Algebraic, Geometric, and Topological Methods in Optimization

Optimal Control for Systems Described by Difference Equations

Shape analysis plays an important role in many applications. In particular, in molecular biology, analyzing molecular shapes is essential to the fundamental problem of understanding how molecules interact. This project aims at developing efficient and effective algorithms to characterize and analyze molecular structures using geometric and topological methods. Two main components of this project are (1) developing novel molecular shape descriptors; and (2) identifying and representing meaningful features based on those descriptors. The project also produces accompanying (visualization) software. Results from this project (09/2006-10/2009) include the following publications. We have also set up web-servers for the software developed in this period, so that our new methods are accessible to a broader scientific community. The web sites are given below as well. In this final technical report, we first list publications and software resulted from this project. We then briefly explain the research conducted and main accomplishments during the period of this project.

Preface: Volume 81

In this paper, a topological preform design optimisation method was proposed based on the evolutionary structural optimisation (ESO) method. In this algorithm, new criteria for element elimination and addition were developed and a boundary smooth technique was employed to improve the boundary quality. Two 2D case problems including forging of aerofoil blade and forging of rail wheel are evaluated using the developed method. The results suggest that this topology optimisation method is an efficient approach for preform design optimisation.

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry.

Dear SIAGA Reader, It is a great pleasure and privilege to welcome you to SIAM Journal on Applied Algebra and Geometry (SIAGA). This journal, with the help of its editorial board, contributors, and readers and the SIAM staff, follows in the outstanding tradition of high-quality SIAM journals. We anticipate that it will become the leading journal dedicated to algebraic, geometric, and topological methods that have strong connection to applications. The journal began accepting submissions in March 2016. By December, it had received 90 manuscripts submitted for publication. Of these, you will find the first nine papers of the initial all-electronic batch published in February 2017. This first group marks the point at which we have established a regular rate of acceptances, building the foundation for a bright future. SIAGA came to exist thanks to the tremendous effort of numerous people. The impetus came from the SIAM Activity Group on Algebraic Geometry, which had its inaugural meeting in Raleigh, North Carolina, in October 2011. Following the second meeting, at Fort Collins, Colorado, in August 2013, a committee was formed to discuss the idea of a new journal. Under the leadership of Frank Sottile and Thorsten Theobald, a proposal was written and then rewritten numerous times. We had strong support from the Journal Committee and SIAM's Board and Council, and the journal received its approval in December 2015. All our invitations to serve on the editorial board were accepted, and SIAGA is now led by a terrific team composed of three Corresponding Editors and 26 Associate Editors. They are doing an amazing job in handling the growing number of submissions in a timely manner. The SIAM staff is a fantastic asset. In particular, Heather Blythe, Mitch Chernoff, and Frank Kunkle have been critical sources of support. The seven cover pictures of SIAGA have been an inspiration for all of us, and we thank Anna Seigal for her SIAM News articles about the mathematics that is expressed in these diagrams. Please enjoy reading the inaugural SIAGA articles. We look forward to receiving your high-quality submissions in the future.

This paper proposes a concurrent multiscale optimization method of macrostructure topology and microstructure shapes for porous structures, aimed at maximizing a specified natural frequency. The multi‐material distribution of the macrostructure and the shape of the microstructures are optimized by topology and shape optimization, respectively. The homogenized properties of the porous materials are calculated using the homogenization method, and the homogenized elastic tensor and density are applied to the macrostructure. The optimum distribution of the porous material in the macrostructure is determined by a multi‐material topology optimization using the generalized solid isotropic material with penalization (GSIMP) method, which is a method to obtain multi‐material topology by implementing the successive binarization. The KS (Kreisselmeier–Steinhauser) function is introduced to solve the repeated natural frequencies issue that lies in the maximization of a specified natural frequency. An area‐constrained multiscale optimization problem is formulated as a distributed‐parameter optimization problem, and the sensitivity functions are derived using the Lagrange multiplier method and the adjoint variable method. Based on the obtained sensitivity functions, the design variables are updated using the H1 gradient method (i.e., a nonparametric shape/topology optimization method). The effectiveness of the proposed method for maximizing a specified natural frequency is confirmed through 2D and 3D numerical examples, in which the influences of sub‐regions of the macrostructure and anisotropic material of the microstructures are also investigated and the results are discussed.

Abstract. Geometric and topological methods are applied to significance testing in the wavelet domain. A geometric test was developed for assigning significance to pointwise significance patches in local wavelet spectra, i.e., contiguous regions of significant wavelet power coefficients with respect to some noise model. This geometric significance test was found to produce results similar to an existing areawise significance test while being more computationally flexible and efficient. The geometric significance test can be readily applied to pointwise significance patches at various pointwise significance levels in wavelet power and coherence spectra. The geometric test determined that features in wavelet power of the North Atlantic Oscillation (NAO) are indistinguishable from a red-noise background, suggesting that the NAO is a stochastic, unpredictable process, which could render difficult the future projections of the NAO under a changing global system. The geometric test did, however, identify features in the wavelet power spectrum of an El Niño index (Niño 3.4) as distinguishable from a red-noise background. A topological analysis of pointwise significance patches determined that holes, deficits in pointwise significance embedded in significance patches, are capable of identifying important structures, some of which are undetected by the geometric and areawise tests. The application of the topological methods to ideal time series and to the time series of the Niño 3.4 and NAO indices showed that the areawise and geometric tests perform similarly in ideal and geophysical settings, while the topological methods showed that the Niño 3.4 time series contains numerous phase-coherent oscillations that could be interacting nonlinearly.

Turnover frame is one of the core connection parts of the bridge detection vehicle. Its performances, including stiffness, manufacturability and durability, are essential for the bridge detection vehicle. Lightweight of turnover frame can decrease the total vehicle weight, which is well conformed to the Chinese national standard GB 1589–2016 (Limits of dimensions, axle load and masses for motor vehicles, trailers and combination vehicles). In this paper, the methods of topology and thickness optimization are utilized to design a new turnover frame. Firstly, the finite element (FE) models of original turnover frame and up mechanism are created, and then the von Mises stress and deformation are obtained. The FE results are validated by the stress experiment of the turnover frame, the maximum error is 7.8% relative to experiment results. Secondly, a novel layout of turnover frame is obtained using solid isotropic microstructure with penalization (SIMP) method under multi-conditions. The thickness optimization is carried out based on topology optimization results. The manufacturability and functionality are considered for optimized results. Finally, stress and durability experiment validations of optimal design are carried out. The stress results are consistent with the simulation, and the structure is in excellent conditions after 600,000 cycles of repeated loads under harsh working conditions in proving ground. By comparing the FE results between the original and optimal turnover frame, it shows that the optimal scheme has 21.5% reduction in total mass while maintaining better stiffness and durability.

A major problem in algebra was to find a formula for the roots of polynomial equations of degree 5 or more, and it was proved by Abel and Ruffini that there was no formula for the roots. Lill developed the ‘graphical method’ by exploring the roots of polynomial equations using a geometric method rather than an algebraic method. To solve arbitrary polynomial equations, certain conditions of the ‘graphical method’ must be satisfied, but Lill did not provide a specific method for this. Beloch expressed quadratic and cubic equations using Lill's method and then discovered a method of satisfying specific conditions using origami. However, when the degree is 4 or more, several conditions must be satisfied simultaneously to solve it with origami, so there has been no research on solving the roots of arbitrary polynomial equations. Therefore, this study selected the following research questions. First, we develop a program that automatically solves quadratic and cubic equations using Lill's method, geogebra, and python. Second, we develop a program that automatically solves arbitrary polynomial equations of degree 4 or higher using Lill's method and python. As a result of the study, we were able to manually find the paths of quadratic and cubic equations using Geogebra and automatically find them using Python and develop a program to solve them using Lill's method. To find the path of an arbitrary nth degree polynomial equation of degree 4 or higher, we used object-oriented programming from the Pygame library to solve it. The significance of this research is that it visualizes the process of approximating the roots of an arbitrary polynomial equation, approaches it with a geometric method rather than an algebraic method, and develops a program to solve arbitrary polynomial equations in a generalized way.

This volume presents an elaborated version of lecture notes for two advanced courses: (Re)Emerging methods in Commutative Algebra and Representation Theory and Building Bridges Between Algebra and Topology, held at the CRM in the spring of 2015. Homological algebra is a rich and ubiquitous area; it is both an active field of research and a widespread toolbox for many mathematicians. Together, these notes introduce recent applications and interactions of homological methods in commutative algebra, representation theory and topology, narrowing the gap between specialists from different areas wishing to acquaint themselves with a rapidly growing field. The covered topics range from a fresh introduction to the growing area of support theory for triangulated categories to the striking consequences of the formulation in the homotopy theory of classical concepts in commutative algebra. Moreover, they also include a higher categories view of Hall algebras and an introduction to the use of idempotent functors in algebra and topology.

Covering an exceptional range of topics, this text provides a unique overview of the Maurer—Cartan methods in algebra, geometry, topology, and mathematical physics. It offers a new conceptual treatment of the twisting procedure, guiding the reader through various versions with the help of plentiful motivating examples for graduate students as well as researchers. Topics covered include a novel approach to the twisting procedure for operads leading to Kontsevich graph homology and a description of the twisting procedure for (homotopy) associative algebras or (homotopy) Lie algebras using the biggest deformation gauge group ever considered. The book concludes with concise surveys of recent applications in areas including higher category theory and deformation theory.

We give an appropriate extension of the concept of “tower of surjections” to arbitrary inverse systems. We introduce a natural closed model structure (in the sense of D. Quillen) on the category of pro-(Simplicial Abelian Groups) and interpret our condition as the definition of fibrant object.

SAMUEL EILENBERG1. Introduction.The title of this article requires some explanation.The term "abstract algebra" was used to indicate that we shall deal with purely algebraic objects like groups, algebras and Lie algebras rather than topological groups, topological algebras, and so on.The method of study is also purely algebraic but is the replica of an algebraic process which has been widely used in topology, thus the words "toplogical methods" could be replaced by "algebraic methods suggested by algebraic topology."These purely algebraic theories do, however, have several applications in topology.The algebraic process borrowed from topology is the following.Consider a sequence of abelian groups {C q ) and homomorphisms 8,such that ôô = 0.In each group C q two subgroups are distinguished Z q = kernel of 8:C q -* 0 +1 , B q = image of ô:C*~l ~» C q

With the rapid development of computers and technology in the 20th century, the topological optimisation (TO) method has spread worldwide in various fields.This novel structural optimisation approach has been applied in many disciplines, including architectural form-finding.Especially Bi-directional Evolutionary Structural Optimisation (BESO), which was proposed in the 1990s, is widely used by thousands of engineers and architects worldwide to design innovative and iconic buildings.To integrate topological optimisation with artificial intelligence (AI) algorithms and to leverage its power to improve the diversity and efficiency of the BESO topological optimisation method, this research explores a non-iterative approach to accelerate the topology optimisation process of structures in architectural form-finding via conditional generative adversarial networks (GANs), which is named as OptiGAN.Trained with topological optimisation results generated through Ameba software, OptiGAN is able to predict a wide range of optimised architectural and structural designs under defined conditions.