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Beatrice Riviere

Publications and source records attributed to Beatrice Riviere.

35 records · Page 2Linked to original sources
arXiv2022

PocketNet: A Smaller Neural Network for Medical Image Analysis

Medical imaging deep learning models are often large and complex, requiring specialized hardware to train and evaluate these models. To address such issues, we propose the PocketNet paradigm to reduce the size of deep learning models by throttling the growth of the number of channels in convolutional neural networks. We demonstrate that, for a range of segmentation and classification tasks, PocketNet architectures produce results comparable to that of conventional neural networks while reducing the number of parameters by multiple orders of magnitude, using up to 90% less GPU memory, and speeding up training times by up to 40%, thereby allowing such models to be trained and deployed in resource-constrained settings.

eess.IV↗
arXiv2022

Discontinuous Galerkin Approximations to Elliptic and Parabolic Problems with a Dirac Line Source

The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.

math.NA↗
arXiv2022

A Sequential Discontinuous Galerkin Method for Two-Phase Flow in Deformable Porous Media

We formulate a numerical method for solving the two-phase flow poroelasticity equations. The scheme employs the interior penalty discontinuous Galerkin method and a sequential time-stepping method. The unknowns are the phase pressures and the displacement. Existence of the solution is proved. Three-dimensional numerical results show the accuracy and robustness of the proposed method.

math.NA↗
arXiv2022

Existence and convergence of a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media

This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first order time extrapolation which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well-posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e. the algorithm converges with first order.

math.NA↗
arXiv2020

A pressure-correction and bound-preserving discretization of the phase-field method for variable density two-phase flows

In this paper, we present an efficient numerical algorithm for solving the time-dependent Cahn--Hilliard--Navier--Stokes equations that model the flow of two phases with different densities. The pressure-correction step in the projection method consists of a Poisson problem with a modified right-hand side. Spatial discretization is based on discontinuous Galerkin methods with piecewise linear or piecewise quadratic polynomials. Flux and slope limiting techniques successfully eliminate the bulk shift, overshoot and undershoot in the order parameter, which is shown to be bound-preserving. Several numerical results demonstrate that the proposed numerical algorithm is effective and robust for modeling two-component immiscible flows in porous structures and digital rocks.

math.NA↗
arXiv2020

A discontinuous Galerkin method for a diffuse-interface model of immiscible two-phase flows with soluble surfactant

A numerical method using discontinuous polynomial approximations is formulated for solving a phase-field model of two immiscible fluids with a soluble surfactant. The scheme recovers the Langmuir adsorption isotherms at equilibrium. Simulations of spinodal decomposition, flow through a cylinder and flow through a sequence of pore throats show the dynamics of the flow with and without surfactant. Finally the numerical method is used to simulate fluid flows in the pore space of Berea sandstone obtained by micro-CT imaging.

physics.comp-ph↗
arXiv2019

A reduced model for solute transport in compliant blood vessels with arbitrary axial velocity profile

We derive a reduced model of solute transport in blood based on the center manifold theory. The derivation is carried out on a convection diffusion equation with general axial and radial velocity profiles in a blood vessel of varying cross section. We couple the resulting one dimensional equation to a reduced model for blood flow in a compliant vessel. In the special case of a no--slip axial velocity profile, we study the dependence of the diffusion coefficient and corresponding numerical solutions on the shape of the profile.

physics.flu-dyn↗
arXiv2018

Numerical analysis of a discontinuous Galerkin method for Cahn-Hilliard-Navier-Stokes equations

In this paper, we derive a theoretical analysis of an interior penalty discontinuous Galerkin methods for solving the Cahn-Hilliard-Navier-Stokes model problem. We prove unconditional unique solvability of the discrete system, obtain unconditional discrete energy dissipation law, and derive stability bounds with a generalized chemical energy density. Convergence of the method is obtained by proving optimal a priori error estimates. Our analysis of the unique solvability is valid for both symmetric and non-symmetric versions of the discontinuous Galerkin formulation.

math.NA↗
arXiv2018

An interior penalty discontinuous Galerkin approach for 3D incompressible Navier--Stokes equation for permeability estimation of porous media

Permeability estimation of porous media from direct solving Navier--Stokes equation has a wide spectrum of applications in petroleum industry. In this paper, we utilize a pressure-correction projection algorithm in conjunction with the interior penalty discontinuous Galerkin scheme for space discretization to build an incompressible Navier--Stokes simulator and to use this simulator to calculate permeability of real rock sample. The proposed method is accurate, numerically robust, and exhibits the potential for tackling realistic problems.

physics.comp-ph↗
arXiv2017

An energy-based equilibrium contact angle boundary condition on jagged surfaces for phase-field methods

We consider an energy-based boundary condition to impose an equilibrium wetting angle for the Cahn-Hilliard-Navier-Stokes phase-field model on voxel-set-type computational domains. These domains typically stem from the micro-CT imaging of porous rock and approximate a (on μm scale) smooth domain with a certain resolution. Planar surfaces that are perpendicular to the main axes are naturally approximated by a layer of voxels. However, planar surfaces in any other directions and curved surfaces yield a jagged/rough surface approximation by voxels. For the standard Cahn-Hilliard formulation, where the contact angle between the diffuse interface and the domain boundary (fluid-solid interface/wall) is 90 degrees, jagged surfaces have no impact on the contact angle. However, a prescribed contact angle smaller or larger than 90 degrees on jagged voxel surfaces is amplified in either direction. As a remedy, we propose the introduction of surface energy correction factors for each fluid-solid voxel face that counterbalance the difference of the voxel-set surface area with the underlying smooth one. The discretization of the model equations is performed with the discontinuous Galerkin method, however, the presented semi-analytical approach of correcting the surface energy is equally applicable to other direct numerical methods such as finite elements, finite volumes, or finite differences, since the correction factors appear in the strong formulation of the model.

physics.flu-dyn↗
arXiv2017

A priori error estimates of Adams-Bashforth discontinuous Galerkin methods for scalar nonlinear conservation laws

In this paper we show theoretical convergence of a second-order Adams-Bashforth discontinuous Galerkin method for approximating smooth solutions to scalar nonlinear conservation laws with E-fluxes. A priori error estimates are also derived for a first-order forward Euler discontinuous Galerkin method. Rates are optimal in time and suboptimal in space; they are valid under a CFL condition.

math.NA↗
arXiv2016

A finite volume/discontinuous Galerkin method for the advective Cahn-Hilliard equation with degenerate mobility on porous domains stemming from micro-CT imaging

A numerical method is formulated for the solution of the advective Cahn-Hilliard (CH) equation with constant and degenerate mobility in three-dimensional porous media with non-vanishing velocity on the exterior boundary. The CH equation describes phase separation of an immiscible binary mixture at constant temperature in the presence of a mass constraint and dissipation of free energy. Porous media/pore-scale problems specifically entail high-resolution images of rocks in which the solid matrix and pore spaces are fully resolved. The interior penalty discontinuous Galerkin method is used for the spatial discretization of the CH equation in mixed form, while a semi-implicit convex-concave splitting is utilized for temporal discretization. The spatial approximation order is arbitrary, while it reduces to a finite volume scheme for the choice of elementwise constants. The resulting nonlinear systems of equations are reduced using the Schur complement and solved via Newton's method. The numerical scheme is first validated using numerical convergence tests and then applied to a number of fundamental problems for validation and numerical experimentation purposes including the case of degenerate mobility. First-order physical applicability and robustness of the numerical method are shown in a breakthrough scenario on a voxel set obtained from a micro-CT scan of a real sandstone rock sample.

math.NA↗
arXiv2016

Comparison of reduced models for blood flow using Runge-Kutta discontinuous Galerkin methods

One-dimensional blood flow models take the general form of nonlinear hyperbolic systems but differ greatly in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we systematically compare several reduced models of blood flow for physiologically relevant vessel parameters, network topology, and boundary data. The models are discretized by a class of Runge-Kutta discontinuous Galerkin methods.

physics.flu-dyn↗
arXiv2015

Numerical method of characteristics for one-dimensional blood flow

Mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one-dimensional nonlinear hyperbolic system describing flow in a single vessel. This model is often simulated by computationally intensive methods like finite elements and discontinuous Galerkin, while some recent applications require more efficient approaches (e.g. for real-time clinical decision support, phenomena occurring over multiple cardiac cycles, iterative solutions to optimization/inverse problems, and uncertainty quantification). Further, the high speed of pressure waves in blood vessels greatly restricts the time step needed for stability in explicit schemes. We address both cost and stability by presenting an efficient and unconditionally stable method for approximating solutions to diagonal nonlinear hyperbolic systems. Theoretical analysis of the algorithm is given along with a comparison of our method to a discontinuous Galerkin implementation. Lastly, we demonstrate the utility of the proposed method by implementing it on small and large arterial networks of vessels whose elastic and geometrical parameters are physiologically relevant.

physics.comp-ph↗