We study the numerical approximation of time-dependent, possibly degenerate, second-order Hamilton-Jacobi-Bellman equations in bounded domains with nonhomogeneous Dirichlet boundary conditions. It is well known that convergence towards the exact solution of the equation, considered here in the viscosity sense, holds if the scheme is monotone, consistent, and stable. While standard finite difference schemes are, in general, not monotone, the so-called semi-Lagrangian schemes are monotone by construction. On the other hand, these schemes make use of a wide stencil and, when the equation is set in a bounded domain, this typically causes an overstepping of the boundary and hence the loss of consistency. We propose here a semi-Lagrangian scheme defined on an unstructured mesh, with a suitable treatment at grid points near the boundary to preserve consistency, and show its convergence for problems where the viscosity solution can even be discontinuous. We illustrate the numerical convergence in several tests, including degenerate and first-order equations.

Variational methods based on optimization strategies are proposed to numerically solve a large family of nonlinear partial differential equations. They are all particular instances of gradient flows with general costs, including the p-Laplace equation and flux-limited equations such as the relativistic heat equation. This is achieved by computing explicit formulas for proximal operators with general costs amenable to efficient numerical approximation. We showcase our numerical approach via validation of the results by recovering the qualitative behavior of particular known cases of this large family of steepest descent evolutions.

We generalize the Brezzi-Rappaz-Raviart approximation theorem, which allows to obtain existence and a priori error estimates for approximations of solutions to some nonlinear partial differential equations. Our contribution lies in the fact that we typically allow for nonlinearities having merely Lipschitz regularity, while previous results required some form of differentiability. This is achieved by making use of the theory of metrically regular mappings, developed in the context of variational analysis. We apply this generalization to derive some quasi-optimal error estimates for finite element approximations to solutions of viscous Hamilton-Jacobi equations and second order mean field games systems.

This paper first introduces a method to approximate the value function of high-dimensional optimal control by neural networks. Based on the established relationship between Pontryagin's maximum principle (PMP) and the value function of the optimal control problem, which is characterized as being the unique solution to an associated Hamilton-Jacobi-Bellman (HJB) equation, we propose an approach that begins by using neural networks to provide a first rough estimate of the value function, which serves as initialization for solving the two point boundary value problem in the PMP and, as a result, generates reliable data. To train the neural network we define a loss function that takes into account this dataset and also penalizes deviations from the HJB equation. In the second part, we address the computation of equilibria in first-order Mean Field Game (MFG) problems by integrating our method with the fictitious play algorithm. These equilibria are characterized by a coupled system of a first-order HJB equation and a continuity equation. To approximate the solution to the continuity equation, we introduce a second neural network that learns the flow map transporting the initial distribution of agents. This network is trained on data generated by solving the underlying ODEs for a batch of initial conditions sampled from the initial distribution of agents. By combining this flow approximation, the previously described method for approximating the value function, and the fictitious play algorithm, we obtain an effective method to tackle high-dimensional deterministic MFGs.

The classical perspective of a function is a construction which transforms a convex function into one that is jointly convex with respect to an auxiliary scaling variable. Motivated by applications in several areas of applied analysis, we investigate an extension of this construct in which the scaling variable is replaced by a nonlinear term. Our construction is placed in the general context of locally convex spaces and it generates a lower semicontinuous convex function under broad assumptions on the underlying functions. Various convex-analytical properties are established and closed-form expressions are derived. Several applications are presented.

This work introduces a new general approach for the numerical analysis of stable equilibria to second order mean field games systems in cases where the uniqueness of solutions may fail. For the sake of simplicity, we focus on a simple stationary case. We propose an abstract framework to study these solutions by reformulating the mean field game system as an abstract equation in a Banach space. In this context, stable equilibria turn out to be regular solutions to this equation, meaning that the linearized system is well-posed. We provide three applications of this property: we study the sensitivity analysis of stable solutions, establish error estimates for their finite element approximations, and prove the local converge of Newton's method in infinite dimensions.

The large gap between the generalization level of state-of-the-art machine learning and human learning systems calls for the development of artificial intelligence (AI) models that are truly inspired by human cognition. In tasks related to image analysis, searching for pixel-level regularities has reached a power of information extraction still far from what humans capture with image-based observations. This leads to poor generalization when even small shifts occur at the level of the observations. We explore a perspective on this problem that is directed to learning the generative process with causality-related foundations, using models capable of combining symbolic manipulation, probabilistic reasoning, and pattern recognition abilities. We briefly review and explore connections of research from machine learning, cognitive science, and related fields of human behavior to support our perspective for the direction to more robust and human-like artificial learning systems.

We study the numerical approximation of a time-dependent variational mean field game system with local couplings and either periodic or Neumann boundary conditions. Following a variational approach, we employ a finite difference discretization and solve the resulting finite-dimensional optimization problem using the Chambolle–Pock primal–dual algorithm. As this involves computing proximal operators and solving ill-conditioned linear systems at each iteration, we propose a general class of parallel-in-time preconditioners based on diagonalization techniques using discrete Fourier transforms. These enable efficient, scalable iterative solvers with robustness across a wide range of viscosities. We further develop fast solvers for the resulting ill-conditioned systems arising at each time step, using exact recursive schemes for structured grids while allowing for other geometries. Numerical experiments confirm the improved performance and parallel scalability of our approach.

Significant medical image visualization and annotation tools, tailored for clinical users, play a crucial role in disease diagnosis and treatment. Developing algorithms for annotation assistance, particularly machine learning (ML)-based ones, can be intricate, emphasizing the need for a user-friendly graphical interface for developers. Many software tools are available to meet these requirements, but there is still room for improvement, making the research for new tools highly compelling. The envisioned tool focuses on navigating sequences of DICOM images from diverse modalities, including Magnetic Resonance Imaging (MRI), Computed Tomography (CT) scans, Ultrasound (US), and X-rays. Specific requirements involve implementing manual annotation features such as freehand drawing, copying, pasting, and modifying annotations. A scripting plugin interface is essential for running Artificial Intelligence (AI)-based models and adjusting results. Additionally, adaptable surveys complement graphical annotations with textual notes, enhancing information provision. The user evaluation results pinpointed areas for improvement, including incorporating some useful functionalities, as well as enhancements to the user interface for a more intuitive and convenient experience. Despite these suggestions, participants praised the application’s simplicity and consistency, highlighting its suitability for the proposed tasks. The ability to revisit annotations ensures flexibility and ease of use in this context.

In this work, we consider a first order mean field games system with non-local couplings. A Lagrange-Galerkin scheme for the continuity equation, coupled with a semi-Lagrangian scheme for the Hamilton-Jacobi-Bellman equation, is proposed to discretize the mean field games system. The convergence of solutions to the scheme towards a solution to the mean field game system is established in arbitrary space dimensions. The scheme is implemented to approximate two mean field games systems in dimension one and two.

In this paper we propose a high-order numerical scheme for time-dependent mean field games systems. The scheme, which is built by combining Lagrange-Galerkin and semi-Lagrangian techniques, is consistent and stable for large time steps compared with the space steps. We provide a convergence analysis for the exactly integrated Lagrange-Galerkin scheme applied to the Fokker-Planck equation, and we propose an implementable version with inexact integration. Finally, we validate the convergence rate of the proposed scheme through the numerical approximation of two mean field games systems.

In this paper, we provide a generalization of the forward-backward splitting algorithm for minimizing the sum of a proper convex lower semicontinuous function and a differentiable convex function whose gradient satisfies a locally Lipschitztype condition. We prove the convergence of our method and derive a linear convergence rate when the differentiable function is locally strongly convex. We recover classical results in the case when the gradient of the differentiable function is globally Lipschitz continuous and an already known linear convergence rate when the function is globally strongly convex. We apply the algorithm to approximate equilibria of variational mean field game systems with local couplings. Compared with some benchmark algorithms to solve these problems, our numerical tests show similar performances in terms of the number of iterations but an important gain in the required computational time.

We consider a deterministic mean field games problem in which a typical agent solves an optimal control problem where the dynamics is affine with respect to the control and the cost functional has a growth which is polynomial with respect to the state variable. In this framework, we construct a mean field game problem in discrete time and finite state space that approximates equilibria of the original game. Two numerical examples, solved with the fictitious play method, are presented.

A perspective function is a construction which combines a base function defined on a given space with a nonlinear scaling function defined on another space and which yields a lower semicontinuous convex function on the product space. Since perspective functions are typically nonsmooth, their use in first-order algorithms necessitates the computation of their proximity operator. This paper establishes closed-form expressions for the proximity operator of a perspective function defined on a Hilbert space in terms of a proximity operator involving its base function and one involving its scaling function.

Recent studies have shown that the microbiome can impact cancer development, progression, and response to therapies suggesting microbiome-based approaches for cancer characterization. As cancer-related signatures are complex and implicate many taxa, their discovery often requires Machine Learning approaches. This review discusses Machine Learning methods for cancer characterization from microbiome data. It focuses on the implications of choices undertaken during sample collection, feature selection and pre-processing. It also discusses ML model selection, guiding how to choose an ML model, and model validation. Finally, it enumerates current limitations and how these may be surpassed. Proposed methods, often based on Random Forests, show promising results, however insufficient for widespread clinical usage. Studies often report conflicting results mainly due to ML models with poor generalizability. We expect that evaluating models with expanded, hold-out datasets, removing technical artifacts, exploring representations of the microbiome other than taxonomical profiles, leveraging advances in deep learning, and developing ML models better adapted to the characteristics of microbiome data will improve the performance and generalizability of models and enable their usage in the clinic.

Bone marrow edema (BME) or bone marrow lesion is the term attributed to an observed signal change within the bone marrow in magnetic resonance imaging (MRI). BME can be originated from multiple mechanisms, with pain being the main symptom. The presence of BME is an unspecific but sensitive sign with a wide differential diagnosis, that may act as a guide that leads to a systematic and correct interpretation of the magnetic resonance examination. An automatic approach for BME detection and quantification aims to reduce the overload of clinicians, decreasing human error and accelerating the time to the correct diagnosis. In this work, the bone region on the MRI slice was split into several patches and a CNN-based model was trained to detect BME in each patch from the MRI slice. The learning model developed achieved an AUC of 0.853 ± 0.056, showing that the CNN-based model can be used to detect BME in the MRI and confirming the patch strategy implemented to deal with the small data size and allowing the neural network to learn the specific information related with the classification task by reducing the region of the image to be considered. A learning model that can help clinicians with BME identification will decrease the time and the error for the diagnosis, and represent the first step for a more objective assessment of the BME.

"Fundamentos de Ginecología y Obstetricia Vol. 5" es una fuente de información indispensable y una referencia esencial para profesionales y estudiantes en el campo de la atención a la salud de la mujer. En este quinto volumen, los autores profundizan en la exploración de enfermedades ginecológicas y obstétricas, proporcionando una visión integral de las últimas investigaciones y avances en la especialidad. Este volumen presenta un enfoque exhaustivo de los trastornos ginecológicos y obstétricos más desafiantes. Temas como la endometriosis avanzada, las complicaciones del embarazo, la infertilidad y los tumores ginecológicos son abordados con un detalle impresionante. Además, la obra destaca los avances más recientes en los campos de la tecnología reproductiva asistida y la oncología ginecológica.

Biomedical engineering has been targeted as a potential research candidate for machine learning applications, with the purpose of detecting or diagnosing pathologies. However, acquiring relevant, high-quality, and heterogeneous medical datasets is challenging due to privacy and security issues and the effort required to annotate the data. Generative models have recently gained a growing interest in the computer vision field due to their ability to increase dataset size by generating new high-quality samples from the initial set, which can be used as data augmentation of a training dataset. This study aimed to synthesize artificial lung images from corresponding positional and semantic annotations using two generative adversarial networks and databases of real computed tomography scans: the Pix2Pix approach that generates lung images from the lung segmentation maps; and the conditional generative adversarial network (cCGAN) approach that was implemented with additional semantic labels in the generation process. To evaluate the quality of the generated images, two quantitative measures were used: the domain-specific Frechet Inception Distance and Structural Similarity Index. Additionally, an expert assessment was performed to measure the capability to distinguish between real and generated images. The assessment performed shows the high quality of synthesized images, which was confirmed by the expert evaluation. This work represents an innovative application of GAN approaches for medical application taking into consideration the pathological findings in the CT images and the clinical evaluation to assess the realism of these features in the generated images.

Emerging evidence of the relationship between the microbiome composition and the development of numerous diseases, including cancer, has led to an increasing interest in the study of the human microbiome. Technological breakthroughs regarding DNA sequencing methods propelled microbiome studies with a large number of samples, which called for the necessity of more sophisticated data-analytical tools to analyze this complex relationship. The aim of this work was to develop a machine learning-based approach to distinguish the type of cancer based on the analysis of the tissue-specific microbial information, assessing the human microbiome as valuable predictive information for cancer identification. For this purpose, Random Forest algorithms were trained for the classification of five types of cancer-head and neck, esophageal, stomach, colon, and rectum cancers-with samples provided by The Cancer Microbiome Atlas database. One versus all and multi-class classification studies were conducted to evaluate the discriminative capability of the microbial data across increasing levels of cancer site specificity, with results showing a progressive rise in difficulty for accurate sample classification. Random Forest models achieved promising performances when predicting head and neck, stomach, and colon cancer cases, with the latter returning accuracy scores above 90% across the different studies conducted. However, there was also an increased difficulty when discriminating esophageal and rectum cancers, failing to differentiate with adequate results rectum from colon cancer cases, and esophageal from head and neck and stomach cancers. These results point to the fact that anatomically adjacent cancers can be more complex to identify due to microbial similarities. Despite the limitations, microbiome data analysis using machine learning may advance novel strategies to improve cancer detection and prevention, and decrease disease burden.

As a scientist, teacher, and author, Joseph J. Pear (1938–2022) made inventive and innovative contributions to the experimental analysis of behavior and applied behavior analysis. He collaborated on the invention of a computer-controlled video system that provided real-time tracking of an animal as it moved around its environment—the first system of its kind—and used this system to study the behavior of pigeons during schedules of reinforcement and autoshaping, Siamese fighting fish during Sidman avoidance tasks, and even lake sturgeon using darkness as a positive reinforcer. A pioneer in early online teaching, he developed a computer-aided version of Keller’s personalized system of instruction, a system known as CAPSI. Pear coauthored with Garry Martin a best-selling behavior modification textbook, and wrote his own books on topics such as the science of learning and psychological systems. Pear’s work exemplified the growing integration between basic and applied behavior analysis.