Automatic Frechet differentiation for the numerical solution of boundary-value problems

A new solver for nonlinear boundary-value problems (BVPs) in Matlab is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton's method in function space, where instead of the usual Jacobian matrices, the derivatives involved are Frechet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Frechet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable to enable a recursive, delayed evaluation of derivatives with forward mode AD. The AD techniques are applied within a new Chebfun class called chebop which allows users to set up and solve nonlinear BVPs in a few lines of code, using the "nonlinear backslash" operator (\). This framework enables one to study the behaviour of Newton's method in function space

Investment Subsidies and Regional Welfare: A Dynamic Framework

Subsidising investment in lagging regions is an important regional policy instrument in many countries. Some argue that this instrument is not specific enough to concentrate the aid towards the regions that are lagging behind most, because investment subsidies benefit capital owners who might reside elsewhere, possibly in very rich places. Checking under which conditions this is true is thus highly policy relevant. The present paper studies regional investment subsidies in a multiregional neoclassical dynamic framework. We set up a model with trade in heterogeneous goods, with a perfectly integrated financial capital market and sluggish adjustment of regional capital stocks. Consumers and investors act under perfect foresight. We derive the equilibrium system, show how to solve it, and simulate actual European regional subsidies in computational applications. We find that the size of the welfare gains depends on the portfolio distribution held by the households. If households own diversified asset portfolios, we find that the supported regions gain roughly the amounts that are allocated to them in the form of investment subsidies. If they only own local capital stocks, a part of the money is lost through the drop in share prices. From the point of view of total welfare, the subsidy is not efficient. It can lead to a welfare loss for the EU as a whole and definitely leads to welfare losses in the rest of the world, from where investment ows to the supported EU regions

A problem-solving environment for the numerical solution of nonlinear algebraic equations

Nonlinear algebraic equations (NAEs) occur in many areas of science and engineering. The process of solving these NAEs is generally difficult, from finding a good initial guess that leads to a desired solution to deciding on convergence criteria for the approximate solution. In practice, Newton's method is the only robust general-purpose method for solving a system of NAEs. Many variants of Newton's method exist. However, it is generally impossible to know a priori which variant of Newton's method will be effective for a given problem.Many high-quality software libraries are available for the numerical solution of NAEs. However, the user usually has little control over many aspects of what the library does. For example, the user may not be able to easily switch between direct and indirect methods for the linear algebra. This thesis describes a problem-solving environment (PSE) called pythNon for studying the effects (e.g., performance) of different strategies for solving systems of NAEs. It provides the researcher, teacher, or student with a flexible environment for rapid prototyping and numerical experiments. In pythNon, users can directly influence the solution process on many levels, e.g., investigation of the effects of termination criteria and/or globalization strategies. In particular, to show the power, flexibility, and ease of use of the pythNon PSE, this thesis also describes the development of a novel forcing-term strategy for approximating the Newton direction efficiently in the pythNon PSE