Verified Integration of ODEs and Flows Using Differential Algebraic Methods on High-Order Taylor Models

A method is developed that allows the verified integration of ODEs based on local modeling with high-order Taylor polynomials with remainder bound. The use of such Taylor models of order n allows convenient automated verified inclusion of functional dependencies with an accuracy that scales with the (n + 1)-st order of the domain and substantially reduces blow-up.Utilizing Schauder's fixed point theorem on certain suitable compact and convex sets of functions, we show how explicit nth order integrators can be developed that provide verified nth order inclusions of a solution of the ODE. The method can be used not only for the computation of solutions through a single initial condition, but also to establish the functional dependency between initial and final conditions, the so-called flow of the ODE. The latter can be used efficiently for a substantial reduction of the wrapping effect.Examples of the application of the method to conventional initial value problems as well as flows are given. The orders of the integration range up to twelve, and the verified inclusions of up to thirteen digits of accuracy have been demanded and obtained.     

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.     

Carrier heating effects in junctions at very low currents

The effects of carrier heating by the electric field are analysed in abrupt $\text{p-n}$ junctions at very small currents. Two features, which are usually neglected, are shown to be of importance. (a) The heat transported by the carriers plays an appreciable part in the energy balance equation, which determines the carrier temperature. (b) The current has, in the junction and in the neighbourhood, two important additional components, one being due to a correction of the diffusion current, and the other being proportional to the gradient in the temperature of the carriers.     

A simplified theory of the p-i-n diode

A simplified theory of the p-i-n diode is developed for the case of heavy injection in the base, and light injection in the end regions. The current I is taken as the fundamental parameter. The carrier densities n₁ and n₂ at the base boundaries are given directly as functions of the current I by means of simple approximate expressions. The approximation improves with increasing base width and increasing current. It is always within 15% of the correct value for base widths of about 2 diffusion lengths or higher. The accuracy is much better still on the voltage drop across the device, as calculated from the approximate equations for the carrier densities within the base.     

Windstorm and storm surges in Europe: loss trends and possible counter-actions from the viewpoint of an international reinsurer

Windstorm disasters (including storm surges) account for about one-third of all natural disasters throughout the world (by number, fatalities and economic losses), but for more than two-thirds of the corresponding insured losses. Trend analyses reveal that major windstorm disasters and the losses generated by them have increased drastically in recent decades. Risk partnership between the state, the affected population and the insurance industry assumes a key role with regard to the windstorm hazard. Scientists, engineers and insurers must work together in formulating their requirements and shaping them in such a way that politicians can derive clearly recognizable policy options (e.g. land-use, restrictions, design-code adjustments) from them. Another important aspect is stepping up the efforts being made towards curbing climate change, which will, otherwise, exacerbate the risk in the future.     

New Methods for High-Dimensional Verified Quadrature

Conventional verified methods for integration often rely on the verified bounding of analytically derived remainder formulas for popular integration rules. We show that using the approach of Taylor models, it is possible to devise new methods for verified integration of high order and in many variables. Different from conventional schemes, they do not require an a-priori derivation of analytical error bounds, but the rigorous bounds are calculated automatically in parallel to the computation of the integral.The performance of various schemes are compared for examples of up to order ten in up to eight variables. Computational expenses and tightness of the resulting bounds are compared with conventional methods.     

Rigorous global optimization of impulsive planet-to-planet transfers in the patched-conics approximation

The rigorous solution of a generic impulsive planet-to-planet transfer by means of a Taylor-model-based global optimizer is presented. Although a planet-to-planet transfer represents the simplest case of interplanetary transfer, its formulation and solution is a challenging task when the rigorous global optimum is sought. A customized ephemeris function is derived from JPL DE405 to allow the Taylor-model evaluation of planets’ positions and velocities. Furthermore, the validated solution of Lambert's problem is addressed for the rigorous computation of transfer fuel consumption. The optimization problem, which consists in finding the optimal launch and transfer time to minimize the required fuel mass, is complex due to the abundance of local minima and relatively high search-space dimension. Its rigorous solution by means of the Taylor-model-based global optimizer COSY-GO is presented considering Earth–Mars and Earth–Venus transfers as test cases.     

The Bethe condition for thermionic emission near an absorbing boundary

The Bethe condition for thermionic emission in Schottky diodes is analysed in terms of the transport of carriers near the top of an absorbing barrier, under forward bias. It is shown that the usually accepted values of the drift velocity v_s and of the carrier density n(0) at the barrier maximum are not correct, v_s being underestimated by a factor of 2, and n(0) being overestimated by about the same factor. Fortunately, these discrepancies compensate in the product n(0)v_s, which yields the correct forward current. It is shown also by means of examples that the Bethe condition is satisfied only over a relatively limited range.