The equation modelling the evolution of a foam (a complex porous medium consisting of a set of gas bubbles surrounded by liquid films) is solved numerically. This model is described by the reaction–diffusion differential equation with a free boundary. Two numerical methods, namely the fixed-point and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method, are proposed for solving the governing equations. The solution of Burgers’ equation is considered as a special case. We present the Crank–Nicolson scheme combined with Newton’s method for the reaction–diffusion differential equation appearing in a foam breaking phenomenon.

Artists have long recognized that trees are self-similar across enormous differences in magnitudes; i.e., they share a common fractal structure - a trunk subdivides into branches which subdivide into more branches which eventually terminate in leaves, flowers, fruits, etc. Artistid Lindenmayer (1971, 1975, 1989, 1990) invented a mathematics based on graph grammar rewriting systems to describe such iteratively branching structures; these were named in honor of him and are referred to as L-systems. With the advent of fractals into computer graphics, numerous artists have similarly produced a wide variety of software packages to illustrate the beauty of fractal/L-system generated plants. Some tree visualizations such as L-Peach (Allen , 2005) do depend very explicitly upon a complex set of precise measurements of a single species of tree. Nonetheless, we felt that there is a need to build a package that allowed scientists (and students) to collect data from actual specimens in the field or laboratory, insert these measurements into an L-system package, and then visually compare actual trees to the computer generated image with such specimens. Furthermore, the effect of variance in parameters helps users evaluate the developmental plasticity both within and between species and varieties. We have developed 3D FractaL Tree (the L is capitalized in honor of Lindemayer) to generate trees based upon measurement of (1) relative lengths of two successive segments averaged over several iterations, (2) the angle theta between bifurcating limbs at successive joints, (3) the number of steps in branching that one must follow to find a branch extending at the same angle as the first one under consideration to determine the phyllotactic angle phi, (4) the average of the summed areas (determined from measurement of diameters) of bifurcations compared to the trunk to determine whether area of flow is preserved (and to consider Poiseuille’s/Murray’s law of laminar flow in a fractal network), (5) the total number of iterative branching from the base to the tip of tree averaged over several counts based on following out different major limbs, (6) an editable L-system rule chosen from a library of branching patterns that roughly correspond to a specimen under consideration, and (7) a degree of stochasticity applied to the above rules to represent some variation over the course of a lifetime. Of course, turned upside down, the computer imagery could be used to represent root structure instead of above ground growth or the bronchial system of a lung, for example. The measurements are recorded and analyzed in a series of worksheets in Microsoft Excel and the results are entered into the graphics engine in a Java application. 3D FractaL Tree produces a rotatable three-dimensional image of the tree which is helpful for examining such characters as self-avoidance (entanglement and breakage), reception of and penetration of sunlight, distances that small herbivores (such as caterpillars) would have to traverse to go from one tip to another, allometric relationships between the convex hull of the crown (as perceived in a top-down projection of the tree) and the trunk’s diameter, and convex hull of the volume distribution of biomass on different subsections of a tree which have been discussed in the Adaptive Geometry of Trees (Horn, 1971) and subsequent research for the past four decades. Besides being able to rotate the three dimensional tree in the x-y, y-z, and x-z planes as well as zoom-in and zoom-out, three different representations are available in 3D FractaL Tree images: wire frame, solid, and transparent. Easy options for editing L-system rules and saving and exporting images are included. 3D FractaL-Tree is published with a Creative Commons license so that it is freely available for downloading, use, and extending with attribution from our Biological ESTEEM Project (http://bioquest.org/esteem).

Plants and animals have highly ordered structure both in time and in space, and one of the main questions of modern developmental biology is the transformation of genetic information into the regular structure of organism. Any multicellular plant begins its development from the universal unicellular state and acquire own species-specific structure in the course of cell divisions, cell growth and death, according to own developmental program. However the cellular mechanisms of plant development are still unknown. The aim of this work was to elaborate and verify the formalistic approach, which would allow to describe and analyze the large data of cellular architecture obtained from the real plants and to reveal the cellular mechanisms of their morphogenesis. Two multicellular embryos of Calla palustris L. (Araceae) was used as a model for the verification of our approach. The cellular architecture of the embryos was reconstructed from the stack of optical and serial sections in three dimensions and described as graphs of genealogy and space adjacency of cells. In result of the comparative analysis of these graphs, a set of regular cell types and highly conservative pattern of cell divisions during five cell generations were found. This mechanism of cellular development of the embryos could be considered as a developmental program, set of rules or grammars applied to the zygote. Also during the comparative analysis the finite plasticity in cell adjacency was described. The structural equivalence and the same morphogenetic potencies of some cells of the embryos were considered as the space-temporal symmetries. The symmetries were represented as a set of regular cell type permutations in the program of development of the embryo cellular architecture. Two groups of cell type permutations were revealed, each was composed of two elements and could be interpreted as the mirror and rotational space symmetries. The results obtained as well as the developed approach can be used in plant tissue modelling based on the real, large and complex structural data.

In this work, the least pointwise upper and/or lower bounds on the state variableon a specified subdomain of a control system under piecewise constant control action are sought.This results in a non-smooth optimization problem in function spaces. Introducing a Moreau-Yosidaregularization of the state constraints, the problem can be solvedusing a superlinearly convergent semi-smooth Newton method.Optimality conditions are derived, convergence of the Moreau-Yosidaregularization is proved, and well-posedness and superlinearconvergence of the Newton method is shown. Numerical examplesillustrate the features of this problem and the proposed approach.

The fully coupled description of blood flow and mass transport in blood vessels requires extremely robust numerical methods. In order to handle the heterogeneous coupling between blood flow and plasma filtration, addressed by means of Navier-Stokes and Darcy's equations,we need to develop a numerical scheme capable to deal with extremely variable parameters, such as the blood viscosity and Darcy's permeability of the arterial walls. In this paper, we describe a finite element method for the approximation of incompressible flow coupled problems. We exploit stabilized mixed finite elements together with Nitsche's type matching conditions that automatically adapt to the coupling of different combinations of coefficients. We study in details the stability of the method using weighted norms, emphasizing the robustness of the stability estimate with respect to the coefficients. We also consider an iterative method to split the coupled heterogeneous problem in possibly homogeneous local problems, and we investigate the spectral properties of suitable preconditioners for the solution of the global as well as local problems. Finally, we present the simulation of the fully coupled blood flow and plasma filtration problems on a realistic geometry of a cardiovascular artery after the implantation of a drug eluting stent (DES). A similar finite element method for mass transport is then employed to study the evolution of the drug released by the DES in the blood stream and in the arterial walls, and the role of plasma filtration on the drug deposition is investigated.

Galerkin discretizations of integral equations in $\mathbb{R}^{d}$ requirethe evaluation of integrals $I = \int_{S^{(1)}}\int_{S^{(2)}}g(x,y){\rm d}y{\rm d}x$ where S (1),S (2) are d-simplices and g has a singularityat x = y. We assume that g is Gevrey smooth for x $\ne$ y andsatisfies bounds for the derivatives which allow algebraic singularitiesat x = y. This holds for kernel functions commonly occurring in integralequations. We construct a family of quadrature rules $\mathcal{Q}_{N}$ usingN function evaluations of g which achieves exponential convergence|I – $\mathcal{Q}_{N}$ | ≤C exp(–r N γ ) with constants r, γ > 0.

We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove thatthe scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be appliedto models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as itshigh resolution and robustness in a number of numerical examples.

We consider a viscoelastic solid in Kelvin-Voigt rheology exhibiting also plasticitywith hardening and coupled with heat-transfer through dissipative heat production by viscoplastic effectsand through thermal expansion and corresponding adiabatic effects. Numerical discretization of the thermodynamically consistent modelis proposed by implicit time discretization, suitable regularization, and finite elements in space. Fine a-priori estimates are derived, and convergence is proved by careful successive limit passage. Computational 3D simulations illustrate an implementation of the method as well as physical effects of residual stresses substantially depending on rate of heat treatment.

One of the most intriguing questions in life science is how living organisms develop and maintain their predominant form and shape via the cascade of the processes of differentiation starting from the single cell. Mathematical modeling of these developmental processes could be a very important tool to properly describe the complex processes of evolution and geometry of morphogenesis in time and space. Here, we summarize the most important biological knowledge on plant development, exploring the different layers of investigation in developmental processes such as plant morphology, genetics, plant physiology, molecular biology and epigenetics. As knowledge on the fundamentals of plant embryogenesis, growth and development is constantly improving, we gather here the latest data on genetic, molecular and hormonal regulation of plant development together with the basic background knowledge. Special emphasis is placed on the regulation of cell cycle progression, on the role of the signal molecules phytohormones in plant development and on the details of plant meristems (loci containing plant stem cells) function. We also explore several proposed biological models regarding regulating plant development. The information presented here could be used as a basis for mathematical modeling and computer simulation of developmental processes in plants.

This paper considers an optimal control problem for a class of controlled hybrid dynamical systems (HDSs) with prescribed switchings. By using Ekeland’s variational principle and a matrix cost functional, a minimum principle for HDSs is derived, which provides a necessary condition of the aforementioned problem. The results given in this paper include both pure continuous systems and pure discrete-time systems as special cases.

Criteria for guaranteeing the existence, uniqueness and asymptotic stability (in the sense of Liapunov) of periodic solutions of a forced Liénard-type equation under certain assumptions are presented. These criteria are obtained by application of the Manásevich–Mawhin continuation theorem, Floquet theory, Liapunov stability theory and some analysis techniques. An example is provided to demonstrate the applicability of our results.

Motivated by the development of efficient Monte Carlo methodsfor PDE models in molecular dynamics,we establish a new probabilistic interpretation of a family of divergence formoperators with discontinuous coefficients at the interfaceof two open subsets of $\mathbb{R}^d$ . This family of operators includes the case of thelinearized Poisson-Boltzmann equation used tocompute the electrostatic free energy of a molecule.More precisely, we explicitly construct a Markov process whoseinfinitesimal generator belongs to this family, as the solution of a SDEincluding a non standard local time term related to the interfaceof discontinuity. We then prove an extendedFeynman-Kac formula for the Poisson-Boltzmann equation.This formula allows us to justifyvarious probabilistic numerical methods toapproximate the free energy of a molecule.We analyse the convergence rate of these simulation procedures andnumerically compare them on idealized molecules models.

This paper discusses analytical and numerical issues related toelliptic equations with random coefficients which are generallynonlinear functions of white noise. Singularity issues are avoidedby using the Itô-Skorohod calculus to interpret the interactionsbetween the coefficients and the solution. The solution is constructedby means of the Wiener Chaos (Cameron-Martin) expansions. Theexistence and uniqueness of the solutions are established underrather weak assumptions, the main of which requires only that theexpectation of the highest order (differential) operator is anon-degenerate elliptic operator. The deterministic coefficientsof the Wiener Chaos expansion of the solution solve a lower-triangularsystem of linear elliptic equations (the propagator). This structureof the propagator insures linear complexity of the related numericalalgorithms. Using the lower triangular structure and linearity of thepropagator, the rate of convergence is derived for a spectral/hp finiteelement approximation. The results of related numerical experiments arepresented.

To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a Lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a Lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear filtering for the mobile signal is therefore those of an acquisition process contaminated by random errors. This will provide a Feynman-Kac distribution flow for the conditional laws and an N particle approximation with a $\mathcal{O}$ $(\frac{1}{\sqrt{N}})$ asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described.

We design a particle interpretation of Feynman-Kac measures on path spacesbased on a backward Markovian representation combined with a traditionalmean field particle interpretation of the flow of their final timemarginals. In contrast to traditional genealogical tree based models, thesenew particle algorithms can be used to compute normalized additivefunctionals “on-the-fly” as well as theirlimiting occupation measures with a given precision degree that does notdepend on the final time horizon.We provide uniform convergence results w.r.t. the time horizon parameter aswell as functional central limit theorems and exponential concentrationestimates, yielding what seems to be the first results of this type for thisclass of models. We also illustrate these results in the context offiltering of hidden Markov models, as well as in computational physics andimaginary time Schroedinger type partial differential equations, with aspecial interest in the numerical approximation of the invariant measureassociated to h-processes.

This paper describes the extension of arecently developed numerical solver for the Landau-LifshitzNavier-Stokes (LLNS) equations to binary mixtures in threedimensions. The LLNS equations incorporate thermal fluctuations intomacroscopic hydrodynamics by using white-noise fluxes. Thesestochastic PDEs are more complicated in three dimensions due to thetensorial form of the correlations for the stochastic fluxes and inmixtures due to couplings of energy and concentration fluxes (e.g.,Soret effect). We present various numerical tests of systems in andout of equilibrium, including time-dependent systems, anddemonstrate good agreement with theoretical results and molecularsimulation.

The paper studies the convergence behavior ofMonte Carlo schemes for semiconductors.A detailed analysis of the systematic error with respect to numerical parameters is performed.Different sources of systematic error are pointed out andillustrated in a spatially one-dimensional test case.The error with respect to the number of simulation particlesoccurs during the calculation of the internal electric field.The time step error, which is related to the splitting of transport andelectric field calculations, vanishes sufficiently fast.The error due to the approximation of the trajectories ofparticles depends on the ODE solver used in the algorithm.It is negligible compared to the other sources of time steperror, when a second order Runge-Kutta solver is used. The error related to the approximate scattering mechanismis the most significant source of error with respect to the time step.

This special volume of the ESAIM Journal, Mathematical Modelling and Numerical Analysis,contains a collection of articles on probabilistic interpretations of some classes of nonlinear integro-differential equations.The selected contributions deal with a wide range of topics in applied probability theory and stochastic analysis, with applications in a variety of scientific disciplines, includingphysics, biology, fluid mechanics, molecular chemistry, financial mathematics and bayesian statistics. In this preface, we provide a brief presentation of the main contributions presented in this special volume. We have also included an introduction to classic probabilistic methods and a presentation of the more recent particle methods, with a synthetic picture of their mathematical foundations and their range of applications.

This paper considers Schrödinger operators, and presents a probabilistic interpretation of the variation (or shape derivative) of the Dirichlet groundstate energy when the associated domain is perturbed. This interpretation relies on the distribution on the boundary of a stopped random process with Feynman-Kac weights. Practical computations require in addition the explicit approximation of the normal derivative of the groundstate on the boundary. We then propose to use this formulation in the case of the so-called fixed node approximation of Fermion groundstates, defined by the bottom eigenelements of the Schrödinger operator of a Fermionic system with Dirichlet conditions on the nodes (the set of zeros) of an initially guessed skew-symmetric function. We show that shape derivatives of the fixed node energy vanishes if and only if either (i) the distribution on the nodes of the stopped random process is symmetric; or (ii) the nodes are exactly the zeros of a skew-symmetric eigenfunction of the operator. We propose an approximation of the shape derivative of the fixed node energy that can be computed with a Monte-Carlo algorithm, which can be referred to as Nodal Monte-Carlo (NMC). The latter approximation of the shape derivative also vanishes if and only if either (i) or (ii) holds.