A tutorial on pseudospectral methods for computational optimal control

[English] This paper is a tutorial introduction to pseudospectral optimal control. With pseudospectral methods, a function is approximated as a linear combination of smooth basis functions, which are often chosen to be Legendre or Chebyshev polynomials. Collocation of the differential-algebraic equations is performed at orthogonal collocation points, which are selected to yield interpolation of high accuracy. Pseudospectral methods directly discretize the original optimal control problem to recast it into a nonlinear programming format. A numerical optimizer is then employed to find approximate local optimal solutions. The paper also briefly describes the functionality and implementation of PSOPT, an open source software package written in C++ that employs pseudospectral discretization methods to solve multi-phase optimal control problems. The software implements the Legendre and Chebyshev pseudospectral methods, and it has useful features such as automatic differentiation, sparsity detection, and automatic scaling. The use of pseudospectral methods is illustrated in two problems taken from the literature on computational optimal control. [Portuguese] Este artigo e um tutorial introdutorio sobre controle otimo pseudo-espectral. Em metodos pseudo-espectrais, uma funcao e aproximada como uma combinacao linear de funcoes de base suaves, tipicamente escolhidas como polinomios de Legendre ou Chebyshev. A colocacao de equacoes algebrico-diferenciais e realizada em pontos de colocacao ortogonal, que sao selecionados de modo a minimizar o erro de interpolacao. Metodos pseudoespectrais discretizam o problema de controle otimo original de modo a converte-lo em um problema de programa cao nao-linear. Um otimizador numerico e entao empregado para obter solucoes localmente otimas. Este artigo tambem descreve sucintamente a funcionalidade e a implementacao de um pacote computacional de codigo aberto escrito em C++ chamado PSOPT. Tal pacote emprega metodos de discretizacao pseudo-spectrais para resolver problemas de controle otimo com multiplas fase. O PSOPT permite a utilizacao de metodos de Legendre ou Chebyshev, e possui caractersticas uteis tais como diferenciacao automatica, deteccao de esparsidade e escalonamento automatico. O uso de metodos pseudo-espectrais e ilustrado em dois problemas retirados da literatura de controle otimo computacional.

The UK Department of National Heritage: a model for the development and integration of leisure policy in the new Europe?

Leisure is in the vanguard of a social and cultural revolution which is replacing the former East/West political bipolarity with a globalised economic system in which the new Europe has a central rôle. Within this revolution, leisure, including recreation, culture and tourism, is constructed as the epitome of successful capitalist development; the very legitimisation of the global transmogrification from a production to a consumption orientation. While acting as a direct encouragement to the political transformation in many eastern European states, it is uncertain how the issue of leisure policy is being handled, given its centrality to the new economic order. This paper therefore examines the experience of western Europe, considering in particular the degree to which the newly-created Department of National Heritage in the UK provides a potential model for leisure development and policy integration in the new Europe. Despite an official rhetoric of support and promotion of leisure activities, reflecting the growing economic significance of tourism and the positive relationship between leisure provision and regional economic development, the paper establishes that in the place of the traditional rôle of the state in promoting leisure interests, the introduction of the Department has signified a shift to the use of leisure to promote the Government's interests, particularly in regenerating citizen rights claims towards the market. While an institution such as the Department of National Heritage may have relevance to emerging states as a element in the maintenance of political hegemony, therefore, it is questionable how far it can be viewed as a promoter or protector of leisure as a signifier of a newly-won political, economic and cultural freedom throughout Europe.

On the numerical integration of a class of singular perturbation problems

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.

Comment on “A modified leapfrog scheme for shallow water equations” by Wen-Yih Sun and Oliver M.T. Sun

A recent paper published in this journal considers the numerical integration of the shallow-water equations using the leapfrog time-stepping scheme [Sun Wen-Yih, Sun Oliver MT. A modified leapfrog scheme for shallow water equations. Comput Fluids 2011;52:69–72]. The authors of that paper propose using the time-averaged height in the numerical calculation of the pressure-gradient force, instead of the instantaneous height at the middle time step. The authors show that this modification doubles the maximum Courant number (and hence the maximum time step) at which the integrations are stable, doubling the computational efficiency. Unfortunately, the pressure-averaging technique proposed by the authors is not original. It was devised and published by Shuman [5] and has been widely used in the atmosphere and ocean modelling community for over 40 years.

Optimal convergence estimates for the trace of the polynomial L2-projection operator on a simplex

In this paper we study convergence of the L2-projection onto the space of polynomials up to degree p on a simplex in Rd, d >= 2. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in Rd from [P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163].

A priori error analysis of space–time Trefftz discontinuous Galerkin methods for wave problems

We present and analyse a space–time discontinuous Galerkin method for wave propagation problems. The special feature of the scheme is that it is a Trefftz method, namely that trial and test functions are solution of the partial differential equation to be discretised in each element of the (space–time) mesh. The method considered is a modification of the discontinuous Galerkin schemes of Kretzschmar et al. (2014) and of Monk & Richter (2005). For Maxwell’s equations in one space dimension, we prove stability of the method, quasi-optimality, best approximation estimates for polynomial Trefftz spaces and (fully explicit) error bounds with high order in the meshwidth and in the polynomial degree. The analysis framework also applies to scalar wave problems and Maxwell’s equations in higher space dimensions. Some numerical experiments demonstrate the theoretical results proved and the faster convergence compared to the non-Trefftz version of the scheme.

Noether-type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.

Integration of steroid hormone initiated membrane action to genomic function in the brain

Estrogen is a ligand for the estrogen receptor (ER), which on binding 17beta-estradiol, functions as a ligand-activated transcription factor and regulates the transcription of target genes. This is the slow genomic mode of action. However, rapid non-genomic actions of estrogen also exist at the cell membrane. Using a novel two-pulse paradigm in which the first pulse rapidly initiates non-genomic actions using a membrane-limited estrogen conjugate (E-BSA), while the second pulse promotes genomic transcription from a consensus estrogen response element (ERE), we have demonstrated that rapid actions of estrogen potentiate the slower transcriptional response from an ERE-reporter in neuroblastoma cells. Since rapid actions of estrogen activate kinases, we used selective inhibitors in the two-pulse paradigm to determine the intracellular signaling cascades important in such potentiation. Inhibition of protein kinase A (PKA), PKC, mitogen activated protein kinase (MAPK) or phosphatidylinositol 3-OH kinase (PI-3K) in the first pulse decreases potentiation of transcription. Also, our data with both dominant negative and constitutive mutants of Galpha subunits show that Galpha(q) initiates the rapid signaling cascade at the membrane in SK-N-BE(2)C neuroblastoma cells. We discuss two models of multiple kinase activation at the membrane Pulses of estrogen induce lordosis behavior in female rats. Infusion of E-BSA into the ventromedial hypothalamus followed by 17beta-estradiol in the second pulse could induce lordosis behavior, demonstrating the applicability of this paradigm in vivo. A model where non-genomic actions of estrogen couple to genomic actions unites both aspects of hormone action.