arXiv subjects

Beatrice Riviere

Publications and source records attributed to Beatrice Riviere.

At least 19 recordsLinked to original sources

arXiv2026

Hybrid coupling with numerics-informed neural networks and the overlapping Schwarz alternating method

We develop a hybrid modeling framework for coupling pre-trained numerics-informed neural networks (NINNs) with classical full order models (FOMs) using the overlapping Schwarz alternating method. We consider the two-dimensional advection-diffusion equation in the advection-dominated, Peclet-number 10^6 regime. We first demonstrate that, unlike the corresponding physics-informed neural network (PINN), a monolithic NINN can be accurately trained on our model problem without domain decomposition. We then employ overlapping multiplicative Schwarz as a deployment mechanism for coupling a pre-trained, subdomain-local NINN with a neighboring FOM, with the NINN weights held fixed throughout the Schwarz iteration. We consider two training approaches for the subdomain-local NINNs: a top-down approach, in which boundary data are obtained from a coupled Schwarz solve on the full domain with a FOM on each subdomain (FOM-FOM Schwarz), and a bottom-up approach, in which boundary traces are generated synthetically on the NINN subdomain without requiring any full-domain solves. The resulting hybrid NINN-FOM solutions agree closely with the corresponding FOM-FOM Schwarz solutions, with the top-down and bottom-up training approaches yielding comparable accuracy.

cs.LG↗

arXiv2026

Proximal Discontinuous Galerkin Methods for Variational Inequalities

We introduce a family of proximal discontinuous Galerkin methods for variational inequalities, focusing on the obstacle problem as a didactic example. Each member of this family is born from applying a different well-known nonconforming finite element discretization to the Bregman proximal point method. We explicitly treat four examples: the symmetric interior penalty discontinuous Galerkin, the enriched Galerkin, the hybridizable interior penalty and the hybrid high-order methods. We formulate a unified analysis framework for this family of methods and prove the existence and uniqueness of solutions, energy dissipation, and error estimates for both the primal and dual variables. Remarkably, the proximal hybrid high-order method with piecewise constant cell unknowns and piecewise affine facet unknowns leads to the first higher-order convergence result for any proximal Galerkin method.

math.NA↗

arXiv2026

Numerical Analysis of a Coupled 3D-1D Transport Problem

A finite element solution coupled with an interior penalty discontinuous Galerkin solution are defined for the approximation of the coupled 3D-1D solute transport problem. Under sufficient regularity for the weak solutions, optimal error bounds are derived for the 3D concentration and 1D concentration, that are optimal with respect to the time step size and the mesh sizes. Numerical results verify the theoretical results.

math.NA↗

arXiv2025

Analysis of a Discontinuous Galerkin Method for Diffusion Problems on Intersecting Domains

The interior penalty discontinuous Galerkin method is applied to solve elliptic equations on either networks of segments or networks of planar surfaces, with arbitrary but fixed number of bifurcations. Stability is obtained by proving a discrete Poincaré's inequality on the hypergraphs. Convergence of the scheme is proved for $H^r$ regularity solution with $1 < r \leq 2$. In the low regularity case ($r \leq 3/2$), a weak consistency result is obtained via generalized lifting operators for Sobolev spaces defined on hypergraphs. Numerical experiments confirm the theoretical results.

math.NA↗

arXiv2025

Adaptive Collocation Point Strategies For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method

Physics-informed neural networks (PINNs) have gained significant attention for solving forward and inverse problems related to partial differential equations (PDEs). While advancements in loss functions and network architectures have improved PINN accuracy, the impact of collocation point sampling on their performance remains underexplored. Fixed sampling methods, such as uniform random sampling and equispaced grids, can fail to capture critical regions with high solution gradients, limiting their effectiveness for complex PDEs. Adaptive methods, inspired by adaptive mesh refinement from traditional numerical methods, address this by dynamically updating collocation points during training but may overlook residual dynamics between updates, potentially losing valuable information. To overcome this limitation, we propose two adaptive collocation point selection strategies utilizing the QR Discrete Empirical Interpolation Method (QR-DEIM), a reduced-order modeling technique for efficiently approximating nonlinear functions. Our results on benchmark PDEs demonstrate that our QR-DEIM-based approaches improve PINN accuracy compared to existing methods, offering a promising direction for adaptive collocation point strategies.

cs.LG↗

arXiv2025

Pre- and Post-Treatment Glioma Segmentation with the Medical Imaging Segmentation Toolkit

Medical image segmentation continues to advance rapidly, yet rigorous comparison between methods remains challenging due to a lack of standardized and customizable tooling. In this work, we present the current state of the Medical Imaging Segmentation Toolkit (MIST), with a particular focus on its flexible and modular postprocessing framework designed for the BraTS 2025 pre- and post-treatment glioma segmentation challenge. Since its debut in the 2024 BraTS adult glioma post-treatment segmentation challenge, MIST's postprocessing module has been significantly extended to support a wide range of transforms, including removal or replacement of small objects, extraction of the largest connected components, and morphological operations such as hole filling and closing. These transforms can be composed into user-defined strategies, enabling fine-grained control over the final segmentation output. We evaluate three such strategies - ranging from simple small-object removal to more complex, class-specific pipelines - and rank their performance using the BraTS ranking protocol. Our results highlight how MIST facilitates rapid experimentation and targeted refinement, ultimately producing high-quality segmentations for the BraTS 2025 challenge. MIST remains open source and extensible, supporting reproducible and scalable research in medical image segmentation.

cs.CV↗

arXiv2025

A Hybridizable Discontinuous Galerkin Method for the Miscible Displacement Problem Under Minimal Regularity

A numerical method based on the hybridizable discontinuous Galerkin method in space and backward Euler in time is formulated and analyzed for solving the miscible displacement problem. Under low regularity assumptions, convergence is established by proving that, up to a subsequence, the discrete pressure, velocity and concentration converge to a weak solution as the mesh size and time step tend to zero. The analysis is based on several key features: an H(div) reconstruction of the velocity, the skew-symmetrization of the concentration equation, the introduction of an auxiliary variable and the definition of a new numerical flux. Numerical examples demonstrate optimal rates of convergence for smooth solutions, and convergence for problems of low regularity.

math.NA↗

arXiv2025

A Priori Generalizability Estimate for a CNN

We formulate truncated singular value decompositions of entire convolutional neural networks. We demonstrate the computed left and right singular vectors are useful in identifying which images the convolutional neural network is likely to perform poorly on. To create this diagnostic tool, we define two metrics: the Right Projection Ratio and the Left Projection Ratio. The Right (Left) Projection Ratio evaluates the fidelity of the projection of an image (label) onto the computed right (left) singular vectors. We observe that both ratios are able to identify the presence of class imbalance for an image classification problem. Additionally, the Right Projection Ratio, which only requires unlabeled data, is found to be correlated to the model's performance when applied to image segmentation. This suggests the Right Projection Ratio could be a useful metric to estimate how likely the model is to perform well on a sample.

cs.CV↗

arXiv2025

Learning Discontinuous Galerkin Solutions to Elliptic Problems via Small Linear Convolutional Neural Networks

In recent years, there has been an increasing interest in using deep learning and neural networks to tackle scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods, such as physics-informed neural networks, depend on automatic differentiation and the sampling of collocation points, which can result in a lack of interpretability and lower accuracy compared to traditional numerical methods. To address this issue, we propose two approaches for learning discontinuous Galerkin solutions to PDEs using small linear convolutional neural networks. Our first approach is supervised and depends on labeled data, while our second approach is unsupervised and does not rely on any training data. In both cases, our methods use substantially fewer parameters than similar numerics-based neural networks while also demonstrating comparable accuracy to the true and DG solutions for elliptic problems.

cs.LG↗

arXiv2024

MIST: A Simple and Scalable End-To-End 3D Medical Imaging Segmentation Framework

Medical imaging segmentation is a highly active area of research, with deep learning-based methods achieving state-of-the-art results in several benchmarks. However, the lack of standardized tools for training, testing, and evaluating new methods makes the comparison of methods difficult. To address this, we introduce the Medical Imaging Segmentation Toolkit (MIST), a simple, modular, and end-to-end medical imaging segmentation framework designed to facilitate consistent training, testing, and evaluation of deep learning-based medical imaging segmentation methods. MIST standardizes data analysis, preprocessing, and evaluation pipelines, accommodating multiple architectures and loss functions. This standardization ensures reproducible and fair comparisons across different methods. We detail MIST's data format requirements, pipelines, and auxiliary features and demonstrate its efficacy using the BraTS Adult Glioma Post-Treatment Challenge dataset. Our results highlight MIST's ability to produce accurate segmentation masks and its scalability across multiple GPUs, showcasing its potential as a powerful tool for future medical imaging research and development.

eess.IV↗

arXiv2024

1D Thermoembolization Model Using CT Imaging Data for Porcine Liver

Objective: Innovative therapies such as thermoembolization are expected to play an important role in improvising care for patients with diseases such as hepatocellular carcinoma. Thermoembolization is a minimally invasive strategy that combines thermal ablation and embolization in a single procedure. This approach exploits an exothermic chemical reaction that occurs when an acid chloride is delivered via an endovascular route. However, comprehension of the complexities of the biophysics of thermoembolization is challenging. Mathematical models can aid in understanding such complex processes and assisting clinicians in making informed decisions. In this study, we used a Hagen Poiseuille 1D blood flow model to predict the mass transport and possible embolization locations in a porcine hepatic artery. Method: The 1D flow model was used on in vivo embolization imaging data of three pigs. The hydrolysis time constant of acid chloride chemical reaction was optimized for each pig, and LOOCV method was used to test the model's predictive ability. Conclusion: This basic model provided a balanced accuracy rate of 66.8% for identifying the possible locations of embolization in the hepatic artery. Use of the model provides an initial understanding of the vascular transport phenomena that are predicted to occur as a result of thermoembolization.

physics.med-ph↗

arXiv2024

Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods like PINNs rely on auto differentiation and sampling collocation points, leading to a lack of interpretability and lower accuracy than traditional numerical methods. As a result, we propose a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small linear convolutional neural networks. Our proposed approach uses substantially fewer parameters than similar finite difference-based approaches while also demonstrating comparable accuracy to the true solution for several selected elliptic and parabolic problems compared to the finite difference method.

cs.LG↗

arXiv2024

A simple and efficient convex optimization based bound-preserving high order accurate limiter for Cahn-Hilliard-Navier-Stokes system

For time-dependent PDEs, the numerical schemes can be rendered bound-preserving without losing conservation and accuracy, by a post processing procedure of solving a constrained minimization in each time step. Such a constrained optimization can be formulated as a nonsmooth convex minimization, which can be efficiently solved by first order optimization methods, if using the optimal algorithm parameters. By analyzing the asymptotic linear convergence rate of the generalized Douglas-Rachford splitting method, optimal algorithm parameters can be approximately expressed as a simple function of the number of out-of-bounds cells. We demonstrate the efficiency of this simple choice of algorithm parameters by applying such a limiter to cell averages of a discontinuous Galerkin scheme solving phase field equations for 3D demanding problems. Numerical tests on a sophisticated 3D Cahn-Hilliard-Navier-Stokes system indicate that the limiter is high order accurate, very efficient, and well-suited for large-scale simulations. For each time step, it takes at most $20$ iterations for the Douglas-Rachford splitting to enforce bounds and conservation up to the round-off error, for which the computational cost is at most $80N$ with $N$ being the total number of cells.

math.NA↗

arXiv2024

A Generalized Surface Loss for Reducing the Hausdorff Distance in Medical Imaging Segmentation

Within medical imaging segmentation, the Dice coefficient and Hausdorff-based metrics are standard measures of success for deep learning models. However, modern loss functions for medical image segmentation often only consider the Dice coefficient or similar region-based metrics during training. As a result, segmentation architectures trained over such loss functions run the risk of achieving high accuracy for the Dice coefficient but low accuracy for Hausdorff-based metrics. Low accuracy on Hausdorff-based metrics can be problematic for applications such as tumor segmentation, where such benchmarks are crucial. For example, high Dice scores accompanied by significant Hausdorff errors could indicate that the predictions fail to detect small tumors. We propose the Generalized Surface Loss function, a novel loss function to minimize Hausdorff-based metrics with more desirable numerical properties than current methods and with weighting terms for class imbalance. Our loss function outperforms other losses when tested on the LiTS and BraTS datasets using the state-of-the-art nnUNet architecture. These results suggest we can improve medical imaging segmentation accuracy with our novel loss function.

eess.IV↗

arXiv2023

Discontinuous Galerkin methods for 3D-1D systems

We propose and analyze discontinuous Galerkin (dG) approximations to 3D-1D coupled systems which model diffusion in a 3D domain containing a small inclusion reduced to its 1D centerline. Convergence to weak solutions of a steady state problem is established via deriving a posteriori error estimates and bounds on residuals defined with suitable lift operators. For the time dependent problem, a backward Euler dG formulation is also presented and analysed. Further, we propose a dG method for networks embedded in 3D domains, which is, up to jump terms, locally mass conservative on bifurcation points. Numerical examples in idealized geometries portray our theoretical findings, and simulations in realistic 1D networks show the robustness of our method.

math.NA↗

arXiv2023

FMG-Net and W-Net: Multigrid Inspired Deep Learning Architectures For Medical Imaging Segmentation

Accurate medical imaging segmentation is critical for precise and effective medical interventions. However, despite the success of convolutional neural networks (CNNs) in medical image segmentation, they still face challenges in handling fine-scale features and variations in image scales. These challenges are particularly evident in complex and challenging segmentation tasks, such as the BraTS multi-label brain tumor segmentation challenge. In this task, accurately segmenting the various tumor sub-components, which vary significantly in size and shape, remains a significant challenge, with even state-of-the-art methods producing substantial errors. Therefore, we propose two architectures, FMG-Net and W-Net, that incorporate the principles of geometric multigrid methods for solving linear systems of equations into CNNs to address these challenges. Our experiments on the BraTS 2020 dataset demonstrate that both FMG-Net and W-Net outperform the widely used U-Net architecture regarding tumor subcomponent segmentation accuracy and training efficiency. These findings highlight the potential of incorporating the principles of multigrid methods into CNNs to improve the accuracy and efficiency of medical imaging segmentation.

eess.IV↗

arXiv2023

Numerical analysis of a hybridized discontinuous Galerkin method for the Cahn-Hilliard problem

The mixed form of the Cahn-Hilliard equations is discretized by the hybridizable discontinuous Galerkin method. For any chemical energy density, existence and uniqueness of the numerical solution is obtained. The scheme is proved to be unconditionally stable. Convergence of the method is obtained by deriving a priori error estimates that are valid for the Ginzburg-Lindau chemical energy density and for convex domains. The paper also contains discrete functional tools, namely discrete Agmon and Gagliardo-Nirenberg inequalities, which are proved to be valid in the hybridizable discontinuous Galerkin spaces.

math.NA↗

arXiv2022

Convergence of a Decoupled Splitting Scheme for the Cahn-Hilliard-Navier-Stokes System

This paper is devoted to the analysis of an energy-stable discontinuous Galerkin algorithm for solving the Cahn-Hilliard-Navier-Stokes equations within a decoupled splitting framework. We show that the proposed scheme is uniquely solvable and mass conservative. The energy dissipation and the $L^\infty$ stability of the order parameter are obtained under a CFL condition. Optimal a priori error estimates in the broken gradient norm and in the $L^2$ norm are derived. The stability proofs and error analysis are based on induction arguments and do not require any regularization of the potential function.

math.NA↗