A general approach to D-optimal designs for weighted univariate polynomial regression models

We study the D-optimal design problem for the common weighted univariate polynomial regression model with efficiency function l. We characterize the efficiency functions for which an explicit solution of the D-optimal design problem is available based on a differential equation for the logarithmic derivative of the efficiency function. In contrast to the common approach which starts with a given efficiency function and derives a differential equation for the supporting polynomial of the D-optimal design, we derive a differential equation for the efficiency function, such that an explicit solution of the D-optimal design problem is possible. The approach is illustrated for various convex design spaces and is depicted in several new examples. Also, this concept incorporates all classical efficiency functions discussed in the literature so far

Uncertainty Relations for Angular Momentum

In this work we study various notions of uncertainty for angular momentum in the spin-s representation of SU(2). We characterize the "uncertainty regions'' given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s. Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s, since they are saturated by states taken from the Holstein-Primakoff approximation. We show that, for all s, all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius sqrt(s(s+1)). Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s. For large s the Maassen-Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r(s), where r(s)/s goes to 1 as s tends to infinity.Comment: 30 pages, 22 figures, 1 cut-out paper model, video abstract available on https://youtu.be/h01pHekcwF

Boundary current fluctuations for the half space ASEP and six vertex model

We study fluctuations of the current at the boundary for the half space asymmetric simple exclusion process (ASEP) and the height function of the half space six vertex model at the boundary at large times. We establish a phase transition depending on the asymptotic density of particles at the boundary, with GSE and GOE limits as well as the Baik--Rains crossover distribution near the critical point. This was previously known for half space last passage percolation, and recently established for the half space log-gamma polymer and KPZ equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto. The proof uses the underlying algebraic structure of these models in a crucial way to obtain exact formulas. In particular, we show a relationship between the half space six vertex model and a half space Hall--Littlewood measure with two boundary parameters, which is then matched to a free boundary Schur process via a new identity of symmetric functions. Fredholm Pfaffian formulas are established for the half space ASEP and six vertex model, indicating a hidden free fermionic structure.Comment: 60 pages, many figures. Comments are welcome

Stochastic Systems with Cumulative Prospect Theory

Stochastic control problems arise in many fields. Traditionally, the most widely used class of performance criteria in stochastic control problems is risk-neutral. More recent attempts at introducing risk-sensitivity into stochastic control problems include the application of utility functions. The decision theory community has long debated the merits of using expected utility for modeling human behaviors, as exemplified by the Allais paradox. Substantiated by strong experimental evidence, Cumulative Prospect Theory (CPT) based performance measures have been proposed as alternatives to expected utility based performance measures for evaluating human-centric systems. Our goal is to study stochastic control problems using performance measures derived from the cumulative prospect theory. The first part of this thesis solves the problem of evaluating Markov decision processes (MDPs) using CPT-based performance measures. A well-known method of solving MDPs is dynamic programming, which has traditionally been applied with an expected utility criterion. When the performance measure is CPT-inspired, several complications arise. Firstly, when solving a problem via dynamic programming, it is important that the performance criterion has a recursive structure, which is not true for all CPT-based criteria. Secondly, we need to prove the traditional optimality criteria for the updated problems (i.e., MDPs with CPT-based performance criteria). The theorems stated in this part of the thesis answer the question: what are the conditions required on a CPT-inspired criterion such that the corresponding MDP is solvable via dynamic programming? The second part of this thesis deals with stochastic global optimization problems. Using ideas from the cumulative prospect theory, we are able to introduce a novel model-based randomized optimization algorithm: Cumulative Weighting Optimization (CWO). The key contributions of our research are: 1) proving the convergence of the algorithm to an optimal solution given a mild assumption on the initial condition; 2) showing that the well-known cross-entropy optimization algorithm is a special case of CWO-based algorithms. To the best knowledge of the author, there is no previous convergence proof for the cross-entropy method. In practice, numerical experiments have demonstrated that a CWO-based algorithm can find a better solution than the cross-entropy method. Finally, in the future, we would like to apply some of the ideas from cumulative prospect theory to games. In this thesis, we present a numerical example where cumulative prospect theory has an unexpected effect on the equilibrium points of the classic prisoner's dilemma game

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Numerical Solution of Optimal Control Problems with Explicit and Implicit Switches

This dissertation deals with the efficient numerical solution of switched optimal control problems whose dynamics may coincidentally be affected by both explicit and implicit switches. A framework is being developed for this purpose, in which both problem classes are uniformly converted into a mixed–integer optimal control problem with combinatorial constraints. Recent research results relate this problem class to a continuous optimal control problem with vanishing constraints, which in turn represents a considerable subclass of an optimal control problem with equilibrium constraints. In this thesis, this connection forms the foundation for a numerical treatment. We employ numerical algorithms that are based on a direct collocation approach and require, in particular, a highly accurate determination of the switching structure of the original problem. Due to the fact that the switching structure is a priori unknown in general, our approach aims to identify it successively. During this process, a sequence of nonlinear programs, which are derived by applying discretization schemes to optimal control problems, is solved approximatively. After each iteration, the discretization grid is updated according to the currently estimated switching structure. Besides a precise determination of the switching structure, it is of central importance to estimate the global error that occurs when optimal control problems are solved numerically. Again, we focus on certain direct collocation discretization schemes and analyze error contributions of individual discretization intervals. For this purpose, we exploit a relationship between discrete adjoints and the Lagrange multipliers associated with those nonlinear programs that arise from the collocation transcription process. This relationship can be derived with the help of a functional analytic framework and by interrelating collocation methods and Petrov–Galerkin finite element methods. In analogy to the dual-weighted residual methodology for Galerkin methods, which is well–known in the partial differential equation community, we then derive goal–oriented global error estimators. Based on those error estimators, we present mesh refinement strategies that allow for an equilibration and an efficient reduction of the global error. In doing so we note that the grid adaption processes with respect to both switching structure detection and global error reduction get along with each other. This allows us to distill an iterative solution framework. Usually, individual state and control components have the same polynomial degree if they originate from a collocation discretization scheme. Due to the special role which some control components have in the proposed solution framework it is desirable to allow varying polynomial degrees. This results in implementation problems, which can be solved by means of clever structure exploitation techniques and a suitable permutation of variables and equations. The resulting algorithm was developed in parallel to this work and implemented in a software package. The presented methods are implemented and evaluated on the basis of several benchmark problems. Furthermore, their applicability and efficiency is demonstrated. With regard to a future embedding of the described methods in an online optimal control context and the associated real-time requirements, an extension of the well–known multi–level iteration schemes is proposed. This approach is based on the trapezoidal rule and, compared to a full evaluation of the involved Jacobians, it significantly reduces the computational costs in case of sparse data matrices