The advent of Network Science was precipitated by an urgent need to decipher simple mechanisms that could explain the formation and growth of natural and man-made networks. The large-scale structural patterns of these networks systematically deviate from stylized models such as random networks or lattices in a variety of ways, which include the existence of phenomena like “small-worldness”...
Consider saddle-point systems of the following form
$$ \hspace{5.5cm}
\mathcal{A}
\left( \begin{array}{c} u \\ v \end{array} \right) =
\left(
\begin{array}{cc}
A & B^T \\ B & 0 \\
\end{array}
\right) \left( \begin{array}{c}
u \\ v \end{array} \right) = \left(
\begin{array}{c} f \\ g \end{array} \right)
\hspace{5.5cm} (1)
$$
where $A \in...
Rational Krylov subspaces have become a fundamental ingredient in numerical linear algebra methods associated with reduction strategies. Nonetheless, a lot of their potential is still to be explored.
In particular, many structural properties of the reduced matrices in these subspaces are not fully understood. We advance in this analysis by deriving decay bounds on the entries of rational...
We consider the problem of computing the square root of a perturbation of the scaled identity matrix, $A = \alpha I_n + UV^{*}$, where $U$ and $V$ are $n \times k$ matrices with $k \leq n$.
This problem arises in various applications, including computer vision and optimization methods for machine learning. We derive a new formula for the pth root of $A$ that involves a weighted sum of...
Various phenomena simulated using partial differential equations (PDEs) give rise to constrained systems of equations. These include models of optimal control with constraints given by elliptic PDEs, as well as fundamental models of fluid dynamics such as the Stokes equations where the constraints correspond to the incompressibility (divergence-free) condition. If these models also depend on...
Wearable healthcare devices, content delivery systems, and smart home devices are pushing applications, data, services and, consequently, large-scale computations away from centralized environments onto the network, closer to the requests. Newly emerging technology of edge computing, where networked, autonomous devices work collaboratively to achieve a common goal through the synergy...
We provide rigorous theoretical bounds for Anderson acceleration (AA) that allow efficient approximate calculations of the residual that reduce communication time and storage space while maintaining convergence. Specifically, we propose a reduced variant of AA, which consists in projecting the least squares to compute the Anderson mixing onto a subspace of reduced dimension. We numerically...
Linear time-varying differential-algebraic equations with extra symmetries such as self-adjoint and skew-adjoint systems are studied. Local and global canonical forms under congruence are presented and used to classify the geometric properties of the flow associated with the differential equation as symplectic or generalized orthogonal flow. As applications, the results are applied to the...
Asynchronous methods refer to parallel iterative procedures where each process performs its task without waiting for other processes to be completed, i.e., with whatever information it has locally available and with no synchronizations with other processes. For the numerical solution of a general partial differential equation on a domain, Schwarz iterative methods use a decomposition of the...
Augmented Lagrangian preconditioner for linearized incompressible Navier-Stokes equations has
been introduced in [1,2]. Over the years it was proved to be an efficient technique to solve highly
non-symmetric algebraic systems having a saddle point structure and resulting from discretizations
of fluid problems. In the talk we review the approach and discuss some of its recent...
When considering the stability under arbitrary switching of a discrete-time linear switched system
$$
x(n + 1) = A_{\sigma(n)} x(n), \qquad
\sigma : \mathbb N \to \left\{ 1, . . . , m \right\},
$$
where $x(0) \in \mathbb{R}^d$, the matrix $A_{\sigma(n)} \in \mathbb{R}^{d \times d}$ belongs to a finite family
$F = \{ A_i \}_{1 \leq i \leq m}$ and $\sigma$ denotes the switching law,...
In this talk we consider the problem of finding effective preconditioners for linear systems of the form $Hx = y$ where
$$
H = A^T D A + \lambda^2 L^T L
$$
where $A$ and $L$ are structured matrices (e.g., Toeplitz), $D$ is a diagonal matrix, and $\lambda$ is a scalar.
These linear systems can arise when iteratively computing approximations to nonlinear inverse problems. Typically in...
Let $A \in \mathbb{C}^{n \times n}$ be a matrix, $v \in \mathbb{C}^n$ a vector, and $f: D \subseteq \mathbb{C} \to \mathbb{C}$ a function defined on the spectrum of $A$. We consider the task of computing $f(A)v$, the action of the matrix function $f(A)$ on the vector $v$. This task arises in many applications and it is a viable alternative to computing $f(A)$ directly, which is impossible if...
