Iñigo Arregui

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.

An Eulerian approach for large displacements of thin shells including geometrical non-linearities

Abstract This paper is devoted to the problem of large displacement of shells, the only non-linearity being geometrical. The geometry of the mid-surface of the shell is determined thanks to the knowledge of nodes and normal vectors, and is continuously updated. Stresses are also carried from one configuration to another, which enables us to linearize the equilibrium equations. It is then possible to build an Eulerian formulation. Moreover, some control strategies are proposed, in order to find limit points and unstable solutions. Finally, we check in some classical examples, the accuracy of this technique.

Numerical solution of an optimal investment problem with proportional transaction costs

This paper mainly concerns the numerical solution of a nonlinear parabolic double obstacle problem arising in a finite-horizon optimal investment problem with proportional transaction costs. The problem is initially posed in terms of an evolutive HJB equation with gradient constraints and the properties of the utility function allow to obtain the optimal investment solution from a nonlinear problem posed in one spatial variable. The proposed numerical methods mainly consist of a localization procedure to pose the problem on a bounded domain, a characteristics method for time discretization to deal with the large gradients of the solution, a Newton algorithm to solve the nonlinear term in the governing equation and a projected relaxation scheme to cope with the double obstacle (free boundary) feature. Moreover, piecewise linear Lagrange finite elements for spatial discretization are considered. Numerical results illustrate the performance of the set of numerical techniques by recovering all qualitative properties proved in Dai and Yi (2009) [6].

Evaluation of the Optimal Utility of Some Investment Projects with Irreversible Environmental Effects

In this work, the authors propose efficient numerical methods to solve mathematical models for different optimal investment problems with irreversible environmental effects. A relevant point is that both the benefits of the environment and the alternative project are uncertain. The cases with instantaneous and progressive transformation of the environment are addressed. In the first case, an augmented Lagrangian active set (ALAS) algorithm combined with finite element methods are proposed as a more efficient technique for the numerical solution to the obstacle problem associated with a degenerated elliptic PDE. In the second case, the mathematical model can be split into two subsequent steps: first we solve numerically a set of parameter dependent boundary value problems (the parameter being the level of progressive transformation), and secondly an evolutive nonstandard obstacle problem is discretized, thus leading to an obstacle problem at each time step. Also, an ALAS algorithm is proposed at each time step. Numerical solutions are validated through qualitative properties theoretically proven in the literature for different examples.

A Monte Carlo approach to American options pricing including counterparty risk

ABSTRACT In this work, we propose a numerical technique to compute the total value adjustment for the pricing of American options when considering counterparty risk. Several linear and nonlinear mathematical models, associated to different choices of the mark-to-market value at default, are deduced and numerically solved, thus leading to approximations of the option price with counterparty risk. The methodology is based on Monte Carlo simulations combined with a dynamic programming strategy. At each time step, an optimal stopping criterion is applied and the decision on either exercising or not the option is taken. We present some numerical tests to illustrate the behaviour of the proposed method.

Total value adjustment for European options with two stochastic factors. Mathematical model, analysis and numerical simulation

Abstract In the present paper we derive novel (non)linear PDE models for pricing European options and the associated total value adjustment (XVA), when incorporating the counterparty risk. The main innovative aspect is the consideration of stochastic spreads instead of less realistic constant spreads previously used in the literature. For the nonlinear model, a rigorous mathematical analysis based on sectorial differential operators allows to state the existence and uniqueness of a solution. Moreover, for the numerical solution we propose an appropriate set of techniques based on the method of characteristics for time discretization, finite element for spatial discretization and fixed point iteration for the nonlinear terms. Finally, numerical examples illustrate the expected behaviour of the option prices and the total value adjustment.

A numerical strategy for telecommunications networks capacity planning under demand and price uncertainty

The massive use of Internet in the last twenty years has created a huge demand for telecommunications networks capacity. In this work, financial option pricing methods are applied to the problem of network investment decision timing. The main innovative aspect is the consideration of two uncertain factors: the capacity demand and the bandwidth price, the evolution of which is modeled by suitable stochastic processes. Thus, we consider the optimal decision problem of upgrading a line in terms of the (highly volatile) uncertain demand for capacity and the price. By using real options pricing methodology, a set of partial differential equation problems are posed and appropriate numerical methods based on characteristics methods combined with finite elements to approximate the solution are proposed. The combination with a dynamic programming strategy gives rise to a global algorithm to help in the decision of optimizing the value of the line.

CVA Computing by PDE Models

In order to incorporate the credit value adjustment (CVA) in derivative contracts, we propose a set of numerical methods to solve a nonlinear partial differential equation [2] modelling the CVA. Additionally to adequate boundary conditions proposals, characteristics methods, fixed point techniques and finite elements methods are designed and implemented. A numerical test illustrates the behavior of the model and methods.

Original article: Adaptive numerical methods for an hydrodynamic problem arising in magnetic reading devices

The mechanical behavior of magnetic reading devices is mainly governed by compressible Reynolds equations when the air bearing modeling approximation is considered. First, the convection dominated feature motivates the use of a characteristics scheme adapted to steady state problems. Secondly, a duality method to treat the particular nonlinear diffusion term is applied. A piecewise linear finite element for spatial discretization has been chosen. Moreover, in certain conditions and devices, strong air pressure gradients arise locally, either due to a strongly convection dominated regime or to the presence of slots in the storage device, for example. In the present work we improve the previous numerical methods proposed to cope with this new setting. Thus, mainly adaptive mesh refinement algorithms based on pressure gradient indicators and appropriate multigrid techniques to solve the linear systems arising at each iteration of the duality method are proposed. Finally, several examples illustrate the performance of the set of numerical techniques.

Numerical solution of a free boundary problem associated to investments with instantaneous irreversible environmental effects

In this work, some numerical methods are proposed to solve a problem which models the joint utility of starting at optimal time an industrial process that provides some benefits but also implies some irreversible effects on the environment. The mathematical model is posed in terms of an obstacle problem associated to an elliptic equation on an unbounded domain. First, following Diaz and Faghloumi [5], an equivalent formulation on a bounded domain is proposed by means of adequate changes of variables and unknown. Next, projected Gauss-Seidel and Lions-Mercier type algorithms to solve the finite elements discretized problem are proposed; the latter is combined with multigrid techniques and adaptive refinement. The algorithms efficiency is first illustrated by some numerical tests on examples with known analytical solution or theoretically stated properties. Finally, some more realistic examples are presented for which there are not known properties of the solution.