Linear Algebra in Dioids: A Survey of Recent Results

This chapter is concerned with the Boolean matrix approach to problems concerning the existence and determination of paths in directed graphs.

In the compiler literature, parsing algorithms for context-free grammars are presented using string rewriting systems or abstract machines such as pushdown automata. Unfortunately, the resulting descriptions can be baroque, and even a basic understanding of some parsing algorithms, such as Earley’s algorithm for general context-free grammars, can be elusive. In this paper, we present a graphical representation of context-free grammars called the Grammar Flow Graph (GFG) that permits parsing problems to be phrased as path problems in graphs; intuitively, the GFG plays the same role for context-free grammars that nondeterministic finite-state automata play for regular grammars. We show that the GFG permits an elementary treatment of Earley’s algorithm that is much easier to understand than previous descriptions of this algorithm. In addition, look-ahead computation can be expressed as a simple inter-procedural dataflow analysis problem, providing an unexpected link between front-end and back-end technologies in compilers. These results suggest that the GFG can be a new foundation for the study of context-free grammars.

Mining web log datasets has been extensively studied using Frequent Pattern Mining (FPM) and its various other forms. Identifyingfrequent patterns in different sequences can help in analyzing the most common sub-sequences (e.g., the pages visitedtogether). However, this approach would not be able to identify general structures spanning over multiple sequences. In responseto understanding general structures, we introduce a new form of sequential pattern mining called super-sequence frequent patternmining (SS-FPM). In contrast to sub-sequences determined by FPM, SS-FPM determines the super-sequences that can containthe common parts from different sequences. This can be useful in understanding the general behavior/flow of users in web usagemining, classifying web pages and users, making predictions etc. In essence, finding frequent super-sequence patterns turnsout to be related to the well-known heaviest (longest) path problem in graphs, which is known to be NP-hard. Accordingly,we transform a given sequential dataset into a sequence graph and formulate the problem as k-hop heaviest path problem. Wethen propose an efficient heuristic called sequence matrix method using dynamic programming techniques. We compared ourmethod to the existing Heavypath method. The results show that our method is more efficient especially on large datasets.

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

In the minimum constraint removal problem, we are given a set of overlapping geometric objects as obstacles in the plane, and we want to find the minimum number of obstacles that must be removed to reach a target point t from the source point s by an obstacle-free path. The problem is known to be intractable and no sub-linear approximations are known even for simple obstacles such as rectangles and disks. The main result of our paper is an approximation framework that gives an O(nα(n))-approximation for polygonal obstacles, where α(n) denotes the inverse Ackermann's function. For pseudodisks and rectilinear polygons, the same technique achieves an O(n)-approximation. The technique also gives O(n)-approximation for the minimum color path problem in graphs. We also present some inapproximability results for the geometric constraint removal problem.

Solution of nonlinear partial differential equations from base equations

Solutions in the process of resolving numerical methods, especially "Solutions of Non Linear Equations", are determined by several criteria including: 1. Solutions will be obtained if the tolerances found from numerical calculations are less than or equal to the specified tolerances, depending on the criteria of the question. 2. The solution will be obtained if the function value f(xn +1) reaches zero (0). 3. The solution will be obtained if the value of xn+1 has a constant value (no change). In some observations of manual efforts to find non-linear equations solutions, students cannot do the calculation perfectly. This is due to lack of understanding of teaching materials or modules and also the algorithms that are difficult to grasp. Moreover, the calculation has to be done for several times iterations in order to obtain the expected results. Comparing the results of calculations done manually using Microsoft Excel and the result of calculations using numerical applications created using C# programming language, it concludes: non-linear equations manual calculation to get one of the roots of non-linear functions tend to not completed in a short time so the error rate and accuracy cannot be measured. The calculations performed using Microsoft Excel, obtained the error rate reaches 0.158% and the accuracy rate reaches 99.842%. Calculations performed using numerical applications using the C# programming language obtained error rate reached 0.004% and accuracy rate reached 99.996%.

Two-dimensional wavelets scheme for numerical solutions of linear and nonlinear Volterra integro-differential equations

A method of deletion of dimensions for a mathematical model which gives the number of variables less than that prescribed by the π-theorem was proposed. In a number of cases, it is possible to exclude from consideration all similarity criteria or, in other words, achieve self-similarity in them. In the framework of deletion of dimensions, this is expressed in a transition from criteria to similarity numbers. Thus, information is reduced without its loss. The limit reduction of the number of variables in the mathematical model makes it possible to use analytical approximation dependences as approximate solutions. Such dependences are obtained from the initial solutions by using the coefficients of stretch, which corresponds to the group methods for solving equations. It was proposed to use solutions of the linearized forms of the original nonlinear equations as initial solutions. Such approach makes it possible to take into account physical character of the change in the studied quantities in the approximation dependences when solution of nonlinear equations cannot be realized using standard functions. Efficiency of the method was illustrated by the example of study of the pendulum motion, which is a counterpart of inertial link of the second order in the theory of automatic control. Solutions were obtained for the cases of presence and absence of environmental resistance. The last variant is interesting by the feasibility of comparing the proposed and available analytical solutions in terms of elliptic integrals.

In this dissertation we develop numerical algorithms and software tools to facilitate parallel solutions of nonlinear programming (NLP) problems. In particular, we address large-scale, block-structured problems with an intrinsic decomposable configuration. These problems arise in a great number of engineering applications, including parameter estimation, optimal control, network optimization, and stochastic programming. The structure of these problems can be leveraged by optimization solvers to accelerate solutions and overcome memory limitations, and we propose variants to two classes of optimization algorithms: augmented Lagrangian (AL) schemes and Schur-complement interior-point methods. The convergence properties of augmented Lagrangian decomposition schemes like the alternating direction method of multipliers (ADMM) and progressive hedging (PH) are well established for convex optimization but convergence guarantees in non-convex settings are still poorly understood. In practice, however, ADMM and PH often perform satisfactorily in complex non-convex NLPs. In this work, we study connections between the method of multipliers (MM), ADMM, and PH to derive benchmarking metrics that explain why PH and ADMM work in practice. We illustrate the concepts using challenging dynamic optimization problems. Our exposition seeks to establish more formalism in benchmarking ADMM, PH, and AL schemes and to motivate algorithmic improvements. The effectiveness of nonlinear interior-point solvers for solving large-scale problems relies quite heavily on the solution of the underlying linear algebra systems. The schur-complement decomposition is very effective for parallelizing the solution of linear systems with modest coupling. However, for systems with large number of coupling variables the schur-complement method does not scale favorably. We implement an approach that uses a Krylov solver (GMRES) preconditioned with ADMM to solve block-structured linear systems that arise in the interior-point method. We show that this ADMM-GMRES approach overcomes the well-known scalability issues of Schur decomposition. One important drawback of using decomposition approaches like ADMM and PH is their convergence rate. Unlike Schur-complement interior-point algorithms that have super-linear convergence, augmented Lagrangian approaches typically exhibit linear and sublinear rates. We exploit connections between ADMM and the Schur-complement decomposition to derive an accelerated version of ADMM. Specifically, we study the effectiveness of performing a Newton-Raphson algorithm to compute multiplier estimates for augmented Lagrangian methods. We demonstrate using two-stage stochastic programming problems that our multiplier update achieves convergence in fewer iterations for MM on general nonlinear problems. In the case of ADMM, the newton update significantly reduces the number of subproblem solves for convex quadratic programs (QPs). Moreover, we show that using newton multiplier updates makes the method robust to the selection of the penalty parameter. Traditionally, state-of-the-art optimization solvers are implemented in low-level programming languages. In our experience, the development of decomposition algorithms in these frameworks is challenging. They present a steep learning curve and can slow the development and testing of new numerical algorithms. To mitigate these challenges, we developed PyNumero, a new open source framework implemented in Python and C++. The package seeks to facilitate development of optimization algorithms for large-scale optimization within a high-level programming environment while at the same time minimizing the computational burden of using Python. The efficiency of PyNumero is illustrated by implementing algorithms for problems arising in stochastic programming and optimal control. Timing results are presented for both serial and parallel implementations. Our computational studies demonstrate that with the appropriate balance between compiled code and Python, efficient implementations of optimization algorithms are achievable in these high-level languages.

We consider the fine-grained complexity of sparse graph problems that currently have O(mn) time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (Minimum Weight Cycle), Radius, Eccentricities, BC (Betweenness Centrality), etc. We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in the O(mn) class. There are many sub-cubic reductions between graph problems in the O(mn) class, but surprisingly few of these preserve sparsity. In the directed case, our results give a partial order on a large collection of problems in the O(mn) class (along with some equivalences), and many of our reductions are very nontrivial. In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to unweighted APSP. We develop a new ‘bit-sampling’ method for these sparse reductions on undirected graphs, which also gives rise to improved or simpler algorithms for cycle finding problems in undirected graphs. We formulate the the notion of MWC hardness, which is based on the assumption that a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in the O(mn) class establish that several problems in this class, including 2-SiSP (second simple shortest path), s-t Replacement Paths, Radius, Eccentricities and BC are MWC hard. Our sparse reductions give MWC hardness a status for the O(mn) class similar to 3SUM hardness for the quadratic class, since they show sub-mn hardness for a large collection of fundamental and well-studied graph problems that have maintained an O(mn) time bound for over half a century. We also identify Eccentricities and BC as key problems in the O(mn) class which are simultaneously MWC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than nk. Our framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse and for which the O(mn) time algorithms are the ones typically used in practice, and not the O(n3) time algorithms.

A class of strongly singular, nonlinear integral equations is considered and sufficient conditions for the uniqueness of the solution and, in the linear and homogeneous case, the nonpositiveness of the associated eigenvalues are obtained. When the singular integral is proportional to a small perturbation parameter, a procedure for the construction of the uniformly valid leading term of the asymptotic solutions for both linear and nonlinear equations is presented. The results are illustrated by means of an example that emerges in the analysis of microcrack growth in reinforced ceramics.

Doctoral thesis had been developed in the period from September 2011 until February 2016 in department of Computer Control of Faculty of Information Technologies, Latvia University of Agriculture. The PhD thesis consists of 145 pages, which include 15 tables, 44 figures, and 12 annexes, which are summarised in separate volume. For the development of this thesis 196 sources were used including 194 in foreign language. Promotion thesis hypothesis: It is possible to upgrade stoichiometric model creation and validation procedures, to decrease calculation time, manual work and manually processed data amount. The aim of PhD thesis: The aim of thesis is to develop stoichiometric computer models based procedures of metabolic engineering task solving procedures to reduce computation and manual work. In order to achieve the aim of the PhD thesis several tasks were completed: 1. To analyse the mathematical regularities between the stoichiometric modelling and molecular processes of metabolism; 2. To research operating principles of stoichiometric model analysis methods;3. To improve stoichiometric model development and validation procedure to reduce time consumption, number of operations and manual processing of data. 4. To develop stoichiometric model analysis methods visualization tool for metabolic engineering tasks. 5. To create stoichiometric model analysis methods sets of procedures for reduction time consumption number of operations and number of analysed solutions.The content of the PhD thesis: it is structured according to the aim and the tasks of PhD thesis consisting of 5 chapters: First chapter includes general description about biological parameters, basic information and metabolism processes, which is used in stoichiometric model as creation, validation, improvement and analysis. Also includes description about systems biology modelling methods, where main text is dedicated for stoichiometric modelling, using linear algebra, matrixes, and mathematical vectors properties for stoichiometric model analysis problems research. Second chapter includes description about 6 different stoichiometric modelling analysis methods. For Each method specific functionality is described, necessary analysis methods input and output data is researched and analyzed how the modelling aim should be combined with analysis methods properties. Third chapter includes existing stoichiometric model creation procedure and using this information and promotion thesis author gain experience creating and upgrading several stoichiometric models, was upgraded. It consists from 5 subclasses, where using procedure in cyclic development process new author made procedure decreases stoichiometric model development time. Are described and classified sthoichiometric model constraints. Fourth chapter includes author developed visualization method for stoichiometric model analysis methods in user friendly interface. In chapter are researched different systems biology used computational tools, vector graphic creation tools, which are used to create first time a visual metabolic network for automatic flux visualization. Fifth chapter includes 12 different type and dificulty procedures or stoichiometric model analysis. These procedure allow improving metabolic engineering purposed task decreasing model analysis computational time, user manual work and solution set amount. The main results, conclusions and future development prospects are described in the conclusions of the PhD thesis.

Tan-cot method is applied to get exact soliton solutions of non-linear partial differential equations notably generalized Benjamin-Bona-Mahony, Zakharov-Kuznetsov Benjamin-Bona-Mahony, Kadomtsov- Petviashvilli Benjamin-Bona-Mahony and Korteweg-de Vries equations, which are important evolution equations with wide variety of physical applications. Elastic behavior and soliton fusion/fission is shown graphically and discussed physically as far as possible. Keywords: Benjamin-Bona-Mahony equation, Tan-Cot method, soliton solutions.

General theory of instantaneous power for multi-phase systems with distortion, unbalance and direct current components