Quasi-Newton Methods for Discretized Non-linear Boundary Problems

The discretization of non-linear boundary problems generallyleads to a finite system of non-linear algebraic equations,and it is to be expected that this latter has special structurearising both from the boundary problem and the method of discretizationused. The numerical solution of the algebraic system representsa serious numerical problem, and it is the point of this paperto indicate that, in certain important cases, special purposequasi-Newton methods can be constructed. We illustrate by consideringa single nonlinear differential equation discretized by collocationand present experimental results which indicate that an improvementin performance can be expected from the special methods.     

Economical experiments: Bayesian efficient experimental design

We propose and implement a Bayesian optimal design procedure. Our procedure takes as its primitives a class of parametric models of strategic behavior, a class of games (experimental designs), and priors on the behavioral parameters. We select the experimental design that maximizes the information from the experiment. We sequentially sample with the given design and models until only one of the models has viable posterior odds. A model which has low posterior odds in a small collection of models will have an even lower posterior odds when compared to a larger class, and hence we can dismiss it. The procedure can be used sequentially by introducing new models and comparing them to the models that survived earlier rounds of experiments. The emphasis is not on running as many experiments as possible, but rather on choosing experimental designs to distinguish between models in the shortest possible time period. We illustrate this procedure with a simple experimental game with one-sided incomplete information.We acknowledge the financial support from NSF grant #SES-9223701 to the California Institute of Technology. We also acknowledge the research assistance of Eugene Grayver who wrote the software for the experiments.     

A GPU based iteration approach to efficiently evaluate radiation symmetry for laser driven inertial confinement fusion

Radiation computation is very important for high energy density experiments design in the laser-driven Inertial Confinement Fusion. The view-factor based models are often used to calculate the radiation on the capsule inside a hohlraum. However, it usually takes much time to solve them when the number of equations is very large.In this paper, an efficient iteration approach GPU is presented. The core idea is: (1) guaranteed symmetry, strictly diagonally dominant, and positive definite properties underlying the models are described, (2) a preconditioned conjugate gradient iteration approach is presented to compute the radiation based on such guaranteed properties, and (3) such approach is then parallelized and implemented for GPU so that the large scale models, especially for the non-linear model, can be efficiently solved in reasonable time.Finally, two experimental targets for Shenguang laser facilities built in China are demonstrated and compared to validate the efficiency of the presented approach. The results show that, the models’ computation (1) can be speeded up with successive over-relax iteration method by eight times as compared with Cholesky factorization based direct approach, (2) can be accelerated more with the preconditioned conjugate gradient iteration approach by almost eight times, and (3) can be further accelerated about 2 to 4 times as it parallelized and run on the GPU, which enables the large scale models, can be efficiently solved in reasonable time on the usual desktop computers.     

Representing knowledge about linear programming formulation

This paper describes a system, LPFORM, that enables users to design linear programming models interactively, using graphics. An output of the system is an algebraic statement of the model and data references that are subsequently used to generate the input for a solver in the standard MPS format. The emphasis of this paper is on the types of knowledge one has on the submodels that make up larger models and how this knowledge can be organized.     

A secular equation for the Jacobian matrix of certain multispecies kinematic flow models

Multispecies kinematic flow models lead to nonlinear systems of conservation laws with a possibly large number of unknowns, the concentrations or the densities of the different species. In recent years, the hyperbolic character of several of these models has been analyzed by considering the characteristic polynomial of the Jacobian matrix of the system. This analysis can be considerably simplified by realizing that the fluxes in these models have a particular algebraic structure that can be exploited within a systematic algebraic framework. The framework can serve to determine the eigenvalues, and even the eigenvectors, of the Jacobian matrix of the system, which allows the use of characteristic‐based high‐resolution shock capturing schemes in numerical simulations.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A rate-dependent algebraic stress model for turbulence

Based on the stress transport model, a rate-dependent algebraic expression for the Reynolds stress tensor is developed. It is shown that the new model includes the normal stress effects and exhibits viscoelastic behavior. Furthermore, it is compatible with recently developed improved models of turbulence. The model is also consistent with the limiting behavior of turbulence in the inertial sublayer and is capable of predicting secondary flows in noncircular ducts. The TEACH code is modified according to the requirements of the rate-dependent model and is used to predict turbulent flow fields in a channel and behind a backward-facing step. The predicted results are compared with the available experimental data and those obtained from the standard k-ε and algebraic stress models. It is shown that the predictions of the new model are in better agreements with the experimental data.     

Category-theoretic models of algebraic computer systems

A computer system is said to be algebraic if it contains nodes that implement unconventional computation paradigms based on universal algebra. A category-based approach to modeling such systems that provides a theoretical basis for mapping tasks to these systems’ architecture is proposed. The construction of algebraic models of general-purpose computations involving conditional statements and overflow control is formally described by a reflector in an appropriate category of algebras. It is proved that this reflector takes the modulo ring whose operations are implemented in the conventional arithmetic processors to the ?ukasiewicz logic matrix. Enrichments of the set of ring operations that form bases in the ?ukasiewicz logic matrix are found.     

Iterative solution of matrix games using grid variational inequalities

A new approach to the approximate solution of matrix games is proposed. It is based on the reduction of the original problem to a variational inequality of a special form. In particular, this makes it possible to design preconditioned iterative methods, which proved to be effective as a tool for the numerical solution of large and ill-conditioned systems of linear algebraic equations.     

Algebraic algorithms for the analysis of mechanical trusses

Infinite periodic lattices can be used as models for analyzing and understanding various properties of mechanical truss constructions with periodic structures. For infinite lattices, the problems of connectivity and stability are nontrivial from the mathematical point of view and have not been addressed adequately in the literature. In this paper, we will present a set of algebraic algorithms, which are based on ideal theory, to solve such problems.

For the understanding of the notion ``complicated three-dimensional lattices', it is essential to have this paper with colored figures.

     

反映强流动曲率效应的非线性湍流模型

首先定性地分析了流线曲率效应对流场湍流结构的影响,然后以U型槽道流为典型算例,对多种湍流模型进行了评估．评估的模型包括:线性涡粘性模型,二阶和三阶非线性涡粘性模型,二阶显式代数应力模型和Reynolds应力模型．评估结果表明,性能良好的三阶非线性涡粘性模型,如黄于宁等人发展的HM模型以及CLS模型,可以较好地描述流线的曲率效应对湍流结构的影响,如凸曲率作用下内壁附近湍流强度的衰减和凹曲率作用下外壁附近湍流的增强,并且较好地确定了管道下游的分离点位置和分离泡长度,其预测的结果和实验符合较好,与Reynolds力模型的结果十分接近,因此可以较好地应用于具有曲率效应的工程湍流的计算．     

An integrated algebraic approach to conflict resolution with three-level preference

An integrated algebraic approach is developed to calculate stabilities in multiple decision maker graph models with three levels of preference. The algebraic approach establishes an integrated paradigm for stability analysis and status quo analysis under different preference structures, such as two-level preference and three-level preference. Difficulties in coding algorithms to analyze stabilities, rooted in their logical representation, led to the development of matrix representations of preference and explicit matrix calculations to determine stabilities. Here, the algebraic approach is used to represent graph models with three levels of preference and to conduct stability analysis for such models. The algebraic approach facilitates the development of new stability concepts and algorithms to calculate them, and reveals an inherent link between status quo analysis and traditional stability analysis. Hence, it will facilitate the design of an integrated decision support system for the graph model for conflict resolution.     

A Topology Based Discretization of PDAEs Describing Water Transportation Networks

We consider partial differential algebraic systems (PDAEs) describing water transportation networks. Similar to the approach in [6], we follow the method of lines for the discretization. However, we do not consider free surface flow models but pressure flow models covering hydraulic shocks. Moreover, we include switching models reflecting the on/off state of pumpes and valves. Aiming at a stable numerical simulation of the PDAEs we present a topology based spatial discretization that results in a differential algebraic system (DAE) of index 1. Furthermore we show that the DAE index can be higher than 1 if the spatial discretization is not adapted to the position of reservoirs and demand nodes within the network. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Low-complexity quantized switching controllers using approximate bisimulation

In this paper, we consider the problem of synthesizing low-complexity controllers for incrementally stable switched systems. For that purpose, we establish a new approximation result for the computation of symbolic models that are approximately bisimilar to a given switched system. The main advantage over existing results is that it allows us to design naturally quantized switching controllers for safety or reachability specifications; these can be pre-computed offline and therefore the online execution time is reduced. Then, we present a technique to reduce the memory needed to store the control law by borrowing ideas from algebraic decision diagrams for compact function representation and by exploiting the non-determinism of the synthesized controllers. We show the merits of our approach by applying it to a simple model of temperature regulation in a building.     

Type theories,toposes and constructive set theory: predicative aspects of AST

We introduce a predicative version of topos (stratified pseudotopos) based on the notion of small maps in algebraic set theory, developed by Joyal and one of the authors. Examples of stratified pseudotoposes can be constructed in Martin-Löf type theory, which is a predicative theory. A stratified pseudotopos admits construction of the internal category of sheaves, which is again a stratified pseudotopos. We also show how to build models of Aczel-Myhill constructive set theory using this categorical structure.     

Mathematical Programming Modelling Using MGG

MGG is a software package for the application of mathematicalprogramming (MP). It complements the Sciconic MP code by providinga facility for developing MP models quickly and efficiently,and it enables changes to be made easily to established models.It is available on a wide range of minis and mainframes anda version of it is available as part of the Micro LP systemon IBM PCs and compatibles. MGG is based on an approach to modellingMP problems which stresses the primacy of the mathematical formulation.The process of MP modelling is divided into two stages: modelpreparation and running of the model. The user writes a mathematicalformulation, which MGG converts to matrix generator and reportwriter programs. This is done once to produce the programs whichcan then be run many times on different data. This paper describesMGG and draws comparisons with other matrix generator and mathematicalprogramming languages. It starts by considering how an MP problemcan be described, and then sets out a methodology for formulation.The MGG language is based upon this approach. A simple exampleis presented which is shown both as an algebraic formulationand in the MGG language. The process of building and runningmodels with MGG is then described. Finally, some comments areoffered on experience of using the software.     

Index-aware model order reduction for differential-algebraic equations

We introduce a model order reduction (MOR) procedure for differential-algebraic equations, which is based on the intrinsic differential equation contained in the starting system and on the remaining algebraic constraints. The decoupling procedure in differential and algebraic part is based on the projector and matrix chain which leads to the definition of tractability index. The differential part can be reduced by using any MOR method, we use Krylov-based projection methods to illustrate our approach. The reduction on the differential part induces a reduction on the algebraic part. In this paper, we present the method for index-1 differential-algebraic equations. We implement numerically this procedure and show numerical evidence of its validity.     

基于区间分析的不等式自动证明

提出了一种基于区间分析的不等式自动证明方法,这一方法可以处理类型较为一般的不等式, 只需对应的函数具有所需的高阶连续可微性质, 而传统的不等式自动证明方法一般仅处理代数类型, 或可最终转化为代数类型的不等式.实际例子显示, 该方法可以解决一些其他方法无法解决的问题.     

Normal forms,linearity, and prime algebraicity over nonflat domains

Using representations of nonflat Scott domains to model type systems, it is natural to wish that they be “linear”, in which case the complexity of the fundamental test for entailment of information drops from exponential to linear, the corresponding mathematical theory becomes much simpler, and moreover has ties to models of computation arising in the study of sequentiality, concurrency, and linear logic. Earlier attempts to develop a fully nonflat semantics based on linear domain representations for a rich enough type system allowing inductive types, were designed in a way that felt rather artificial, as it featured certain awkward and counter‐intuitive properties; eventually, the focus turned on general, nonlinear representations. Here we try to turn this situation around, by showing that we can work linearly in a systematic way within the nonlinear model, and that we may even restrict to a fully linear model whose objects are in a bijective correspondence with the ones of the nonlinear and are easily seen to form a prime algebraic domain. To obtain our results we study mappings of finite approximations of objects that can be used to turn approximations into normal and linear forms.     

Fuzzy sets and type 2 under algebraic product and algebraic sum

The concept of fuzzy sets of type 2 has been proposed by L.A. Zadeh as an extension of ordinary fuzzy sets. A fuzzy set of type 2 can be defined by a fuzzy membership function, the grade (or fuzzy grade) of which is taken to be a fuzzy set in the unit interval [0, 1] rather than a point in [0, 1].This paper investigates the algebraic properties of fuzzy grades (that is, fuzzy sets of type 2) under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring. Moreover, the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well-known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join, meet and algebraic product.     

On Numbers which are Mahler Measures

We investigate which algebraic numbers can be Mahler measures. Adler and Marcus showed that these must be Perron numbers. We prove that certain integer multiples of every Perron number are Mahler measures. The results of Boyd give some necessary conditions on Perron number to be a measure. These do not include reciprocal algebraic integers, so it would be of interest to find one which is not a Mahler measure. We prove a result in this direction. Finally, we show that for every non-negative integer k there is a cubic algebraic integer having norm 2 such that precisely the kth iteration of its Mahler measure is an integer.