Experimental design for parameter estimation of two time-scale model of photosynthesis and photoinhibition in microalgae

A method for parameter estimation of two time-scale model of photosynthesis and photoinhibition is presented.     

A new fundamental solution for differential Riccati equations arising in control

The matrix differential Riccati equation (DRE) is ubiquitous in control and systems theory. The presence of the quadratic term implies that a simple linear-systems fundamental solution does not exist. Of course it is well-known that the Bernoulli substitution may be applied to obtain a linear system of doubled size. Here, however, tools from max-plus analysis and semiconvex duality are brought to bear on the DRE. We consider the DRE as a finite-dimensional solution to a deterministic linear/quadratic control problem. Taking the semiconvex dual of the associated semigroup, one obtains the solution operator as a max-plus integral operator with quadratic kernel. The kernel is equivalently represented as a matrix. Using the semigroup property of the dual operator, one obtains a matrix operation whereby the kernel matrix propagates as a semigroup. The propagation forward is through some simple matrix operations. This time-indexed family of matrices forms a new fundamental solution for the DRE. Solution for any initial condition is obtained by a few matrix operations on the fundamental solution and the initial condition. In analogy with standard-algebra linear systems, the fundamental solution can be viewed as an exponential form over a certain idempotent semiring. This fundamental solution has a particularly nice control interpretation, and might lead to improved DRE solution speeds.     

A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity

We introduce a new box-scheme, called ``hermitian box-scheme' on the model of the one-dimensional Poisson problem. The scheme combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil, which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L ² norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from 1 to 2.5 in the discrete H ¹ norm. This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work.     

Keller's Box-Scheme for the One-Dimensional Stationary Convection-Diffusion Equation

u ,∇u)=f, is to take the average onto the same mesh of the two equations of the mixed form, the conservation law div p=f and the constitutive law p=ϕ(u,∇u). In this paper, we perform the numerical analysis of two Keller-like box-schemes for the one-dimensional convection-diffusion equation cu _x −ɛu _xx =f. In the first one, introduced by B. Courbet in [9,10], the numerical average of the diffusive flux is upwinded along the sign of the velocity, giving a first order accurate scheme. The second one is fourth order accurate. It is based onto the Euler-MacLaurin quadrature formula for the average of the diffusive flux. We emphasize in each case the link with the SUPG finite element method. Received June 7, 2001; revised October 2, 2001     

Numerical simulation of multi-species biofilms in porous media for different kinetics

There are bacteria that can form strong biofilms in porous media. These biofilms can be used as biobarriers to restrict the flow of pollutants. For certain contaminants, a second species of bacteria that can actually react with the contaminants can be added to the biobarrier to actually degrade the pollutants. We propose some mathematical models for the formation of these reacting biobarriers under different hypotheses, and numerically solve the resulting equations for the flow, transport and reactions. Qualitative comparisons with some experimental results are also given.     

On construction and simulation of count data models

The mean–variance range and the shape flexibility are important measures of the applicability of a count data model. This paper develops a method for constructing nonnegative integer-valued random variables with any interval domain, any theoretically possible mean–variance pair, and different shapes. The basic tool is a simple mean-preserving discretization procedure for random variables. Two corresponding variate generation algorithms are derived, and shown to be comparable to the alias method. As an application, our method enables production of count data models with full under- and over-dispersion flexibility and desired shape.     

Spectral projected subgradient with a momentum term for the Lagrangean dual approach

The Lagrangean dual problem, with a non-differentiable convex objective function, is usually solved by using the subgradient method, whose convergence is guaranteed if the optimal value of the dual objective function is known. In practice, this optimal value is approximated by a previously computed bound. In this work, we combine the subgradient method with a different choice of steplength, based on the recently developed spectral projected gradient method, that does not require either exact or approximated estimates of the optimal value. We also add a momentum term to the subgradient direction that accelerates the convergence process towards global solutions. To illustrate the behavior of our new algorithm we solve Lagrangean dual problems associated with integer programming problems. In particular, we present and discuss encouraging numerical results for set covering problems and generalized assignment problems.     

A Note on the Dual Treatment of Higher-Order Regularization Functionals

In this paper, we apply the dual approach developed by A. Chambolle for the Rudin-Osher-Fatemi model to regularization functionals with higher order derivatives. We emphasize the linear algebra point of view by consequently using matrix-vector notation. Numerical examples demonstrate the differences between various second order regularization approaches.     

On the use of velocity feedback for robot impact control

Most of the existing impact control strategies have relied on integral action and robot joint velocity feedback to achieve the desired damping effect. The limitations of such approaches are discussed in this paper. Based on a robot tip velocity reconstruction scheme, a positive tip velocity feedback method was introduced. This helped overcome the non-colocation problem due to the higher order arm dynamics and enhanced the impact suppression. A control strategy which includes the velocity feedback and an integral control was then developed and implemented on an industrial robot.     

An Iterative Substructuring Method for div-stable Finite Element Approximations of the Oseen Problem

We apply an iterative substructuring algorithm with transmission conditions of Robin–Robin type to the discretized Oseen problem appearing as a linearized variant of the incompressible Navier–Stokes equations. Here we consider finite element approximations using velocity/pressure pairs which satisfy the Babuška–Brezzi stability condition. After proving well-posedness and strong convergence of the method, we derive an a-posteriori error estimate which controls convergence of the discrete subdomain solutions to the global discrete solution by measuring the jumps of the velocities at the interface. Additionally we obtain information how to design a parameter of the Robin interface condition which essentially influences the convergence speed. Numerical experiments confirm the theoretical results and the applicability of the method. Received February 18, 2000; revised February 21, 2001     

On the numerical approximation of a degenerated hyperbolic system

The heat exchanger in a heat pump system may be conveniently described by a degenerated hyperbolic system, namely the zero Mach-number limit of the Euler equations. This leads to a mixed hyperbolic/parabolic system with coupled time-dependent boundary conditions. We propose a method-of-lines discretisation by using an upwinding scheme. We derive stability estimates for the linearisation with frozen coefficients. The resulting differential–algebraic equation has a perturbation index of 2 and a weak instability with respect to the space step size. The latter property is validated experimentally even for the nonlinear system. In contrast, the perturbation index did not exceed one in the numerical experiments.     

Two-dimensional motion-planning for nonholonomic robots using the bump-surfaces concept

In this paper, a new method is introduced for finding a near-optimal path of a nonholonomic robot moving in a 2D environment cluttered with static obstacles. The method is based on the Bump-Surfaces concept and is able to deal with robots represented by a translating and rotating rigid body. The proposed approach is applied to car-like robots.     

On the use of piecewise-linear functions in suboptimal feedback control laws

The synthesis of a suboptimal feedback control law for second-order nonlinear systems with a quadratic performance index is considered. The synthesis methods developed for this problem generally consist of different ways for determining the coefficients of a power series which approximates the optimal control laws. A method using piecewise-linear functions is described.     

Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time

In many real-life applications of optimal control problems with constraints in form of partial differential equations (PDEs), hyperbolic equations are involved which typically describe transport processes. Since hyperbolic equations usually propagate discontinuities of initial or boundary conditions into the domain on which the solution lives or can develop discontinuities even in the presence of smooth data, problems of this type constitute a severe challenge for both theory and numerics of PDE constrained optimization.     

A Method for Approximate Inversion of the Hyperbolic CDF

It has been observed by E. Eberlein and U. Keller that the hyperbolic distribution fits logarithmic rates of returns of a stock much better than the normal distribution. We give a method for sampling from the hyperbolic distribution by the inversion method, which is suited for simulation using low discrepancy point sets. Instead of directly inverting the cumulative distribution function (CDF) we provide an approximation of the inverse function which is simple to obtain by standard numerical methods and which is fast to compute. Received May 16, 2002; revised November 5, 2002 Published online: December 12, 2002 RID="*" ID="*" Partly supported by the Austrian Science Fund Project S8305.     

On the design of feedback control for relative stability

A basic theorem on relative stability is generalized and carefully interpreted. Two new theorems are then presented which simplify the calculation of feedback gains to achieve a prescribed damping ratio.