Eulerian-Lagrangian method of cells based on localized adjoint method

The localized adjoint method, when applied using an Eulerian-Lagrangian frame, has been quite successful in treating advection-dominated transport. The resulting methodology is known as ELLAM. In previous work, bilinear functions were used as test functions. In this paper, local constant functions are used instead, leading to procedures which are appealing because, in addition to other advantages of ELLAM methods, they ensure local mass conservation, are easy to apply and can be combined without difficulty with existing solute-transport codes which are based on finite volumes. In addition, the procedures for deriving the algorithms presented here are used as an illustration of a general methodology for treating numerically partial differential equations, which is advocated by the authors. Such methodology consists in identifying the information about the sought solution which is contained in the approximate one and then using this insight to choose the interpolation procedure to be applied. © 1994 John Wiley & Sons, Inc.     

Eulerian-Lagrangian localized adjoint methods for convection-diffusion equations and their convergence analysis

We develop and analyze Eulerian-Lagrangian localized adjointmethods (ellam) for convection-diffusion problems. The formulationuses space-time elements, with edges oriented along Lagrangianflow paths, in a time-marching scheme, where space-time testfunctions are chosen to satisfy a local adjoint condition. Thisallows Eulerian-Lagrangian concepts to be applied in a systematicmass-conservative manner to problems with general boundary conditions.In one space dimension with constant velocity, all combinationsof inflow and outflow Dirichlet, Neumann, or flux boundary conditionsare carefully considered, compared and discussed based on bothanalysis and numerical experiments. In some cases, the discreteunknowns include influxes, outfluxes, or resolution of the outflowingsolution finer than the time-step size. Optimal-order errorestimates in all cases and some superconvergence results areobtained. Numerical results show the strong potential of thesemethods and verify the theoretical estimates. Implementationsfor variable-coefficient problems in one and multiple spacedimensions, considered in detail elsewhere, are outlined.     

Optimal-order convergence rates for Eulerian-Lagrangian localized adjoint methods for reactive transport and contamination in groundwater

Eulerian–Langrangian localized adjoint methods (ELLAM) were developed to solve convection–diffusion–reaction equations governing contaminant transport in groundwater flowing through a porous medium, subject to various combinations of boundary conditions. In this article, we prove optimal-order error estimates and some superconvergence results for the ELLAM schemes. In contrast to many existing estimates for a variety of numerical methods, which often contain the temporal derivatives of the exact solution, our error estimates contain the total derivatives of the exact solution but do not involve any temporal derivatives of the exact solution. © 1995 John Wiley & Sons, Inc.     

The randomized Kaczmarz method with mismatched adjoint

This paper investigates the randomized version of the Kaczmarz method to solve linear systems in the case where the adjoint of the system matrix is not exact—a situation we refer to as “mismatched adjoint”. We show that the method may still converge both in the over- and underdetermined consistent case under appropriate conditions, and we calculate the expected asymptotic rate of linear convergence. Moreover, we analyze the inconsistent case and obtain results for the method with mismatched adjoint as for the standard method. Finally, we derive a method to compute optimized probabilities for the choice of the rows and illustrate our findings with numerical examples.     

Eulerian–Lagrangian localized adjoint methods for reactive transport with biodegradation

The microbial degradation of organic contaminants in the subsurface holds significant potential as a mechanism for in-situ remediation strategies. The mathematical models that describe contaminant transport with biodegradation involve a set of advective–diffusive–reactive transport equations. These equations are coupled through the nonlinear reaction terms, which may involve reactions with all of the species and are themselves coupled to growth equations for the subsurface bacterial populations. In this article, we develop Eulerian–Lagrangian localized adjoint methods (ELLAM) to solve these transport equations. ELLAM are formulated to systematically adapt to the changing features of governing partial differential equations. The relative importance of retardation, advection, diffusion, and reaction is directly incorporated into the numerical method by judicious choice of the test functions that appear in the weak form of the governing equation. Different ELLAM schemes for linear variable–coefficient advective–diffusive–reactive transport equations are developed based on different operator splittings. Specific linearization techniques are discussed and are combined with the ELLAM schemes to solve the nonlinear, multispecies transport equations. © 1995 John Wiley & Sons, Inc.     

Airline pricing and fare product differentiation: A new theoretical framework

A basic premise in the development of yield management has been that the differentiated fare products offered by airlines are targeted to distinct segments of the total demand for air travel in a market, each of which compete for space on a fixed capacity aircraft. Such representations of differential pricing assume that the airline can segment its demand perfectly and without cost to consumers, and further, ignore the dependence of the demand for a given fare product on the price levels and characteristics of the other available fare products. In this paper, we introduce a new ‘generalised cost’ model of fare product differentiation that incorporates the relationships between available airline fare products as well as the cost incurred by consumers of accepting more restrictions. We extend the model to incorporate the diversion of passengers to lower-priced fare products as a result of their ability to meet the additional restrictions imposed by airlines, and then demonstrate how seat inventory control can be used to induce diverting passengers to ‘sell up’ to higher-priced fare products by applying booking limits. An example of how the model can be used for joint fare product price level optimisation is presented, along with a numerical example that illustrates the extent to which the conventional model of price discrimination over-estimates passenger demand and, in turn, total airline revenues.     

The complete matrix ring over an adjoint regular ring is adjoint regular

We prove that the ring of all n×n matrices over an adjoint regular ring is adjoint regular, thus confirming a longstanding conjecture in the theory of adjoint semigroups.     

Approximate dual Gabor atoms via the adjoint lattice method

Regular Gabor frames for \({\boldsymbol {L}{^{2}}(\mathbb {R}^d)}\) are obtained by applying time-frequency shifts from a lattice in \(\boldsymbol {\Lambda } \vartriangleleft {\mathbb {R}^{d} \times \mathbb {\widehat {R}}}\) to some decent so-called Gabor atom g, which typically is something like a summability kernel in classical analysis, or a Schwartz function, or more generally some \(g \in {\boldsymbol {S}_{0}(\mathbb {R}^{d})}\) . There is always a canonical dual frame, generated by the dual Gabor atom \({\widetilde g}\) . The paper promotes a numerical approach for the efficient calculation of good approximations to the dual Gabor atom for general lattices, including the non-separable ones (different from \({a\mathbb {Z}^{d}\,{\times }\,b\mathbb {Z}^{d}}\) ). The theoretical foundation for the approach is the well-known Wexler-Raz biorthogonality relation and the more recent theory of localized frames. The combination of these principles guarantees that the dual Gabor atom can be approximated by a linear combination of a few time-frequency shifted atoms from the adjoint lattice \(\boldsymbol {\Lambda }\circ\) . The effectiveness of this approach is justified by a new theoretical argument and demonstrated by numerical examples.     

The application of adjoint method for shape optimization in Stokes–Oseen flow

This paper presents a numerical method for shape optimization of a body immersed in an incompressible viscous flow governed by Stokes–Oseen equations. The purpose of this work is to optimize the shape that minimizes a given cost functional. Based on the continuous adjoint method, the shape gradient of the cost functional is derived by involving a Lagrangian functional with the function space parametrization technique. Then, a gradient‐type algorithm is applied to the shape optimization problem. The numerical examples indicate the proposed algorithm is feasible and effective in low Reynolds number flow. Copyright © 2016 John Wiley & Sons, Ltd.     

An Eulerian-Lagrangian approach for incompressible fluids: Local theory

We study a formulation of the incompressible Euler equations in terms of the inverse Lagrangian map. In this formulation the equations become a first order advective nonlinear system of partial differential equations.

     

Solution of the advective-dispersive transport equation using a least squares collocation,Eulerian-Lagrangian method

Numerical solution of the advective-dispersive transport equation is difficult when advection dominates. Difficulties arise because of the first-order spatial derivatives which can be elminated by a local coordinate transformation to the characteristic lines of the first order hyperbolic portion of the equation. The resulting differential equation is discretized using a finite difference in time and finite elements in space employing cubic Hermite basis functions. The residuals at individual collocation points are then computed. The sum of the squares of the residuals is minimized to form the necessary set of algebraic equations. The method has performed well in one-dimensional test problems.     

One-sided elasticities and technical efficiency in multi-output production: A theoretical framework

One of the concepts that have sparked considerable interest in the theory of production and efficiency is that of returns to scale (RTS). Economics researchers typically define RTS using the notion of elasticity. Considerable research activity on RTS has also been observed by management science researchers, who utilize the methodology of Data Envelopment Analysis (DEA) to gain insights on RTS. In this paper, we present a theoretical framework that integrates existing economics and management science literature on RTS, and provides a solid foundation for research work in this area. Our framework defines, discusses, and proposes an approach to measure input- and output-oriented elasticities, and one-sided RTS. We demonstrate how the work done in DEA is a special case of our framework, and discuss the conditions under which the resulting two left-hand, and the two right-hand elasticities can be equal. Future research directions are also discussed.     

Nodal adjoint method for multigroup diffusion in heterogeneous slab reactors

A variational adjoint nodal method is proposed to yield an approximation theory for the eigenvalues of the multigroup neutron diffusion boundary value problem of a heterogeneous quasicritical one-dimensional slab reactor. Semianalytical eigensolutions are constructed for the associated whole-reactor group nodal fluxes. The method appears to be mathematically more consistent, computationally more straightforward, and practically more convenient than alternative nodal or finite element schemes. © 1992 John Wiley & Sons, Inc.     

Application of regularization technique to variational adjoint method: A case for nonlinear convection-diffusion problem

A discrete assimilation system for a one-dimensional variable coefficient convection-diffusion equation is constructed. The variational adjoint method combined with the regularization technique is employed to retrieve the initial condition and diffusion coefficient with the aid of a set of simulated observations. Several numerical experiments are performed: (a) retrieving both the initial condition and diffusion coefficient jointly (Experiment JR), (b) retrieving either of them separately (Experiment SR), (c) retrieving only the diffusion coefficient with the iteration count increased to 800 (Experiment NoR-SR), and (d) retrieving only the diffusion coefficient with the consideration of a regularization term based on the Experiment NoR-SR (Experiment AdR-SR). The results indicate that within the limit of 100 iterations, the retrieval quality of the Experiment SR is better than those from the Experiment JR. Compared with the initial condition, the diffusion coefficient is a little difficult to retrieve, whereas we still achieve the desired result by increasing the iterations or integrating the regularization term into the cost functional for the improvement with respect to the diffusion coefficient. Further comparisons between the Experiment NoR-SR and AdR-SR show that the regularization term can really help not only improve the precision of retrieval to a large extent, but also speed up the convergence of solution, even if some perturbations are imposed on those observations.     

Towards an integrated theory of mathematics conceptual learning and instructional design: The Learning Through Activity theoretical framework

We discuss the theoretical framework of the Learning Through Activity research program. The framework includes an elaboration of the construct of mathematical concept, an elaboration of Piaget’s reflective abstraction for the purpose of mathematics pedagogy, further development of a distinction between two stages of conceptual learning, and a typology of different reverse concepts. The framework also involves instructional design principles built on those constructs, including steps for the design of task sequences, development of guided reinvention, and ways of promoting reversibility of concepts. This article represents both a synthesis of prior work and additions to it.     

The uniqueness of the adjoint operation I

Let H be a complex, infinite-dimensional Hilbert space. Let B(H) denote the set of bounded linear operators on H. This paper contains a nonlinear characterization of the adjoint operation on B(H). The statement of this result is:THEOREM:Let h: B(H) B(H)be a function such that h(I)0.Then h(ST)=h(T)h(S)and h(S)S0for all elements Sand Tof B(H)if and only if h(S)=S^* for all S B(H).     

The uniqueness of the adjoint operation. II

Let H be a complex, finite-dimensional Hilbert space, and let L(H) denote the set of linear transformations mapping H into itself. For certain interesting subsets A(H) of L(H) [nonsingular transformations and L(H) are examples], the functions h: A(H) → L(H) which have the properties h(ST) = h(T)h(S) and h(S)S ⩾ 0 are characterized.     

Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework

The article contributes to the ongoing discussion on ways to deal with the diversity of theories in mathematics education research. It introduces and systematizes a collection of case studies using different strategies and methods for networking theoretical approaches which all frame (qualitative) empirical research. The term ‘networking strategies’ is used to conceptualize those connecting strategies, which aim at reducing the number of unconnected theoretical approaches while respecting their specificity. The article starts with some clarifications on the character and role of theories in general, before proposing first steps towards a conceptual framework for networking strategies. Their application by different methods as well as their contribution to the development of theories in mathematics education are discussed with respect to the case studies in the ZDM-issue.