Marcos Raydan

We adapt an optimization technique, originally developed for quadratic matrix model updating problems, to construct optimal stabilizers for linearized partial differential equations (PDEs). The proposed approach is flexible, allowing the optimization of mass, damping, and stiffness operators either individually or simultaneously. It is also straightforward to implement within standard numerical frameworks. We analyze the numerical and practical properties of the method, highlighting its robustness even with low-order discretizations and its capacity to produce interpretable stabilization operators. The effectiveness and versatility of the approach are demonstrated through numerical experiments on several representative linearized PDEs, including the Klein–Gordon, Boussinesq, Benjamin–Bona–Mahony, and Korteweg–de Vries equations. Our results indicate that the method provides a unified, efficient, and modular tool for PDE stabilization.

Several strategies are described and analyzed to speed-up gradient-type methods when applied to the minimization of strictly convex quadratics and strictly convex functions. The proposed techniques focus on relaxing the traditional optimal step length associated with gradient methods, including the steepest descent (SD) and the minimal residual (MR) methods. Such a relaxation avoids the well-known negative zigzag effect and allows the iterates to move in the entire space which in turn implies that every so often the search direction approaches some eigenvector of the underlying Hessian matrix. The proposed speedups then rely on taking advantage of the properties of the Lanczos method once a search direction that approaches an eigenvector has been identified in order to accelerate the convergence towards the global minimizer. After analyzing the proposed strategies, we illustrate them on the global minimization of strictly convex functions.

Several strategies are described and analyzed to speed-up gradient-type methods when applied to the minimization of strictly convex quadratics and strictly convex functions. The proposed techniques focus on relaxing the traditional optimal step length associated with gradient methods, including the steepest descent (SD) and the minimal residual (MR) methods. Such a relaxation avoids the well-known negative zigzag effect and allows the iterates to move in the entire space which in turn implies that every so often the search direction approaches some eigenvector of the underlying Hessian matrix. The proposed speedups then rely on taking advantage of the properties of the Lanczos method once a search direction that approaches an eigenvector has been identified in order to accelerate the convergence towards the global minimizer. After analyzing the proposed strategies, we illustrate them on the global minimization of strictly convex functions.

High-dimensional omics data often contain more variables than observations, which negatively impacts the performance of classical data analysis methods. Dimensionality reduction is typically addressed through variable selection strategies that incorporate a penalty term into the model. While effective for selecting task-specific variables, this approach may not be optimal when the goal is to preserve the dataset structure and the overall biological information for multiple downstream analyses. In such cases, a priori unsupervised variable selection is preferable. In this study, we evaluate several unsupervised variable selection approaches to derive a representative subset of the original dataset. Building on the performance assessment results, we introduce TRIM-IT, a novel tool for unsupervised variable selection, clustering, survival analysis, and differential gene expression analysis. Applied to glioblastoma (GBM) data, TRIM-IT identified three clusters that correlate with tumor histology, exhibit distinct survival curves, and display unique molecular profiles with genes potentially serving as biomarkers. The tool is available for reproduction and adaptation to other studies.

Several manifold optimization schemes are presented and analyzed for solving a specialized inverse structured symmetric matrix problem with prescribed spectrum. Some entries in the desired matrix are assigned in advance and cannot be altered. The rest of the entries are free, some of them preferably away from zero. The reconstructed matrix must satisfy these requirements and its eigenvalues must be the given ones. This inverse eigenvalue problem is related to the problem of determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern. Our optimization schemes are based on considering the eigenvector matrix as the only unknown and iteratively moving on the manifold of orthogonal matrices, forcing the additional structural requirements through a change of variables and a convenient differentiable objective function in the space of square matrices. We propose Riemannian gradient-type methods combined with two different well-known retractions, and with two well-known constrained optimization strategies: penalization and augmented Lagrangian. We also present a block alternating technique that takes advantage of a proper separation of variables. Convergence properties of the penalty alternating approach are established. Finally, we present initial numerical results to demonstrate the effectiveness of our proposals.

Several manifold optimization schemes are presented and analyzed for solving a specialized inverse structured symmetric matrix problem with prescribed spectrum. Some entries in the desired matrix are assigned in advance and cannot be altered. The rest of the entries are free, some of them preferably away from zero. The reconstructed matrix must satisfy these requirements and its eigenvalues must be the given ones. This inverse eigenvalue problem is related to the problem of determining the graph, with weights on the undirected edges, of the matrix associated with its sparse pattern.Our optimization schemes are based on considering the eigenvector matrix as the only unknown and iteratively moving on the manifold of orthogonal matrices, forcing the additional structural requirements through a change of variables and a convenient differentiable objective function in the space of square matrices. We propose Riemannian gradient-type methods combined with two different well-known retractions, and with two well-known constrained optimization strategies: penalization and augmented Lagrangian. We also present a block alternating technique that takes advantage of a proper separation of variables. Convergence properties of the penalty alternating approach are established. Finally, we present initial numerical results to demonstrate the effectiveness of our proposals.