Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

The use of a formal sensitivity analysis on epidemic models with immune protection from maternally acquired antibodies

This paper considers the outcome of a formal sensitivity analysis on a series of epidemic model structures developed to study the population level effects of maternal antibodies. The analysis is used to compare the potential influence of maternally acquired immunity on various age and time domain observations of infection and serology, with and without seasonality. The results of the analysis indicate that time series observations are largely insensitive to variations in the average duration of this protection, and that age related empirical data are likely to be most appropriate for estimating these characteristics

Towards Reduced-order Model Accelerated Optimization for Aerodynamic Design

The adoption of mathematically formal simulation-based optimization approaches within aerodynamic design depends upon a delicate balance of affordability and accessibility. Techniques are needed to accelerate the simulation-based optimization process, but they must remain approachable enough for the implementation time to not eliminate the cost savings or act as a barrier to adoption. This dissertation introduces a reduced-order model technique for accelerating fixed-point iterative solvers (e.g. such as those employed to solve primal equations, sensitivity equations, design equations, and their combination). The reduced-order model-based acceleration technique collects snapshots of early iteration (pre-convergent) solutions and residuals and then uses them to project to significantly more accurate solutions, i.e. smaller residual. The technique can be combined with other convergence schemes like multigrid and adaptive timestepping. The technique is generalizable and in this work is demonstrated to accelerate steady and unsteady flow solutions; continuous and discrete adjoint sensitivity solutions; and one-shot design optimization solutions. This final application, reduced-order model accelerated one-shot optimization approach, in particular represents a step towards more efficient aerodynamic design optimization. Through this series of applications, different basis vectors were considered and best practices for snapshot collection procedures were outlined. The major outcome of this dissertation is the development and demonstration of this reduced-order model acceleration technique. This work includes the first application of the reduced-order model-based acceleration method to an explicit one-shot iterative optimization process