Improving Model-Based Software Synthesis: A Focus on Mathematical Structures

Computer hardware keeps increasing in complexity. Software design needs to keep up with this. The right models and abstractions empower developers to leverage the novelties of modern hardware. This thesis deals primarily with Models of Computation, as a basis for software design, in a family of methods called software synthesis. We focus on Kahn Process Networks and dataflow applications as abstractions, both for programming and for deriving an efficient execution on heterogeneous multicores. The latter we accomplish by exploring the design space of possible mappings of computation and data to hardware resources. Mapping algorithms are not at the center of this thesis, however. Instead, we examine the mathematical structure of the mapping space, leveraging its inherent symmetries or geometric properties to improve mapping methods in general. This thesis thoroughly explores the process of model-based design, aiming to go beyond the more established software synthesis on dataflow applications. We starting with the problem of assessing these methods through benchmarking, and go on to formally examine the general goals of benchmarks. In this context, we also consider the role modern machine learning methods play in benchmarking. We explore different established semantics, stretching the limits of Kahn Process Networks. We also discuss novel models, like Reactors, which are designed to be a deterministic, adaptive model with time as a first-class citizen. By investigating abstractions and transformations in the Ohua language for implicit dataflow programming, we also focus on programmability. The focus of the thesis is in the models and methods, but we evaluate them in diverse use-cases, generally centered around Cyber-Physical Systems. These include the 5G telecommunication standard, automotive and signal processing domains. We even go beyond embedded systems and discuss use-cases in GPU programming and microservice-based architectures

Fast exact algorithms for optimization problems in resource allocation and switched linear systems

University of Minnesota Ph.D. dissertation.June 2019. Major: Industrial Engineering. Advisor: Qie He. 1 computer file (PDF); x, 138 pages.Discrete optimization is a branch of mathematical optimization where some of the decision variables are restricted to real values in a discrete set. The use of discrete decision variables greatly expands the scope and capacity of mathematical optimization models. In the era of big data, efficiency and scalability are increasingly important in evaluating the performance of an algorithm. However, discrete optimization problems usually are challenging to solve. In this thesis, we develop new fast exact algorithms for discrete optimization problems arising in the field of resource allocation and switched linear systems. The first problem is the discrete resource allocation problem with nested bound constraints. It is a fundamental problem with a wide variety of applications in search theory, economics, inventory systems, etc. Given $B$ units of resource and $n$ activities, each of which associated with a convex allocation cost $f_i(\cdot)$, we aim to find an allocation of resources to the $n$ activities, denoted by \bm{x} \in \Ze^n, to minimize the total allocation cost $\sum\limits_{i = 1}^{n} f_i(x_i)$ subject to the total amount of resource constraint as well as lower and upper bound constraints on total resource allocated to subsets of activities. We develop a $\Theta(n^2\log\frac{B}{n})$-time algorithm for it. It is an infeasibility-guided divide-and-conquer algorithm and the worst-case complexity is usually not achieved. Numerical experiments demonstrate that our algorithm significantly outperforms a state-of-the-art optimization solver and the performance of our algorithm is competitive compared to the algorithm with the best worst-case complexity for this problem in the literature. The second problem is the minimum convex cost network flow problem on the dynamic lot size network. In the dynamic lot size network, there are one source node and $n$ sink nodes with demand $d_i, i = 1, \dots, n$. Let $B = \sum_{i=0}^{n}d_i$ be the total demand. We aim to find a flow $\bm{x}$ to minimize the total arc cost and satisfy all the flow balance and capacity constraints. Many optimization models in the literature can be seen as special cases of this problem, including dynamic lot-sizing problem and speed optimization. It is also a generalization of the first problem. We develop the Scaled Flow-improving Algorithm. For the continuous problem, our algorithm finds a solution that is at most $\epsilon$ away from an optimal solution in terms of the infinity norm in $O(n^2\log{\frac{B}{n\epsilon}})$ time. For the integer problem, our algorithm terminates in $O(n^2\log\frac{B}{n})$ time. Our algorithm has the best worst-case complexity in the literature. In particular, it solves the discrete resource allocation problem with nested bound constraints in $O(n\log{n}\log\frac{B}{n})$ time and it also achieves the best worst-case complexity for that problem. We conduct extensive numerical experiments on instances with a variety of convex objectives. The numerical result demonstrates the efficiency of our algorithm in solving large-sized instances. The last problem is the optimal control problem in switched linear systems. We consider the following dynamical system that consists of several linear subsystems: $K$ matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the $K$ matrices and the given vector.This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We prove the problem is NP-hard. We propose a simple exact algorithm for this problem. Our algorithm runs in polynomial time when the given set of matrices has the oligo-vertex property, a concept we introduce for a set of matrices. We derive several easy-to-verify sufficient conditions for a set of matrices to have the oligo-vertex property. In particular, we show that a pair of binary matrices has the oligo-vertex property. Numerical results demonstrate the clear advantage of our algorithm in solving large-sized instances of the problem over one state-of-the-art global solver. We also pose several open questions on the oligo-vertex property and discuss its potential connection with the finiteness property of a set of matrices, which may be of independent interest

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2