Stefan Kiel

A verified method for solving piecewise smooth initial value problems

Abstract In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VALENCIA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

Uses of GPU Powered Interval Optimization for Parameter Identification in the Context of SO Fuel Cells

Abstract In this paper, we discuss parameter identification for models based on ordinary differential equations in the context of solid oxide fuel cells. In this case, verified methods (e.g. interval analysis), which provide a guarantee of correctness for the computed result, can be of great help for dealing with the appearing uncertainty and for devising accurate control strategies. Moreover, interval arithmetic can be used to discard infeasible areas of parameter space in a natural way and so to improve the results of traditional numerical algorithms. We describe a simulation environment interfacing different verified and floating point based approaches and show how the interchangeability between different techniques enhances parameter identification. Additionally, we give details on a possible parallelization of our version of the global interval optimization algorithm on the CPU and the GPU. The applicability of the method and the features of the environment are demonstrated with the help of different fuel cell models.

Verified distance computation between non-convex superquadrics using hierarchical space decomposition structures

Verified distance computation is an important task in various application domains. In some domains a proof of correctness is crucial. In this paper, we show how we can apply the methods provided by our uniform framework for verified geometric computations to derive verified bounds on the distance between non-convex objects. The framework features a layered structure enabling the algorithm to run independently whether the objects are described by implicit functions or parametric ones or by polyhedrons. The approach is based on the use of adaptively constructed hierarchical decompositions of the models. As a practical example we use various scenarios occurring in automatic surgery assistance systems for total hip replacement (THR). To ensure that an implant selected by the system fits into the patient’s femoral shaft, we have to derive verified bounds on the distance between them. In this case, the models are either superquadrics or polyhedrons, both of which can be non-convex We first show how to increase the enclosure quality of implicit objects by incorporating interval contractors into the hierarchical space decomposition. Next, we describe the construction of a decomposition structure for parametric objects. After that, we present an improvement of the case selector for computing the distance between interval tree nodes, yielding tighter results. We also show how to integrate surface normals into the algorithm if first-order information is available and how to accelerate the solving process by incorporating information gained by non-verified floating-point solvers. Finally, we provide numerical results for all distance query types occurring during the THR procedure and examine whether it is advisable to perform the computation on the implicit model or on the parametric one if both are available. Further numerical results are presented for test cases involving contractors in the decomposition structures.

A Comparison of verified distance computation between implicit objects using different arithmetics for range enclosure

This paper describes a new algorithm for computing verified bounds on the distance between two arbitrary fat implicit objects. The algorithm dissects the objects into axis-aligned boxes by constructing an adaptive hierarchical decomposition during runtime. Actual distance computation is performed on the cubes independently of the original object’s complexity. As the whole decomposition process and the distance computation are carried out using verified techniques like interval arithmetic, the calculated bounds are rigorous. In the second part of the paper, we test our algorithm using 18 different test cases, split up into 5 groups. Each group represents a different level of complexity, ranging from simple surfaces like the sphere to more complex surfaces like the Kleins bottle. The algorithm is independent of the actual technique for range bounding, which allows us to compare different verified arithmetics. Using our newly developed uniform framework for verified computations, we perform tests with interval arithmetic, centered forms, affine arithmetic and Taylor models. Finally, we compare them based on the time needed for deriving verified bounds with a user defined accuracy.

Verified spatial subdivision of implicit objects using implicit linear interval estimations

In this paper we describe the LIETree, a new data structure for verified spatial decomposition of implicit objects. The LIETree is capable of utilizing implicit linear interval estimations for calculating a verified enclosure of the implicit functions codomain. Furthermore, it uses consistency techniques to tighten the object enclosure. Overall, it delivers improved accuracy and uses fewer nodes than common uniform subdivision schemes using interval or affine arithmetic for enclosure.

A Flexible Environment for Accurate Simulation, Optimization, and Verification of SOFC Models

The technology of solid oxide fuel cells is one of the important topics in modern engineering. One research direction is to design robust and accurate control strategies for this kind of fuel cells based on models spatially discretized into ordinary differential equations. To allow users to employ new models and techniques easily in combination with different verified and traditional tools, we implement the environment VERICELL. It features an intuitive graphic interface for construction of fuel cell models from predefined building blocks. In this paper, we describe modeling and verification possibilities available in VERICELL for the analysis of solid oxide fuel cells.

Control-Oriented Models for SO Fuel Cells from the Angle of V&V: Analysis, Simplification Possibilities, Performance

In this paper, we take a look at the analysis and parameter identification for control-oriented, dynamic models for the thermal subsystem of solid oxide fuel cells (SOFC) from the systematized point of view of verification and validation (V&V). First, we give a possible classification of models according to their verification degree which depends, for example, on the kind of arithmetic used for both formulation and simulation. Typical SOFC models, consisting of several coupled differential equations for gas preheaters and the temperature distribution in the stack module, do not have analytical solutions because of spatial nonlinearity. Therefore, in the next part of the paper, we describe in detail two possible ways to simplify such models so that the underlying differential equations can be solved analytically while still being sufficiently accurate to serve as the basis for control synthesis. The simplifying assumption is to approximate the heat capacities of the gases by zero-order polynomials (or first-oder polynomials, respectively) in the temperature. In the last, application-oriented part of the paper, we identify the parameters of these models as well as compare their performance and their ability to reflect the reality with the corresponding characteristics of models in which the heat capacities are represented by quadratic polynomials (the usual case). For this purpose, the framework UniVerMeC (Unified Framework for Verified GeoMetric Computations) is used, which allows us to employ different kinds of arithmetics including the interval one. This latter possibility ensures a high level of reliability of simulations and of the subsequent validation. Besides, it helps to take into account bounded uncertainty in measurements.