Carlos Vázquez

An upwind numerical approach for an American and European option pricing model

The numerical solution of several mathematical models arising in financial economics for the valuation of both European and American call options on different types of assets is considered. All the models are based on the Black-Scholes partial differential equation. In the case of European options a numerical upwind scheme for solving the boundary backward parabolic partial differential equation problem is presented. When treating with American options an additional inequality constraint leads to a discretized linear complementarity problem. In each case, the numerical approximation of option values is computed by means of optimization algorithms. In particular, Uzawas method allows to compute the optimal exercise boundary which corresponds to the classical concept of moving boundary in continuum mechanics.

An Average Flow Model of the Reynolds Roughness Including a Mass-Flow Preserving Cavitation Model

An average Reynolds equation for predicting the effects of deterministic periodic roughness, taking Jakobsson, Floberg, and Olsson mass flow preserving cavitation model into account, is introduced based upon the double scale analysis approach. This average Reynolds equation can be used both for a microscopic interasperity cavitation and a macroscopic one. The validity of such a model is verified by numerical experiments both for one-dimensional and two-dimensional roughness patterns.

ON A DOUBLY NONLINEAR PARABOLIC OBSTACLE PROBLEM MODELLING ICE SHEET DYNAMICS

This paper deals with the weak formulation of a free (moving) boundary problem arising in theoretical glaciology. Considering shallow ice sheet flow, we present the mathematical analysis and the numerical solution of the second order nonlinear degenerate parabolic equation modelling, in the isothermal case, the ice sheet non-Newtonian dynamics. An obstacle problem is then deduced and analyzed. The existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method. Then the solutions are numerically computed with a method of characteristics and a duality algorithm to deal with the resulting variational inequalities. The weak framework we introduce and its analysis (both qualitative and numerical) are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension; they can be successfully applied to more realistic and sophisticated models related to other geophysical settings.

Numerical simulation of a lubricated hertzian contact problem under imposed load

Abstract A lot of lubrication problems can be reduced to a contact between an elastic sphere and a rigid plane which leads to a Reynolds–Hertz nonlinear coupled model, where the cavitation phenomenon, the lubricant piezoviscosity and the balance between an imposed load on the device and the hydrodynamic load, must be taken into account. Last aspects add some difficulties: a new piezoviscous nonlinearity and a nonlocal constraint on the main unknown. In this paper, the proposed coupled problem is modelled, normalized and numerically solved. Thus, a numerical scheme which combines fixed point algorithms, the method of characteristics, duality techniques and finite element approximations is developed. Finally, when using a numerical flux conservation test, some computed results for real data are presented.

Numerical computation of free boundary problems in elastohydrodynamic lubrication

Abstract An alternative algorithm has been developed for computing the behavior of thin fluid films in two elastohydrodynamic lubrication problems. The presence of elasticity, lubrication, and cavitation leads to a nonlinear coupled system of partial differential equations. The hydrodynamic part of both problems is governed by the well-known Reynolds equation combined with the cavitation model of Elrod-Adams, which motivates the appearance of a free boundary. Elastic deformations are taken into account by means of the Hertz equation in rolling ball contact problems or the elastic hinged plate biharmonic equation in the case of journal-bearing devices with thin bearing. A numerical method decoupling the hydrodynamic part of the problem and the elastic one is presented. This method also involves an upwind scheme to discretize the lubrication model and finite element approximations.

Numerical approach of temperature distribution in a free boundary model for polythermal ice sheets

Summary. A numerical method for solving the thermal subproblem appearing in the modelization of polythermal ice sheets is described. This thermal problem mainly involves three nonlinearities: a reaction term due to the viscous dissipation, a Signorini boundary condition associated to the geothermic flux and an enthalpy term issued from the two phase Stefan formulation of the polythermal regime. The stationary temperature is obtained as the limit of an evolutive problem which is discretized in time with an upwind characteristics scheme and in space with finite elements. The nonlinearities are solved either by Newton-Raphson method or by duality techniques applied to maximal monotone operators. The application of the algorithms provides the dimensionless temperature distribution approximation and allows to identify the cold and temperate ice regions.

Shaping protein distributions in stochastic self-regulated gene expression networks

In this work, we study connections between dynamic behavior and network parameters, for self-regulatory networks. To that aim, a method to compute the regions in the space of parameters that sustain bimodal or binary protein distributions has been developed. Such regions are indicative of stochastic dynamics manifested either as transitions between absence and presence of protein or between two positive protein levels. The method is based on the continuous approximation of the chemical master equation, unlike other approaches that make use of a deterministic description, which as will be shown can be misleading. We find that bimodal behavior is a ubiquitous phenomenon in cooperative gene expression networks under positive feedback. It appears for any range of transcription and translation rate constants whenever leakage remains below a critical threshold. Above such a threshold, the region in the parameters space which sustains bimodality persists, although restricted to low transcription and high translation rate constants. Remarkably, such a threshold is independent of the transcription or translation rates or the proportion of an active or inactive promoter and depends only on the level of cooperativity. The proposed method can be employed to identify bimodal or binary distributions leading to stochastic dynamics with specific switching properties, by searching inside the parameter regions that sustain such behavior.

Finite element solution of a Reynolds–Koiter coupled problem for the elastic journal–bearing

Abstract The aim of this work is to present a numerical algorithm for solving a new elastohydrodynamic free-boundary problem which models the lubricant pressure behaviour by means of Reynolds equation, jointly with the free-boundary model of Elrod–Adams for cavitation and the elastic deformation of the bearing with the Koiter shell model; a non-local constraint on the pressure is imposed. The proposed algorithm uncouples the hydrodynamic and elastic parts to achieve a fixed point. For the two-dimensional problem, a combination of finite elements, transport-diffusion and duality methods is performed. For the three-dimensional elastic problem, a finite element discretization is applied to a mixed formulation for the Koiter model.

Stochastic modeling and numerical simulation of gene regulatory networks with protein bursting

Gene expression is inherently stochastic. Advanced single-cell microscopy techniques together with mathematical models for single gene expression led to important insights in elucidating the sources of intrinsic noise in prokaryotic and eukaryotic cells. In addition to the finite size effects due to low copy numbers, translational bursting is a dominant source of stochasticity in cell scenarios involving few short lived mRNA transcripts with high translational efficiency (as is typically the case for prokaryotes), causing protein synthesis to occur in random bursts. In the context of gene regulation cascades, the Chemical Master Equation (CME) governing gene expression has in general no closed form solution, and the accurate stochastic simulation of the dynamics of complex gene regulatory networks is a major computational challenge. The CME associated to a single gene self regulatory motif has been previously approximated by a one dimensional time dependent partial integral differential equation (PIDE). However, to the best of our knowledge, multidimensional versions for such PIDE have not been developed yet. Here we propose a multidimensional PIDE model for regulatory networks involving multiple genes with self and cross regulations (in which genes can be regulated by different transcription factors) derived as the continuous counterpart of a CME with jump process. The model offers a reliable description of systems with translational bursting. In order to provide an efficient numerical solution, we develop a semilagrangian method to discretize the differential part of the PIDE, combined with a composed trapezoidal quadrature formula to approximate the integral term. We apply the model and numerical method to study sustained stochastic oscillations and the development of competence, a particular case of transient differentiation attained by certain bacterial cells under stress conditions. We found that the resulting probability distributions are distinguishable from those characteristic of other transient differentiation processes. In this way, they can be employed as markers or signatures that identify such phenomena from bacterial population experimental data, for instance. The computational efficiency of the semilagrangian method makes it suitable for purposes like model identification and parameter estimation from experimental data or, in combination with optimization routines, the design of gene regulatory networks under molecular noise.

NUMERICAL APPROACH OF THERMOMECHANICAL COUPLED PROBLEMS WITH MOVING BOUNDARIES IN THEORETICAL GLACIOLOGY

In this work we propose a new numerical method for solving a thermomechanical coupled problem which arises as a mathematical model for the evolution of ice sheets profiles and the corresponding temperatures. From the accumulation–ablation ratio, the atmospheric temperature and the geothermic flux, the ice sheet profile is the solution of a moving boundary problem governed by a nonlinear convection–diffusion equation. Moreover, the mathematical model which governs the temperature distribution involves three nonlinearities: a reaction term due to the viscous dissipation, a Signorini boundary condition associated to the geothermic flux and an enthalpy term issued from the two-phase Stefan formulation of the polythermal regime. The problems are discretized in time with upwind characteristics schemes and in space with finite elements. The nonlinearities are solved either by fixed point methods or by duality techniques. Several numerical simulation examples involving real data sets issued from the Antartic ice sheet are shown.