approach for describing the features of these classical schemes, that in the space of matrices represent
an interesting research area with special features to be explored.
• Secant method for nonlinear matrix problems.
P. Van Dooren S. P. Bhattacharyya, R. H. Chan, V. Olshevsky and A. Routray (Editors).
Lecture Notes in Electrical Engineering. Numerical Linear Algebra in Signals,
Systems and Control. (2011) (with M. Raydan).
Abstract (click to view)
Nonlinear matrix equations arise in different scientific topics,
such as applied statistics and control theory, among others. Standard
approaches to solve them include and combine some variations of
Newton's method, matrix factorizations, and reduction to generalized
eigenvalue problems. fIn this paper we explore the use of secant methods
in the space of matrices, that represent a new approach with interesting
features.
For the special problem of computing the inverse or the pseudoinverse of a
given matrix, we propose a specialized secant method for which we establish stability
and qsuperlinear convergence, and for which we also present some numerical results.
In addition, for solving
quadratic matrix equations, we discuss several issues, and present
preliminary and encouraging numerical experiments.
• Inexact Newton with Krylov projection and recycling for Riccati equations.
[PDF file]
Update of poster presented at the Conference on Numerical Linear Algebra: Perturbation, Performance, and Portability
(A Conference in Celebration of G.W. (Pete) Stewart's 70th Birthday), July 1920, 2010 Austin, Texas.
(with Daniel Szyld).
Abstract (click to view)
Our purpose is to solve Riccati equations which arise in many applications in Control Theory. It
is standard to seek lowrank solutions (for storage issues). Since this is a nonlinear matrix equation,
Newton’s method is a successful strategy; it requires the solution of a Lyapunov equation at each iteration. In
this work in progress, we propose to use an Inexact Newton method to solve the Riccati equation and
a Krylov projection method to solve the Lyapunov equation with a recycling process.
• A new inversionfree method for a rational matrix
equation. Linear Algebra and Its Applications, 433 pp. 64–71. (2010) (with M. Raydan).
Abstract (click to view)
Motivated by the classical NewtonSchulz method for finding the inverse of a
nonsingular matrix, we develop a new inversionfree method for
obtaining the minimal Hermitian positive definite solution of the
matrix rational equation $X + A^*X^{1}A = I$, where $I$ is the
identity matrix and $A$ is a given nonsingular matrix.
We present convergence results and discuss stability properties
when the method starts from the available matrix $AA^*$. We also
present numerical results to compare our proposal with some
previously developed inversionfree techniques for solving the
same rational matrix equation.
• A secant method for the matrix sign function.
(Submitted for publication).
Abstract (click to view)
A secant method
has been recently proposed for solving nonlinear matrix problems with
interesting features, including low computational cost and qsuperlinear
convergence for special cases. In this work, we analyze this secant method for
the special problem of computing the sign of a given matrix. The global and
$q$superlinear convergence of the proposed secant method are proved from
specialized initial guesses, and the numerical stability is also established.
We complement our analysis with several numerical experiments that show the
advantages of using the secant method over the wellknown variant of Newton's
method for the same problem.
• Specialized and hybrid Newton schemes for the matrix pth root.
Applied Mathematical Sciences, Vol. 2, 49, pp. 2401
 2424 (2008) (with M. Raydan and B. De Abreu).
Abstract (click to view)
We discuss different variants of Newton's method for computing the pth root
of a given matrix. A suitable implementation is presented for solving the
Sylvester equation, that appears at every Newton's iteration, via Kronecker products.
This approach is quadratically convergent and stable, but too expensive in computational
cost. In contrast we propose and analyze some specialized versions that exploit
the commutation of the iterates with the given matrix.
These versions are relatively inexpensive but have either stability problems
or stagnation problems when good precision is required.
Hybrid versions are presented to take advantage of the best features in
both approaches. Preliminary and encouraging numerical results are presented
for p=3 and p=5.
• Block Linear Method for LargeScale Sylvester Equations. Computational and
Applied Mathemataics, Vol. 27(1), pp. 4759 (2008).
Abstract (click to view)
We present and analyze new iterative schemes for solving the
largescale Sylvester equation AXXB=C where X \in \R^{n x p}, and
A,B and C are given matrices. These new schemes are based on
fixed point iterations and some recently developed methods for
solving block linear systems of equations. Our schemes are
flexible in the sense that for solving the block linear system, at
each iteration, any available method (direct or iterative) can be
used. We present a convergence analysis under some hypothesis on
the matrices A and B. We also present encouraging numerical
results for largescale problems. In particular, the new schemes
are compare favorably with schemes based on using block Krylov
subspace method directly on the Sylvester equation and with
recently developed method based in the construction of a low rank
approximation of the matrix C.
• Selective Alternating Projections to Find the Nearest SDD+ Matrix. Applied
Mathematics and Computation, Vol. 145, pp. 205220 (2003) (with J. Moreno, R.
Escalante and M. Raydan).
Abstract (click to view)
We extend and improve recently proposed algorithms to solve the problem
of minimizing the distance from a given matrix to the
cone of symmetric and diagonally dominant matrices with positive diagonal
(SDD^{+}). We present a variety of criteria to select a subset of the
supporting hyperplanes of the faces of SDD^{+}, and also of the polar
cone (SDD^+)^o, to then apply Dykstra's alternating projection method.
These selections reduce the number of projections and therefore reduce
the required computational work. In all our new algorithms, the symmetry
and the diagonal dominance of the obtained matrix are guaranteed.
Preliminary numerical experiments indicate that some of the
selection criteria produce a significant reduction in CPU time.
Recent Presentation at Conference:
•
Inexact Newton with Krylov projection and recycling for Riccati equations
Update of poster presented at the Conference on Numerical Linear Algebra: Perturbation, Performance, and Portability
(A Conference in Celebration of G.W. (Pete) Stewart's 70th Birthday), July 1920, 2010 Austin, Texas.
(with Daniel Szyld).
•
Inexact Newton and Krylov methods for Riccati equations
SIAM Annual Meeting, Pittsburgh, PA, 12–16 July 2010. (with Daniel Szyld).
•
A simplified Newton's method for a rational matrix problem
SIAM Conference on Applied Linear Algebra, Monterey, California, 25–29 October 2009.(with Marcos Raydan).
•
A simplified Newton's method for a rational matrix problem
Mathematics and Scientific Computing Seminar, Temple University. Philadelphia, PA, September 2009. (with Marcos Raydan).
Research projects:
•
Newton and quasiNewton methods for solving nonlinear matrix problems.
Leader.
Project CDCDUCV03.00.66.40.2007. (20072009)
•
Numerical Optimization Techniques for Inverse Seismic Problems
Collaborator.
Project UCV97003769, Agenda Petroleo, CONICIT. Collaborator. (20002006)
Teaching:
Análisis Numérico.
Cálculo Científico I
Matemáticas Discretas I.
[PDF file: Nota de Docencia]
Writings:
 Oración para un buen plan de formación
[PDF file].
