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4D Embryogenesis image analysis using PDE methods of image processing

Paul Bourgine, Róbert Čunderlík, Olga Drblíková-Stašová, Karol Mikula, Mariana Remešíková, Nadine Peyriéras, Barbara Rizzi, Alessandro Sarti (2010)


In this paper, we introduce a set of methods for processing and analyzing long time series of 3D images representing embryo evolution. The images are obtained by in vivo scanning using a confocal microscope where one of the channels represents the cell nuclei and the other one the cell membranes. Our image processing chain consists of three steps: image filtering, object counting (center detection) and segmentation. The corresponding methods are based on numerical solution of nonlinear PDEs, namely...

A new reconstruction-enhanced discontinuous Galerkin method for time-dependent problems

Kučera, Václav (2010)

Programs and Algorithms of Numerical Mathematics

This work is concerned with the introduction of a new numerical scheme based on the discontinuous Galerkin (DG) method. We propose to follow the methodology of higher order finite volume schemes and introduce a reconstruction operator into the DG scheme. This operator constructs higher order piecewise polynomial reconstructions from the lower order DG scheme. Such a procedure was proposed already in [2] based on heuristic arguments, however we provide a rigorous derivation, which justifies the increased...

A sixth-order finite volume method for the 1D biharmonic operator: Application to intramedullary nail simulation

Ricardo Costa, Gaspar J. Machado, Stéphane Clain (2015)

International Journal of Applied Mathematics and Computer Science

A new very high-order finite volume method to solve problems with harmonic and biharmonic operators for onedimensional geometries is proposed. The main ingredient is polynomial reconstruction based on local interpolations of mean values providing accurate approximations of the solution up to the sixth-order accuracy. First developed with the harmonic operator, an extension for the biharmonic operator is obtained, which allows designing a very high-order finite volume scheme where the solution is...

Adaptive mesh refinement strategy for a non conservative transport problem

Benjamin Aymard, Frédérique Clément, Marie Postel (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

Long time simulations of transport equations raise computational challenges since they require both a large domain of calculation and sufficient accuracy. It is therefore advantageous, in terms of computational costs, to use a time varying adaptive mesh, with small cells in the region of interest and coarser cells where the solution is smooth. Biological models involving cell dynamics fall for instance within this framework and are often non conservative to account for cell division. In that case...

Adaptive multiresolution methods

Margarete O. Domingues, Sônia M. Gomes, Olivier Roussel, Kai Schneider (2011)

ESAIM: Proceedings

These lecture notes present adaptive multiresolution schemes for evolutionary PDEs in Cartesian geometries. The discretization schemes are based either on finite volume or finite difference schemes. The concept of multiresolution analyses, including Harten’s approach for point and cell averages, is described in some detail. Then the sparse point representation method is discussed. Different strategies for adaptive time-stepping, like local scale dependent time stepping and time step control, are...

Adaptive Multiresolution Methods: Practical issues on Data Structures, Implementation and Parallelization*

K. Brix, S. Melian, S. Müller, M. Bachmann (2011)

ESAIM: Proceedings

The concept of fully adaptive multiresolution finite volume schemes has been developed and investigated during the past decade. Here grid adaptation is realized by performing a multiscale decomposition of the discrete data at hand. By means of hard thresholding the resulting multiscale data are compressed. From the remaining data a locally refined grid is constructed. The aim of the present work is to give a self-contained overview on the construction of an appropriate multiresolution analysis using...

Aerodynamic Computations Using a Finite Volume Method with an HLLC Numerical Flux Function

L. Remaki, O. Hassan, K. Morgan (2011)

Mathematical Modelling of Natural Phenomena

A finite volume method for the simulation of compressible aerodynamic flows is described. Stabilisation and shock capturing is achieved by the use of an HLLC consistent numerical flux function, with acoustic wave improvement. The method is implemented on an unstructured hybrid mesh in three dimensions. A solution of higher order accuracy is obtained by reconstruction, using an iteratively corrected least squares process, and by a new limiting procedure....

An unconditionally stable finite element-finite volume pressure correction scheme for the drift-flux model

Laura Gastaldo, Raphaèle Herbin, Jean-Claude Latché (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of...

Applications of approximate gradient schemes for nonlinear parabolic equations

Robert Eymard, Angela Handlovičová, Raphaèle Herbin, Karol Mikula, Olga Stašová (2015)

Applications of Mathematics

We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergence which shows its second order accuracy. We present also numerical experiments related to image filtering by time-delayed Perona-Malik and tensor diffusion...

Block-structured Adaptive Mesh Refinement - Theory, Implementation and Application

Ralf Deiterding (2011)

ESAIM: Proceedings

Structured adaptive mesh refinement (SAMR) techniques can enable cutting-edge simulations of problems governed by conservation laws. Focusing on the strictly hyperbolic case, these notes explain all algorithmic and mathematical details of a technically relevant implementation tailored for distributed memory computers. An overview of the background of commonly used finite volume discretizations for gas dynamics is included and typical benchmarks to quantify accuracy and performance of the dynamically...

Comparison of the 3D Numerical Schemes for Solving Curvature Driven Level Set Equation Based on Discrete Duality Finite Volumes

Dana Kotorová (2014)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

In this work we describe two schemes for solving level set equation in 3D with a method based on finite volumes. These schemes use the so-called dual volumes as in [Coudiére, Y., Hubert, F.: A 3D discrete duality finite volume method for nonlinear elliptic equations Algoritmy 2009 (2009), 51–60.], [Hermeline, F.: A finite volume method for approximating 3D diffusion operators on general meshes Journal of Computational Physics 228, 16 (2009), 5763–5786.], where they are used for the nonlinear elliptic...

Finite volume method in curvilinear coordinates for hyperbolic conservation laws⋆

A. Bonnement, T. Fajraoui, H. Guillard, M. Martin, A. Mouton, B. Nkonga, A. Sangam (2011)

ESAIM: Proceedings

This paper deals with the design of finite volume approximation of hyperbolic conservation laws in curvilinear coordinates. Such coordinates are encountered naturally in many problems as for instance in the analysis of a large number of models coming from magnetic confinement fusion in tokamaks. In this paper we derive a new finite volume method for hyperbolic conservation laws in curvilinear coordinates. The method is first described in a general...

Finite volume schemes for the generalized subjective surface equation in image segmentation

Karol Mikula, Mariana Remešíková (2009)


In this paper, we describe an efficient method for 3D image segmentation. The method uses a PDE model – the so called generalized subjective surface equation which is an equation of advection-diffusion type. The main goal is to develop an efficient and stable numerical method for solving this problem. The numerical solution is based on semi-implicit time discretization and flux-based level set finite volume space discretization. The space discretization is discussed in details and we introduce three...

Gaussian curvature based tangential redistribution of points on evolving surfaces

Medľa, Matej, Mikula, Karol (2017)

Proceedings of Equadiff 14

There exist two main methods for computing a surface evolution, level-set method and Lagrangian method. Redistribution of points is a crucial element in a Lagrangian approach. In this paper we present a point redistribution that compress quads in the areas with a high Gaussian curvature. Numerical method is presented for a mean curvature flow of a surface approximated by quads.

High order approximation of probabilistic shock profiles in hyperbolic conservation laws with uncertain initial data

Christoph Schwab, Svetlana Tokareva (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We analyze the regularity of random entropy solutions to scalar hyperbolic conservation laws with random initial data. We prove regularity theorems for statistics of random entropy solutions like expectation, variance, space-time correlation functions and polynomial moments such as gPC coefficients. We show how regularity of such moments (statistical and polynomial chaos) of random entropy solutions depends on the regularity of the distribution law of the random shock location of the initial data....

Hyperbolic relaxation models for granular flows

Thierry Gallouët, Philippe Helluy, Jean-Marc Hérard, Julien Nussbaum (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work we describe an efficient model for the simulation of a two-phase flow made of a gas and a granular solid. The starting point is the two-velocity two-pressure model of Baer and Nunziato [Int. J. Multiph. Flow16 (1986) 861–889]. The model is supplemented by a relaxation source term in order to take into account the pressure equilibrium between the two phases and the granular stress in the solid phase. We show that the relaxation process can be made thermodynamically coherent with an...

Is GPU the future of Scientific Computing ?

Georges-Henri Cottet, Jean-Matthieu Etancelin, Franck Perignon, Christophe Picard, Florian De Vuyst, Christophe Labourdette (2013)

Annales mathématiques Blaise Pascal

These past few years, new types of computational architectures based on graphics processors have emerged. These technologies provide important computational resources at low cost and low energy consumption. Lots of developments have been done around GPU and many tools and libraries are now available to implement efficiently softwares on those architectures.This article contains the two contributions of the mini-symposium about GPU organized by Loïc Gouarin (Laboratoire de Mathématiques d’Orsay),...

Lagrangian evolution approach to surface-patch quadrangulation

Martin Húska, Matej Medl'a, Karol Mikula, Serena Morigi (2021)

Applications of Mathematics

We present a method for the generation of a pure quad mesh approximating a discrete manifold of arbitrary topology that preserves the patch layout characterizing the intrinsic object structure. A three-step procedure constitutes the core of our approach which first extracts the patch layout of the object by a topological partitioning of the digital shape, then computes the minimal surface given by the boundaries of the patch layout (basic quad layout) and then evolves it towards the object boundaries....

Modelling and simulation of liquid-vapor phase transition in compressible flows based on thermodynamical equilibrium∗

Gloria Faccanoni, Samuel Kokh, Grégoire Allaire (2012)

ESAIM: Mathematical Modelling and Numerical Analysis

In the present work we investigate the numerical simulation of liquid-vapor phase change in compressible flows. Each phase is modeled as a compressible fluid equipped with its own equation of state (EOS). We suppose that inter-phase equilibrium processes in the medium operate at a short time-scale compared to the other physical phenomena such as convection or thermal diffusion. This assumption provides an implicit definition of an equilibrium EOS...

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