Verification of Invertibility of Complicated Functions over Large Domains

A new method to decide the invertibility of a given high-dimensional function over a domain is presented. The problem arises in the field of verified solution of differential algebraic equations (DAEs) related to the need to perform projections of certain constraint manifolds over large domains. The question of invertibility is reduced to a verified linear algebra problem involving first partials of the function under consideration. Different from conventional approaches, the elements of the resulting matrices are Taylor models for the derivatives of the functions.The linear algebra problem is solved based on Taylor model methods, and it will be shown the method is able to decide invertibility with a conciseness that often goes substantially beyond what can be obtained with other interval methods. The theory of the approach is presented. Comparisons with three other interval-based methods are performed for practical examples, illustrating the applicability of the new method.     

Development of vegetable composition databases based on available data for probabilistic nutrient and contaminant intake assessments

Dietary intake assessment can be considered as a two-step process consisting of collecting and evaluating food composition data, and combining these data with food consumption data. The purpose of this study was to develop databases of nutrient and contaminant concentrations in organic and conventional vegetables and potatoes based on internationally available secondary data. Databases, as described here, are important in (1) comparing the composition of similar foods (organic versus conventional) and (2) assessing probabilistically the combined intake of nutrients and contaminants when nutritional and toxicological dimensions of food consumption are being studied simultaneously.     

European consumers’ perceived importance of qualifying and disqualifying nutrients in food choices

Providing nutrition information through food labels is increasingly important in helping consumers making better informed food choices. Major questions are whether consumers perceive specific nutrients as valuable in food choices, and whether their perception differs for qualifying and disqualifying nutrients. Consumers placing high importance on nutrition are expected to use nutritional labels more. This paper reports on the perceived importance of qualifying nutrients (fibre, vitamins/minerals) and disqualifying nutrients (energy, fat, saturated fat, salt, sugars), and possible differences between consumer groups. A pan-European consumer survey (n = 4828) was conducted with samples representative for age, gender and region in Belgium, France, Italy, Norway, Poland and Spain. Overall, consumers perceive the nutritional value of foods as important when selecting foods, and even more important for qualifying than for disqualifying nutrients. Statistically significant but small differences are observed between countries. A higher perceived importance is reported by women, older respondents, dieters and more health conscious respondents. The effects of children in the household, education and BMI are very small. For age and gender compared to health consciousness, group differences are more significant regarding disqualifying nutrients than qualifying nutrients. Small effect sizes are found for dieting on the perceived importance of qualifying nutrients. Implications for nutrition policy makers and food industries are discussed.     

Controlling the Wrapping Effect in the Solution of ODEs for Asteroids

During the last decade, substantial progress has been made in fighting the wrapping effect in self-validated integrations of linear systems. However, it is still the main problem limiting the applicability of such methods to the long-term integration of non-linear systems. Here we show how high-order self-validated methods can successfully overcome this obstacle.We study and compare the validated integration of a Kepler problem with conventional and high-order methods represented by AWA and Taylor models, respectively. We show that this simple model problem exhibits significant wrapping that is particularly difficult to control for conventional first-order methods. It will become clear that utilizing high-order methods with shrink wrapping allows the system to be analyzed in a fully validated context over large integration times. By comparing high-order Taylor model integrations with Taylor model methods subjected to an artificial wrapping effect, we show that utilizing high-order methods to propagate initial conditions is indeed the foremost reason for the successful suppression of the wrapping effect.To further demonstrate that high-order Taylor model methods can be used for the integration of complicated non-linear systems, we summarize results obtained from a fully verified and self-validated orbit integration of the near earth asteroid 1997 XF11. Since this asteroid will have several close encounters with Earth, its analysis is an important application of reliable computations.     

Verified High-Order Inversion of Functional Depedencies and Interval Newton Methods

A new method for computing verified enclosures of the inverses of given functions over large domains is presented. The approach is based on Taylor Model methods, and the sharpness of the enclosures scales with a high order of the domain. These methods have applications in the solution of implicit equations and the Taylor Model based integration of Differential Algebraic Equations (DAE) as well as other tasks where obtaining verified high-order models of inverse functions is required. The accuracy of Taylor model methods has been shown to scale with the (n + 1)-st order of the underlying domain, and as a consequence, they are particularly well suited to model functions over relatively large domains. Moreover, since Taylor models can control the cancellation and dependency problems (see Makino, K. and Berz, M.: Efficient Control of the Dependency Problem Based on Taylor Model Methods, Reliable Computing 5(1) (1999)) that often affect regular interval techniques, the new method can successfully deal with complicated multidimensional problems. As an application of these new methods, a high-order extension of the standard Interval Newton method that converges approximately with the (n + 1)-st order of the underlying domain is developed. Several examples showing various aspects of the practical behavior of the methods are given. This revised version was published online in August 2006 with corrections to the Cover Date.