arXiv subjects

Ken Trotti

Publications and source records attributed to Ken Trotti.

8 recordsLinked to original sources

arXiv2026

A Non-Monotone Preconditioned Trust-Region Method for Neural Network Training

Training deep neural networks at scale can benefit from domain decomposition, where the network is split into subdomains trained in parallel and coupled by a global trust-region mechanism. Building on the Additively Preconditioned Trust-Region Strategy (APTS), we propose a non-monotone variant with a nonlinear additive Schwarz preconditioner that combines parallel subdomain corrections with global coarse-space directions. A windowed acceptance criterion allows controlled objective increases, avoiding needless rejection of effective coarse steps. The resulting non-monotone APTS (NAPTS) preserves accuracy while reducing CPU time by 30\% and cutting rejected steps to one third of those in APTS.

math.OC↗

arXiv2026

Protect the Brain When Treating the Heart: Feasibility of 2.5D U-Net for Real-Time Gaseous Microemboli Detection

Gaseous microemboli (GME) represent a common complication of cardiac structural interventions across both surgical and transcatheter approaches. Intraoperative transesophageal echocardiography (TEE) represents a convenient methodology to monitor and visualize the presence of circulating GME. However, their detection and quantification are far from trivial due to operator-dependent view, high velocity, and objects with similar structure in the background. Here, we propose a feasibility study based on a 2.5D U-Net architecture to detect GME in space-time connected data. We applied and tested such an architecture on a pilot dataset of eight TEE recordings ($60$ fps, $600\times 800$ pixels) from eight different patients undergoing cardiac surgery, resulting in improved detection of moving GMEs against the background with respect to classical spot detection algorithms and 2D U-Net, yet retaining real-time execution speed with respect to more complex deep-learning architectures. Under leave-one-patient-out cross-validation, the selected model achieved strong detection performance under a three-pixel radius-tolerant grace-zone evaluation, with a precision of 92.55\% and recall of 80.54\%, corresponding to radius-tolerant Intersection over Union (IoU) and Dice coefficients of 73.95\% and 84.13\%, respectively. Complementarily, strict pixel-based segmentation metrics were also computed, yielding an IoU of 41.74\% and a Dice coefficient of 57.98\%. The selected model achieved an average inference time of $0.12 s$ per batch on the tested hardware. To assess specificity on unseen data, we additionally evaluated the model on an external GME-negative TEE dataset, where it produced predominantly empty or near-empty masks, indicating a low rate of spurious detections. These results support the technical feasibility of real-time GME segmentation.

cs.LG↗

arXiv2025

An Additively Preconditioned Trust Region Strategy for Machine Learning

Modern machine learning, especially the training of deep neural networks, depends on solving large-scale, highly nonconvex optimization problems, whose objective function exhibit a rough landscape. Motivated by the success of parallel preconditioners in the context of Krylov methods for large scale linear systems, we introduce a novel nonlinearly preconditioned Trust-Region method that makes use of an additive Schwarz correction at each minimization step, thereby accelerating convergence. More precisely, we propose a variant of the Additively Preconditioned Trust-Region Strategy (APTS), which combines a right-preconditioned additive Schwarz framework with a classical Trust-Region algorithm. By decomposing the parameter space into sub-domains, APTS solves local non-linear sub-problems in parallel and assembles their corrections additively. The resulting method not only shows fast convergence; due to the underlying Trust-Region strategy, it furthermore largely obviates the need for hyperparameter tuning.

math.NA↗

arXiv2025

Data-Parallel Neural Network Training via Nonlinearly Preconditioned Trust-Region Method

Parallel training methods are increasingly relevant in machine learning (ML) due to the continuing growth in model and dataset sizes. We propose a variant of the Additively Preconditioned Trust-Region Strategy (APTS) for training deep neural networks (DNNs). The proposed APTS method utilizes a data-parallel approach to construct a nonlinear preconditioner employed in the nonlinear optimization strategy. In contrast to the common employment of Stochastic Gradient Descent (SGD) and Adaptive Moment Estimation (Adam), which are both variants of gradient descent (GD) algorithms, the APTS method implicitly adjusts the step sizes in each iteration, thereby removing the need for costly hyperparameter tuning. We demonstrate the performance of the proposed APTS variant using the MNIST and CIFAR-10 datasets. The results obtained indicate that the APTS variant proposed here achieves comparable validation accuracy to SGD and Adam, all while allowing for parallel training and obviating the need for expensive hyperparameter tuning.

cs.LG↗

arXiv2024

A note on the convergence of multigrid methods for the Riesz-space equation and an application to image deblurring

In the past decades, a remarkable amount of research has been carried out regarding fast solvers for large linear systems resulting from various discretizations of fractional differential equations (FDEs). In the current work, we focus on multigrid methods for a Riesz-space FDE whose theoretical convergence analysis of such multigrids is currently limited to the two-grid method. Here we provide a detailed theoretical convergence study in the case of V-cycle and W-cycle. Moreover, we discuss its use combined with a band approximation and we compare the result with both $\tau$ and circulant preconditionings. The numerical tests include 2D problems as well as the extension to the case of a Riesz-FDE with variable coefficients. Finally, we apply the best-performing method to an image deblurring problem with Tikhonov regularization.

math.NA↗

arXiv2023

Parallel Trust-Region Approaches in Neural Network Training: Beyond Traditional Methods

We propose to train neural networks (NNs) using a novel variant of the ``Additively Preconditioned Trust-region Strategy'' (APTS). The proposed method is based on a parallelizable additive domain decomposition approach applied to the neural network's parameters. Built upon the TR framework, the APTS method ensures global convergence towards a minimizer. Moreover, it eliminates the need for computationally expensive hyper-parameter tuning, as the TR algorithm automatically determines the step size in each iteration. We demonstrate the capabilities, strengths, and limitations of the proposed APTS training method by performing a series of numerical experiments. The presented numerical study includes a comparison with widely used training methods such as SGD, Adam, LBFGS, and the standard TR method.

math.NA↗

arXiv2023

A domain splitting strategy for solving PDEs

In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the $d$-dimensional advection-diffusion equation, our proposal is a two-level algorithm that merges the solutions obtained from the discretization of the equation over highly anisotropic submeshes to compute an initial approximation of the fine solution. The algorithm then iteratively refines the initial guess by leveraging the structure of the residual. Performing costly calculations on anisotropic submeshes enable us to reduce the dimensionality of the problem by one, and the merging process, which involves the computation of solutions over disjoint domains, allows for parallel implementation.

math.NA↗

arXiv2022

Multigrid for two-sided fractional differential equations discretized by finite volume elements on graded meshes

It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As consequence of this, and due to the conservative nature of the problem, we adopt a finite volume elements discretization approach over a generic non-uniform mesh. We focus on grids mapped by a smooth function which consist in a combination of a graded mesh near the singularity and a uniform mesh where the solution is smooth. Such a choice gives rise to Toeplitz-like discretization matrices and thus allows a low computational cost of the matrix-vector product and a detailed spectral analysis. The obtained spectral information is used to develop an ad-hoc parameter free multigrid preconditioner for GMRES, which is numerically shown to yield good convergence results in presence of graded meshes mapped by power functions that accumulate points near the singularity. The approximation order of the considered graded meshes is numerically compared with the one of a certain composite mesh given in literature that still leads to Toeplitz-like linear systems and is then still well-suited for our multigrid method. Several numerical tests confirm that power graded meshes result in lower approximation errors than composite ones and that our solver has a wide range of applicability.

math.NA↗