Frank Uhlig

Mark Colley, Sebastian Pickl, Frank Uhlig et al.

The automation of the driving task affects both the primary driving task and the automotive user interfaces. The liberation of user interface space and cognitive load on the driver allows for new ways to think about driving. Related work showed that activities such as sleeping, watching TV, or working will become more prevalent in the future. However, social aspects according to Maslow's hierarchy of needs have not yet been accounted for. We provide insights of a focus group with N=5 experts in automotive user experience revealing current practices such as social need fulfillment on journeys and sharing practices via messengers and a user study with N=12 participants of a first prototype supporting these needs in various automation levels showing good usability and high potential to improve user experience.

Frank Uhlig, Yunong Zhang

This paper adapts look-ahead and backward finite difference formulas to compute future eigenvectors and eigenvalues of piecewise smooth time-varying symmetric matrix flows $A(t)$. It is based on the Zhang Neural Network (ZNN) model for time-varying problems and uses the associated error function $E(t) = A(t)V(t) - V(t) D(t)$ or $e_i(t) = A(t)v_i(t) -\la_i(t)v_i(t)$ with the Zhang design stipulation that $\dot E(t) = - ηE(t)$ or $\dot e_i(t) = - ηe_i(t)$ with $η> 0$ so that $E(t)$ and $e(t)$ decrease exponentially over time. This leads to a discrete-time differential equation of the form $P(t_k) \dot z(t_k) = q(t_k)$ for the eigendata vector $z(t_k)$ of $A(t_k)$. Convergent look-ahead finite difference formulas of varying error orders then allow us to express $z(t_{k+1})$ in terms of earlier $A$ and $z$ data. Numerical tests, comparisons and open questions complete the paper.

Frank Uhlig

In this paper a new and different neural network, called Zhang Neural Network (ZNN) is appropriated from discrete time-varying matrix problems and applied to the angle parameter-varying matrix field of values (FoV) problem. This problem acts as a test bed for newly discovered convergent 1-step ahead finite difference formulas of high truncation orders. The ZNN method that uses a look-ahead finite difference scheme of error order 6 gives us 15+ accurate digits of the FoV boundary in record time when applied to hermitean matrix flows $A(t)$.

Frank Uhlig

Zhang Neural Networks rely on convergent 1-step ahead finite difference formulas of which very few are known. Those which are known have been constructed in ad-hoc ways and suffer from low truncation error orders. This paper develops a constructive method to find convergent look-ahead finite difference schemes of higher truncation error orders. The method consists of seeding the free variables of a linear system comprised of Taylor expansion coefficients followed by a minimization algorithm for the maximal magnitude root of the formula's characteristic polynomial. This helps us find new convergent 1-step ahead finite difference formulas of any truncation error order. Once a polynomial has been found with roots inside the complex unit circle and no repeated roots on it, the associated look-ahead ZNN discretization formula is convergent and can be used for solving any discretized ZNN based model. Our method recreates and validates the few known convergent formulas, all of which have truncation error orders at most 4. It also creates new convergent 1-step ahead difference formulas with truncation error orders 5 through 8.