arXiv subjects

Angeles Martinez

Publications and source records attributed to Angeles Martinez.

9 recordsLinked to original sources

arXiv2026

Spectral Analysis of Block Diagonally Preconditioned Multiple Saddle-Point Matrices with Inexact Schur Complements

We derive eigenvalue bounds for symmetric block-tridiagonal multiple saddle-point systems preconditioned with block-diagonal Schur complement matrices. This analysis applies to an arbitrary number of blocks and accounts for the case where the Schur complements are approximated, generalizing the findings in [Bergamaschi et al., Linear Algebra and its Applications, 2026]. Numerical experiments are carried out to validate the proposed estimates.

math.NA↗

arXiv2025

Triangular preconditioners for double saddle point linear systems arising in the mixed form of poroelasticity equations

In this paper, we study a class of inexact block triangular preconditioners for double saddle-point symmetric linear systems arising from the mixed finite element and mixed hybrid finite element discretization of Biot's poroelasticity equations. We develop a spectral analysis of the preconditioned matrix, showing that the complex eigenvalues lie in a circle of center $(1,0)$ and radius smaller than 1. In contrast, the real eigenvalues are described in terms of the roots of a third-degree polynomial with real coefficients. The results of numerical experiments are reported to show the quality of the theoretical bounds and illustrate the efficiency of the proposed preconditioners used with GMRES, especially in comparison with similar block diagonal preconditioning strategies along with the MINRES iteration.

math.NA↗

arXiv2024

Spectral analysis of block preconditioners for double saddle-point linear systems with application to PDE-constrained optimization

In this paper, we describe and analyze the spectral properties of a symmetric positive definite inexact block preconditioner for a class of symmetric, double saddle-point linear systems. We develop a spectral analysis of the preconditioned matrix, showing that its eigenvalues can be described in terms of the roots of a cubic polynomial with real coefficients. We illustrate the efficiency of the proposed preconditioners, and verify the theoretical bounds, in solving large-scale PDE-constrained optimization problems.

math.NA↗

arXiv2023

A non-monotone trust-region method with noisy oracles and additional sampling

In this work, we introduce a novel stochastic second-order method, within the framework of a non-monotone trust-region approach, for solving the unconstrained, nonlinear, and non-convex optimization problems arising in the training of deep neural networks. The proposed algorithm makes use of subsampling strategies which yield noisy approximations of the finite sum objective function and its gradient. To effectively control the resulting approximation error, we introduce an adaptive sample size strategy based on inexpensive additional sampling. Depending on the estimated progress of the algorithm, this can yield sample size scenarios ranging from mini-batch to full sample functions. We provide convergence analysis for all possible scenarios and show that the proposed method achieves almost sure convergence under standard assumptions for the trust-region framework. We report numerical experiments showing that the proposed algorithm outperforms its state-of-the-art counterpart in deep neural network training for image classification and regression tasks while requiring a significantly smaller number of gradient evaluations.

math.OC↗

arXiv2023

Some preconditioning techniques for a class of double saddle point problems

In this paper, we describe and analyze the spectral properties of a number of exact block preconditioners for a class of double saddle point problems. Among all these, we consider an inexact version of a block triangular preconditioner providing extremely fast convergence of the FGMRES method. We develop a spectral analysis of the preconditioned matrix showing that the complex eigenvalues lie in a circle of center (1,0) and radius 1, while the real eigenvalues are described in terms of the roots of a third order polynomial with real coefficients. Numerical examples are reported to illustrate the efficiency of inexact versions of the proposed preconditioners, and to verify the theoretical bounds.

math.NA↗

arXiv2022

On the efficiency of Stochastic Quasi-Newton Methods for Deep Learning

While first-order methods are popular for solving optimization problems that arise in large-scale deep learning problems, they come with some acute deficiencies. To diminish such shortcomings, there has been recent interest in applying second-order methods such as quasi-Newton based methods which construct Hessians approximations using only gradient information. The main focus of our work is to study the behaviour of stochastic quasi-Newton algorithms for training deep neural networks. We have analyzed the performance of two well-known quasi-Newton updates, the limited memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) and the Symmetric Rank One (SR1). This study fills a gap concerning the real performance of both updates and analyzes whether more efficient training is obtained when using the more robust BFGS update or the cheaper SR1 formula which allows for indefinite Hessian approximations and thus can potentially help to better navigate the pathological saddle points present in the non-convex loss functions found in deep learning. We present and discuss the results of an extensive experimental study which includes the effect of batch normalization and network's architecture, the limited memory parameter, the batch size and the type of sampling strategy. we show that stochastic quasi-Newton optimizers are efficient and able to outperform in some instances the well-known first-order Adam optimizer run with the optimal combination of its numerous hyperparameters.

cs.LG↗

arXiv2020

Parallel Newton-Chebyshev Polynomial Preconditioners for the Conjugate Gradient method

In this note we exploit polynomial preconditioners for the Conjugate Gradient method to solve large symmetric positive definite linear systems in a parallel environment. We put in connection a specialized Newton method to solve the matrix equation X^{-1} = A and the Chebyshev polynomials for preconditioning. We propose a simple modification of one parameter which avoids clustering of extremal eigenvalues in order to speed-up convergence. We provide results on very large matrices (up to 8 billion unknowns) in a parallel environment showing the efficiency of the proposed class of preconditioners.

math.NA↗

arXiv2020

Compact Quasi-Newton preconditioners for SPD linear systems

In this paper preconditioners for the Conjugate Gradient method are studied to solve the Newton system with symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of SR1 and BFGS low-rank updates. We develop conditions under which the SR1 update maintains the preconditioner SPD. Spectral analysis of the SR1 preconditioned Jacobians shows an improved eigenvalue distribution as the Newton iteration proceeds. A compact matrix formulation of the preconditioner update is developed which reduces the cost of its application and is more suitable for parallel implementation. Some notes on the implementation of the corresponding Inexact Newton method are given and numerical results on a number of model problems illustrate the efficiency of the proposed preconditioners.

math.NA↗

arXiv2013

Efficiently preconditioned Inexact Newton methods for large symmetric eigenvalue problems

In this paper we propose an efficiently preconditioned Newton method for the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners based on the BFGS update formula is proposed, for the Preconditioned Conjugate Gradient solution of the linearized Newton system to solve $A \mathbf{u} = q(\mathbf{u}) \mathbf{u}$, $q(\mathbf{u})$ being the Rayleigh Quotient. We give theoretical evidence that the sequence of preconditioned Jacobians remains close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to one million unknowns account for the efficiency of the proposed algorithm which reveals competitive with the Jacobi-Davidson method on all the test problems.

math.NA↗