Rigid body dynamics with unilateral constraints, collisions and Coulomb friction has been the subject of investigation and controversy for the past century, due to the failure to obtain solutions in certain situations. Recent extensions... more
Rigid body dynamics with unilateral constraints, collisions and Coulomb friction has been the subject of investigation and controversy for the past century, due to the failure to obtain solutions in certain situations. Recent extensions to the theory of ordinary differential equations, notably J.J Moreau’s measure differential equations can be used to give a clearer idea of the issues in these problems. Novel numerical methods have had to be developed to solve these problems, and to prove the convergence of the numerical trajectories to solutions of the true rigid body problem.
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The Traveling Salesman Problem is a conceptually simple problem that is computationally difficult due to the size of the search space, which grows factorially with the number of cities. Beam-ACO is an Ant Colony Optimization heuristic... more
The Traveling Salesman Problem is a conceptually simple problem that is computationally difficult due to the size of the search space, which grows factorially with the number of cities. Beam-ACO is an Ant Colony Optimization heuristic that combines classical ACO with beam search. Beam-ACO is quite effective at finding high quality approximate solutions but it is more computationally demanding than the more classical ACO algorithms. In this work we propose a parallel version of Beam-ACO based on work-stealing. Our parallel Beam-ACO algorithm runs both the ant search and beam evaluation and pruning in parallel. Our experiments verify both that Beam-ACO is indeed one of the most effective ACO metaheuristics and that our parallel Beam-ACO is faster than more traditional parallelization schemes such as multi-colony or ant parallel.
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Single-cell responses are shaped by the geometry of signalling kinetic trajectories carved in a multidimensional space spanned by signalling protein abundances. It is, however, challenging to assay a large number (more than 3) of... more
Single-cell responses are shaped by the geometry of signalling kinetic trajectories carved in a multidimensional space spanned by signalling protein abundances. It is, however, challenging to assay a large number (more than 3) of signalling species in live-cell imaging, which makes it difficult to probe single-cell signalling kinetic trajectories in large dimensions. Flow and mass cytometry techniques can measure a large number (4 to more than 40) of signalling species but are unable to track single cells. Thus, cytometry experiments provide detailed time-stamped snapshots of single-cell signalling kinetics. Is it possible to use the time-stamped cytometry data to reconstruct single-cell signalling trajectories? Borrowing concepts of conserved and slow variables from non-equilibrium statistical physics we develop an approach to reconstruct signalling trajectories using snapshot data by creating new variables that remain invariant or vary slowly during the signalling kinetics. We app...
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Research Interests: Mathematics, Applied Mathematics, Computer Science, Convergence, Contact Mechanics, and 13 moreNewton Method, Siam, Singular perturbation problems, PARTIAL DIFFERENTIAL EQUATION, Port Hamiltonian system, Equation of Motion, Rigid Body Dynamics, Homogeneity, Differential equation, Music Information Dynamics, Lagrangian, Constrained Optimization, and Complementarity Problem
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A number of problems arising from dynamical systems and other areas leads to problems of computing eigenvalues/vectors and singular value decompo-sisitions of products of matrices. A number of recent algorithmsbanel, Brown and Kennel have... more
A number of problems arising from dynamical systems and other areas leads to problems of computing eigenvalues/vectors and singular value decompo-sisitions of products of matrices. A number of recent algorithmsbanel, Brown and Kennel have been devised to compute Schur forms and SVD's of products in terms of the factors. A common connection between them is the use of recursive QR factorisations and related methods. Relationships between these methods, and their accuracy, is discussed. Generalised eigen-and singular value decompositions can also be understood in this framework. 1. Why product algorithms? Product algorithms are algorithms to compute factorisations of products of matrices that works with the product in terms of its factors. There are a number of reasons why this is a good thing to do. In terms of the design of an algorithm, by working in terms of the factors, at each stage of the algorithm the data is close in form to the true data of the original problem. This often means that meaningful backward error results can be proven. These product algorithms can be very important where long products are involved, such as arise from dynamical systems. These are often studied to obtain estimates of Lyapunov exponents, and related quantities such as bounds of (fractal) dimensions of attractors and topological or Kolmogorov entropy. Long products typically have exponentially diverging eigenvalues and singular values. Multiplying out such long products will often result in the (exponentially) smaller eigenvalues being swamped by the roundoo errors from the larger eigenval-ues. As with the common criticism of the normal equations approach to solving least squares problems, the problem comes from the much poorer conditioning obtained from multiplying matrices and then working with the product rather than its factors.
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The meshless method plays an important role in solving problems in computational mechanics where conventional computational methods are not well suited. In this paper, we examine the property of the kernel matrix and investigate the... more
The meshless method plays an important role in solving problems in computational mechanics where conventional computational methods are not well suited. In this paper, we examine the property of the kernel matrix and investigate the convergence and timing performance of some well-known Krylov subspace methods on solving linear systems from meshless discretizations.
... Lilia Yerosheva, Shannon K. Kuntz, Peter M. Kogge, Jay B. Brockman. Page: 3. Influence of Array Allocation Mechanisms on Memory System Energy. R. Athavale, Narayanan Vijaykrishnan, Mahmut T. Kandemir, Mary Jane Irwin. Page: 3. A... more
... Lilia Yerosheva, Shannon K. Kuntz, Peter M. Kogge, Jay B. Brockman. Page: 3. Influence of Array Allocation Mechanisms on Memory System Energy. R. Athavale, Narayanan Vijaykrishnan, Mahmut T. Kandemir, Mary Jane Irwin. Page: 3. A PIM-based Multiprocessor System. ...
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This book introduces students to the art and craft of writing proofs, beginning with the basics of writing proofs and logic, and continuing on with more in-depth issues and examples of creating proofs in different parts of mathematics, as... more
This book introduces students to the art and craft of writing proofs, beginning with the basics of writing proofs and logic, and continuing on with more in-depth issues and examples of creating proofs in different parts of mathematics, as well as introducing proofs-of-correctness for algorithms. The creation of proofs is covered for theorems in both discrete and continuous mathematics, and in difficulty ranging from elementary to beginning graduate level.Just beyond the standard introductory courses on calculus, theorems and proofs become central to mathematics. Students often find this emphasis difficult and new. This book is a guide to understanding and creating proofs. It explains the standard “moves” in mathematical proofs: direct computation, expanding definitions, proof by contradiction, proof by induction, as well as choosing notation and strategies.
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The core of scientific computing is designing, writing, testing, debugging and modifying numerical software for application to a vast range of areas: from graphics, meteorology and chemistry to engineering, biology and finance.... more
The core of scientific computing is designing, writing, testing, debugging and modifying numerical software for application to a vast range of areas: from graphics, meteorology and chemistry to engineering, biology and finance. Scientists, engineers and computer scientists need to write good code, for speed, clarity, flexibility and ease of re-use. Oliveira and Stewart's style guide for numerical software points out good practices to follow, and pitfalls to avoid. By following their advice, readers will learn how to write efficient software, and how to test it for bugs, accuracy and performance. Techniques are explained with a variety of programming languages, and illustrated with two extensive design examples, one in Fortran 90 and one in C++: other examples in C, C++, Fortran 90 and Java are scattered throughout the book. This manual of scientific computing style will be an essential addition to the bookshelf and lab of everyone who writes numerical software.
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ABSTRACT The standard Signorini contact condition is integrated against a given function ψ over the boundary to obtain a simplified model of contact suitable for impact problems. An implicit method (implicit mid-point rule for elasticity,... more
ABSTRACT The standard Signorini contact condition is integrated against a given function ψ over the boundary to obtain a simplified model of contact suitable for impact problems. An implicit method (implicit mid-point rule for elasticity, and implicit Euler for the contact conditions) is proposed to numerically solve the simplified model, and some properties of the solution are obtained. The results are only partial at this stage, but they seem to indicate that contact forces in elastic impacts are considerably more regular than general measures.
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Measure differential inclusions were introduced by J. J. Moreau to study sweeping processes, and have since been used to study rigid body dynamics and impulsive control problems. The basic formulation of an MDI is “d µ / d ?(t) ? K(t)”... more
Measure differential inclusions were introduced by J. J. Moreau to study sweeping processes, and have since been used to study rigid body dynamics and impulsive control problems. The basic formulation of an MDI is “d µ / d ?(t) ? K(t)” where µ is a vector measure, ? an unsigned measure, and K(·) is a set-valued map with closed, convex
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ABSTRACT In this paper, we formulate a time-discretization using the implicit Euler method for contact conditions and the midpoint rule for the elastic part of the equations. The energy functional is defined, and convergence for the... more
ABSTRACT In this paper, we formulate a time-discretization using the implicit Euler method for contact conditions and the midpoint rule for the elastic part of the equations. The energy functional is defined, and convergence for the time-discretization is investigated. Our time-discretization leads to energy dissipation. Using this time discretization and the finite element method with B-spline basis functions, we compute numerical solutions. We show that there is a converging subsequence, and the limit of any such converging subsequence is a solution of the dynamic impact problem. In order to solve the linear complementarity problem that arises in the numerical method, we use a smoothed guarded Newton method. We also investigate numerically the question of whether the numerical solutions converge strongly to their limit and if energy is conserved for the limit. Our numerical results give some evidence that this is so.
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ABSTRACT We consider dynamic frictionless impact problems of elastic materials formulated in abstract settings. The contact conditions for the impact problem are Signorini-type complementarity conditions. Using time discretization and... more
ABSTRACT We consider dynamic frictionless impact problems of elastic materials formulated in abstract settings. The contact conditions for the impact problem are Signorini-type complementarity conditions. Using time discretization and Galerkin approximation, we investigate the convergence of numerical fully discrete trajectories to a solution of the continuous-time problem. In this way we establish the existence of solutions for a class of impact problems, some of which have been previously studied, while others have not. Most of the impact problems to which this theory can be applied are “thick” obstacle problems, although it can also be applied to a number of boundary or “thin” obstacle problems. The crucial assumption for the theory is that the cone of possible contact forces satisfies a strong pointedness condition, which can usually be related to a Sobolev embedding condition.
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Optimization problems recently studied in liver kinetics feature noncompact constraints and highly nonunique solutions. Here a wider class of problems with these properties is considered. It is shown that it is sufficient to solve the... more
Optimization problems recently studied in liver kinetics feature noncompact constraints and highly nonunique solutions. Here a wider class of problems with these properties is considered. It is shown that it is sufficient to solve the problem with an additional, compact constraint. From the solutions of this restricted problem all solutions of the original problem can be obtained. By means of this result and Pontryagin’s maximum principle, solutions are found for some problems of liver-kinetic interest.
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In this work, we consider the dynamic frictionless Euler–Bernoulli equation with the Signorini contact conditions along the length of a thin beam. The existence of solutions is proved based on the penalty method. Employing energy... more
In this work, we consider the dynamic frictionless Euler–Bernoulli equation with the Signorini contact conditions along the length of a thin beam. The existence of solutions is proved based on the penalty method. Employing energy functional with the penalty method, we bound integral of contact forces over space and time. Hölder continuity of the fundamental solution plays an important role
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Uniqueness is shown for a class of index-one differential variational inequalities (DVIs). This result assumes that a certain matrix function is symmetric positive definite everywhere. Examples are given to show that this matrix being... more
Uniqueness is shown for a class of index-one differential variational inequalities (DVIs). This result assumes that a certain matrix function is symmetric positive definite everywhere. Examples are given to show that this matrix being merely positive definite is not sufficient for uniqueness, even though this is sufficient for existence.
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Research Interests: Mathematics, Applied Mathematics, Computer Science, Game Theory, Mathematical Programming, and 15 moreModeling, Numerical Analysis, Inequality, Discontinuity, Dynamical System, Evolutionary Algorithm, Euclidean space, Indexation, Differential equation, Boundary Value Problem, Hybrid System, Engineering System, Complementarity Problem, Euler Equation, and Algebraic equation
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... to the solutions of the continuous-time problems. For a treatment of convergence to the discrete-time approximation to the continuous-time solution in the rigid-body case, see Stewart Ref. [45]. The main goal of this paper is to ...
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ABSTRACT We formulate a dynamic frictionless contact problem with linear viscoelasticity of Kelvin-Voigt type, based on the Signorini contact conditions. We show existence of solutions, and investigate the possibility for obtaining an... more
ABSTRACT We formulate a dynamic frictionless contact problem with linear viscoelasticity of Kelvin-Voigt type, based on the Signorini contact conditions. We show existence of solutions, and investigate the possibility for obtaining an energy balance. Employing time discretization and the finite element method, we compute numerical solutions. Our numerical scheme is implemented with non-smooth Newton’s method which solves the complementarity problem. The numerical results support the idea that the energy losses in the limit of the numerical solution are equal to the losses due to viscosity.