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Solving large systems of linear equations on GPUs
(Springer Verlag, 2018-01-01) Llano-Ríos T.F.; Ocampo-García J.D.; Yepes-Ríos J.S.; Correa-Zabala F.J.; Trefftz C.; Llano-Ríos T.F.; Ocampo-García J.D.; Yepes-Ríos J.S.; Correa-Zabala F.J.; Trefftz C.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Graphical Processing Units (GPUs) have become more accessible peripheral devices with great computing capacity. Moreover, GPUs can be used not only to accelerate the graphics produced by a computer but also for general purpose computing. Many researchers use this technique on their personal workstations to accelerate the execution of their programs and have often encountered that the amount of memory available on GPU cards is typically smaller than the amount of memory available on the host computer. We are interested in exploring approaches to solve problems with this restriction. Our main contribution is to devise ways in which portions of the problem can be moved to the memory of the GPU to be solved using its multiprocessing capabilities. We implemented on a GPU the Jacobi iterative method to solve systems of linear equations and report the details from the results obtained, analyzing its performance and accuracy. Our code solves a system of linear equations large enough to exceed the card’s memory, but not the host memory. Significant speedups were observed, as the execution time taken to solve each system is faster than those obtained with Intel® MKL and Eigen, libraries designed to work on CPUs. © Springer Nature Switzerland AG 2018.
Solving Large Systems of Linear Equations on GPUs
(Springer, 2018-08-19) Ocampo, Juan Diego; Yepes, Johan; Zabala F.J.C.; Trefftz, Christian; Ocampo, Juan Diego; Yepes, Johan; Zabala F.J.C.; Trefftz, Christian; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Graphical Processing Units (GPUs) have become more accessible peripheral devices with great computing capacity. Moreover, GPUs can be used not only to accelerate the graphics produced by a computer but also for general purpose computing.
Implementing an active learning platform to support student learning in a numerical analysis course
(Institute of Electrical and Electronics Engineers Inc., 2017-01-01) Zabala F.J.C.; Parker H.E.; Vieira C.; Zabala F.J.C.; Parker H.E.; Vieira C.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Classroom instruction in the 21st century needs to incorporate innovative, research-based pedagogies. The engineering classroom is currently experiencing a shift towards more active learning activities due to both advances in educational research, and advances in technologies that enable practices such as the flipped classroom model. Given that course transformation is a gradual process that begins at the level of the instructor, educators need access to the essential tools and training in order to introduce these changes into the curricula. This paper introduces a course re-design based on Self-Determination Theory and Constructivism; and outlines effectively implemented active learning strategies using the flipped classroom model. The data were collected from a Numerical Analysis course, which is an important course across several engineering disciplines at Universidad EAFIT. This course enables engineering students to solve complex problems using mathematical and computational methods. This paper describes the implementation of an online active learning platform called "Numérico Interactivo" for two related engineering courses: Numerical Analysis (NA) and Numerical Processes (NP). The platform was available to all students, but only NA implemented it using a flipped classroom model. NP made the platform available as an optional course tool. Informed by SDT principles, "Numérico Interactivo" includes a variety of instructional materials such as explanations, examples, frequently asked questions (FAQ), self-assessment tools, and evaluation. This study compares the two courses in terms of: (1) students' perceptions about the instructional materials of the course; (2) students' use of the platform; and (3) students' perceived usefulness of the different elements within the platform. Results suggest that students in the NA course found the classroom sessions and the homework assignments more useful as compared to the students enrolled in the NP course. In addition, in the NA course students used the platform more often for class preparation and to study before each module. The way in which the platform was implemented in NA also increased student motivation in the course. Overall, the results suggest that "Numérico Interactivo" is useful to implement course re-designs into engineering and computing education courses, but such tools need to be guided by active learning practices so that students can fully benefit from them. © 2017 IEEE.
Solving stochastic epidemiological models using computer algebra
(SPIE-INT SOC OPTICAL ENGINEERING, 2011-01-01) Hincapie, D.; Ospina, J.; Hincapie, D.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Mathematical modeling in Epidemiology is an important tool to understand the ways under which the diseases are transmitted and controlled. The mathematical modeling can be implemented via deterministic or stochastic models. Deterministic models are based on short systems of non-linear ordinary differential equations and the stochastic models are based on very large systems of linear differential equations. Deterministic models admit complete, rigorous and automatic analysis of stability both local and global from which is possible to derive the algebraic expressions for the basic reproductive number and the corresponding epidemic thresholds using computer algebra software. Stochastic models are more difficult to treat and the analysis of their properties requires complicated considerations in statistical mathematics. In this work we propose to use computer algebra software with the aim to solve epidemic stochastic models such as the SIR model and the carrier-borne model. Specifically we use Maple to solve these stochastic models in the case of small groups and we obtain results that do not appear in standard textbooks or in the books updated on stochastic models in epidemiology. From our results we derive expressions which coincide with those obtained in the classical texts using advanced procedures in mathematical statistics. Our algorithms can be extended for other stochastic models in epidemiology and this shows the power of computer algebra software not only for analysis of deterministic models but also for the analysis of stochastic models. We also perform numerical simulations with our algebraic results and we made estimations for the basic parameters as the basic reproductive rate and the stochastic threshold theorem. We claim that our algorithms and results are important tools to control the diseases in a globalized world. © 2011 Copyright Society of Photo-Optical Instrumentation Engineers (SPIE).
Symbolic solution for generalized quantum cylindrical wells using computer algebra
(2008-01-01) Pulgarin, E.Y.V.; Pulgarin, E.Y.V.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
This paper present how to solve the problem of cylindrical quantum wells with potential energy different from zero and with singularity of the energy on the axis of the cylinder. The solution to the problem was obtained using methods of computer algebra. The results depend of Bessel and Kummer functions. This paper present energy levels and wave functions in some of the cases with an exactly form and in other cases with an approximated form, this form depended on the possibility of integrating the special functions and calculating the zeros of these functions. Here we can see the power of the method in the applications concerning complex problems of quantum mechanics, and the possibility of being able to apply this method in order to solve other problems in science and also in engineering.
Possible topological quantum computation via khovanov homology: D-brane topological quantum computer
(SPIE-INT SOC OPTICAL ENGINEERING, 2009-01-01) Vélez, M.; Ospina, J.; Vélez, M.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
A model of a D-Brane Topological Quantum Computer (DBTQC) is presented and sustained. The model isbased on four-dimensional TQFTs of the Donaldson-Witten and Seiber-Witten kinds. It is argued that the DBTQC is able to compute Khovanov homology for knots, links and graphs. The DBTQC physically incorporates the mathematical process of categorification according to which the invariant polynomials for knots, links and graphs such as Jones, HOMFLY, Tutte and Bollobás-Riordan polynomials can be computed as the Euler characteristics corresponding to special homology complexes associated with knots, links and graphs. The DBTQC is conjectured as a powerful universal quantum computer in the sense that the DBTQC computes Khovanov homology which is considered like powerful that the Jones polynomial. © 2009 SPIE.
Embedding a logical Theory of Constructions in agda
(2009-01-01) Bove, A.; Dybjer, P.; Sicard-Ra?irez, A.; Bove, A.; Dybjer, P.; Sicard-Ra?irez, A.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
We propose a new way to reason about general recursive functional in the dependently typed programming language Agda,is based on Martin-Löf's intuitionistic type theory. We show to embed an external programming logic, Aczel's Logical Theory Constructions (LTC) inside Agda. To this end we postulate existence of a domain of untyped functional programs and the rules for these programs. Furthermore, we represent the notions in LTC (intuitionistic predicate logic with equality,totality predicates) as inductive notions in Agda. To illustrate approach we specify an LTC-style logic for PCF, and show to prove the termination and correctness of a general recursive for computing the greatest common divisor of two numbers. © 2009.
Possible quantum algorithms for the Bollobás-Riordan-Tutte polynomial of a ribbon graph
(SPIE-INT SOC OPTICAL ENGINEERING, 2008-01-01) Velez, M.; Ospina, J.; Velez, M.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Three possible quantum algorithms, for the computation of the Bollobás-Riordan-Tutte polynomial of a given ribbon graph, are presented and discussed. The first possible algorithm is based on the spanning quasi-trees expansion for generalized Tutte polynomials of generalized graphs and on a quantum version of the Binary Decision Diagram (BDD) for quasi-trees . The second possible algorithm is based on the relation between the Kauffman bracket and the Tutte polynomial; and with an application of the recently introduced Aharonov-Arad-Eban-Landau quantum algorithm. The third possible algorithm is based on the relation between the HOMFLY polynomial and the Tutte polynomial; and with an application of the Wocjan-Yard quantum algorithm. It is claimed that these possible algorithms may be more efficient that the best known classical algorithms. These three algorithms may have interesting applications in computer science at general or in computational biology and bio-informatics in particular. A line for future research based on the categorification project is mentioned.
Two qutrits universal quantum gates from the nine-dimensional unitary solutions of the Yang-Baxter equation
(SPIE-INT SOC OPTICAL ENGINEERING, 2007-04-25) Vélez, M.; Ospina, J.; Vélez, M.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Using the Kauffman-Lomonaco method, some two-qutrits universal quantum gates are derived from the nine-dimensional unitary solutions of the Yang-Baxter equations associated with algebraic structures like the partial transpose operator and the dihedral group, which admit three dimensional representations. The Yang-Baxterization method given by Zhang-Kauffman-Ge is continuously used to obtain two-qutrits quantum gates and certain Hamiltonians for the evolution of the quantum gates are obtained, being such Hamiltonians interpreted as physical Hamiltonians of chain of particles of spin 1. Finally, the generalization for systems of two qudits is presented in the case of Yang-Baxterization of representations of braided monoidal algebra like the BH algebra and the bicolored Birman-Wenzl-Muraki algebra For these algebras the corresponding two-qudits quantum gates are constructed jointly with the associated Hamiltonians interpreted like physical chains of particles with spin d . It is conjectured that the derived two-qdits quantum gates and the Hamiltonians may be implemented over bi-dimensional lattice systems like anyons systems or more generally over any physical systems ruled by the Yang-Baxter equations.
The Critical Proportion of Immune Individuals needed to Control Hepatitis B
(SPIE-INT SOC OPTICAL ENGINEERING, 2016-05-13) Ospina, J.; Hincapié-Palacio, D.; Ospina, J.; Hincapié-Palacio, D.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
We estimate the critical proportion of immunity (Pc) to control hepatitis B in Medellin - Colombia, based on a random population survey of 2077 individuals of 6-64 years of age. The force of infection (Fi) was estimated according to empirical data of susceptibility by age S(a), assuming a quadratic expression. Parameters were estimated by adjusting data to a nonlinear regression. Fi was defined by -(ds(a)/da)/s(a) and according to the form of the empirical curve S(a) we assume a quadratic expression given by S(a)= Ea2+Ba+C. Then we have the explicit expression for the accumulated Fi by age given by F(a) = -a(Ea+B)/c. The expression of average infection age A is obtained as A = L + EL3/(3C)+BL2/(2C) and the basic reproductive number R-0 is obtained as R-0 = 1 + 6C/(6C+2EL2+3BL). From the las result we obtain the Pc given by Pc=6C/(12C+2EL2+3BL). Numerical simulations were performed with the age-susceptibility proportion and initial values (a=0.02, b=20, c=100), obtaining an adjusted coefficient of multiple determination of 64.83%. According to the best estimate, the algebraic expressions for S(a) and the Fi were derived. Using the result of Fi, we obtain A = 30, L = 85; R-0 CI 95%: 1.42 - 1.64 and Pc: 0-0.29. These results indicate that at the worst case, to maintain control of the disease should be immunes at least 30% of susceptible individuals. Similar results were obtained by sex and residential area.
Solution of the wave equation for an oscillating punctual source by means of the finite elements method
(SPIE-INT SOC OPTICAL ENGINEERING, 2016-01-01) Marín-Suárez, M.; Marín-Suárez, M.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Finite Elements Method (FEM) and Finite Differences Method (FDM) in the Euler implicit approach, were used in order to determine the bi-dimensional electromagnetic field of an oscillating punctual source inside a square geometry with three reflective boundaries and an absorbing one. Although the main approach consist in solving the wave equation for the electric field, it was decided to use the approach of electromagnetic potentials V and A because those potentials give a simpler solution for both, the electric and magnetic fields. Besides, a comparison with theory was made, solving for an electromagnetic field without boundaries produced by an oscillating punctual source.
Dynamics of an Electrochemical Biosensor for the Detection of Toxic Substances in Water
(SPIE-INT SOC OPTICAL ENGINEERING, 2016-01-01) Simon, L.; Ospina, J.; Simon, L.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
A proposed analytical method focuses on electrolyte transport to the electrode of an electrochemical cell. The recombinant Escherichia coli whole-cell biosensor detects toxicity in water based on a set of biochemical reactors. Previous contributions elucidated the kinetics of product formation and validated a mathematical model for its diffusion in the chamber. This work introduces an approach to investigate the dynamics of the probe using Laplace transforms and an effective time constant. The transfer function between the electrolyte production and the total current revealed a faster response for larger electrode radii. Both the first-order and effective time constants increased with the chamber height and radius. Separation of variables yields closed-form solutions and helps estimate the kinetics of p-aminophenol generation. When the bacteria were exposed to phenol concentrations of 1.6, 8.3 and 16 ppm, the corresponding overall rate constants were 5.11x10(-7), 1.13x10(-6) and 1.99x10(-6) (product concentration unit/s(2)), respectively. In addition to parameter estimation, the method can be applied to perform sensitivity analysis and aid manufacturers in meeting design specifications of biosensors.
The epidemic threshold theorem with social and contact heterogeneity
(SPIE-INT SOC OPTICAL ENGINEERING, 2008-03-17) Palacio, D.H.; Giraldo, J.O.; Arias, R.D.G.; Palacio, D.H.; Giraldo, J.O.; Arias, R.D.G.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
The threshold theorem of an epidemic SIR model was compared when infectious and susceptible individuals have homogeneous mixing and heterogeneous social status and when individuals of random networks have contact heterogeneity. Particularly the effect of vaccination in such models is considered when: individuals or nodes are exposed to impoverished, vaccination and loss of immunity. An equilibrium analysis and local stability of small perturbations about the equilibrium values were implemented using computer algebra. Numerical simulations were executed in order to describe the dynamic of transmission of diseases and changes of the basic reproductive rate. The implications of these results are examined around the threats to the global public health security.
Basic reproductive rate of a spatial epidemic model using computer algebra software
(2005-01-01) Doracelly Hincapié, P.; Juan Ospina, G.; Doracelly Hincapié, P.; Juan Ospina, G.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Using computer algebra software we obtain the basic reproductive rate corresponding to the propagation of a directly transmitted disease in a circular habitat when the disease is endemic at the boundary. The method used is the Laplace Transform Technique and calculus of residues. The results that were obtained include both the explicit form of the R0 for the boundary condition that was considered, as the explicit symbolic solution of the model equation. The method that was used can be extended to other more complex problems such as indirectly transmitted diseases with one or more intermediary hosts or effects of genetic, immunological, geographical or social heterogeneity in the human population. This application indicates that the computer algebra software for symbolic computation has a very promissory future in mathematical epidemiology.
Mathematical model for dengue with three states of infection
(SPIE-INT SOC OPTICAL ENGINEERING, 2012-01-01) Hincapie, D.; Ospina, J.; Hincapie, D.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
A mathematical model for dengue with three states of infection is proposed and analyzed. The model consists in a system of differential equations. The three states of infection are respectively asymptomatic, partially asymptomatic and fully asymptomatic. The model is analyzed using computer algebra software, specifically Maple, and the corresponding basic reproductive number and the epidemic threshold are computed. The resulting basic reproductive number is an algebraic synthesis of all epidemic parameters and it makes clear the possible control measures. The microscopic structure of the epidemic parameters is established using the quantum theory of the interactions between the atoms and radiation. In such approximation, the human individual is represented by an atom and the mosquitoes are represented by radiation. The force of infection from the mosquitoes to the humans is considered as the transition probability from the fundamental state of atom to excited states. The combination of computer algebra software and quantum theory provides a very complete formula for the basic reproductive number and the possible control measures tending to stop the propagation of the disease. It is claimed that such result may be important in military medicine and the proposed method can be applied to other vector-borne diseases. © 2012 SPIE.
Numerical simulations of a possible hypercomputational quantum algorithm
(SPRINGER-VERLAG WIEN, 2005-01-01) Sicard, A; Ospina, J; Velez, M; Sicard, A; Ospina, J; Velez, M; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
The hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that cannot be computed or solved by a Turing machine. Several numerical simulations of a possible hypercomputational algorithm based on quantum computations previously constructed by the authors are presented. The hypercomputability of our algorithm is based on the fact that this algorithm could solve a classically non-computable decision problem, the Hilbert's tenth problem. The numerical simulations were realized for three types of Diophantine equations: with and without solutions in non-negative integers, and without solutions by way of various traditional mathematical packages.
Dynamic analysis and performance evaluation of the BIAcore surface plasmon resonance biosensor
(SPIE-INT SOC OPTICAL ENGINEERING, 2015-01-01) Simon, Laurent; Ospina, Juan; Simon, Laurent; Ospina, Juan; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Solution procedures were proposed to analyze nonlinear mass transport through an optical biosensor. A generalized collocation technique was applied to predict the dynamic behavior of an analyte along the flow chamber as a result of convection, diffusion and chemical reaction. The method estimated the effective time constants for reaching average steady-state concentrations of the free and bound analytes in the cell. When diffusion in the direction of flow was neglected, a closed-form solution, based on double Laplace transforms, was obtained after linearizing the original system. In both models, an increase in the sample diffusion coefficient lowered the effective time constant. This approach may help researchers evaluate the performance of biosensors and meet specific design criteria. © 2015 SPIE.
Analysis of a generalized model for influenza including differential susceptibility due to immunosuppression
(SPIE-INT SOC OPTICAL ENGINEERING, 2014-01-01) Hincapié, D.; Ospina, J.; Hincapié, D.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Recently, a mathematical model of pandemic influenza was proposed including typical control strategies such as antivirals, vaccination and school closure; and considering explicitly the effects of immunity acquired from the early outbreaks on the ulterior outbreaks of the disease. In such model the algebraic expression for the basic reproduction number (without control strategies) and the effective reproduction number (with control strategies) were derived and numerically estimated. A drawback of this model of pandemic influenza is that it ignores the effects of the differential susceptibility due to immunosuppression and the effects of the complexity of the actual contact networks between individuals. We have developed a generalized model which includes such effects of heterogeneity. Specifically we consider the influence of the air network connectivity in the spread of pandemic influenza and the influence of the immunosuppresion when the population is divided in two immune classes. We use an algebraic expression, namely the Tutte polynomial, to characterize the complexity of the contact network. Until now, The influence of the air network connectivity in the spread of pandemic influenza has been studied numerically, but not algebraic expressions have been used to summarize the level of network complexity. The generalized model proposed here includes the typical control strategies previously mentioned (antivirals, vaccination and school closure) combined with restrictions on travel. For the generalized model the corresponding reproduction numbers will be algebraically computed and the effect of the contact network will be established in terms of the Tutte polynomial of the network. © 2014 Copyright SPIE.
Analysis of the boundary conditions for pyramidal spine fins in extended surface heat transfer
(WITPress, 2014-01-01) Carranza, R.G.; Ospina, J.; Carranza, R.G.; Ospina, J.; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
Many studies are conducted for pyramidal spine fins relating to temperature profiles and fin efficiencies. However, it is found that a deeper look into the boundary conditions is required. The conditions at the base are simple enough, but the conditions at the tip are more complicated. Despite what condition is applied to the tip, the value at the tip actually has a specific value that is a function of the square root of the Biot number. This result is proven using basic principles from calculus, like the Limit and L’Hospital’s rule. © 2014 WIT Press.
Topological and geometrical quantum computation in cohesive Khovanov homotopy type theory
(SPIE-INT SOC OPTICAL ENGINEERING, 2015-05-21) Ospina, Juan; Ospina, Juan; Universidad EAFIT. Departamento de Ciencias; Lógica y Computación
The recently proposed Cohesive Homotopy Type Theory is exploited as a formal foundation for central concepts in Topological and Geometrical Quantum Computation. Specifically the Cohesive Homotopy Type Theory provides a formal, logical approach to concepts like smoothness, cohomology and Khovanov homology; and such approach permits to clarify the quantum algorithms in the context of Topological and Geometrical Quantum Computation. In particular we consider the so-called a "open-closed stringy topological quantum computera" which is a theoretical topological quantum computer that employs a system of open-closed strings whose worldsheets are open-closed cobordisms. The open-closed stringy topological computer is able to compute the Khovanov homology for tangles and for hence it is a universal quantum computer given than any quantum computation is reduced to an instance of computation of the Khovanov homology for tangles. The universal algebra in this case is the Frobenius Algebra and the possible open-closed stringy topological quantum computers are forming a symmetric monoidal category which is equivalent to the category of knowledgeable Frobenius algebras. Then the mathematical design of an open-closed stringy topological quantum computer is involved with computations and theorem proving for generalized Frobenius algebras. Such computations and theorem proving can be performed automatically using the Automated Theorem Provers with the TPTP language and the SMT-solver Z3 with the SMT-LIB language. Some examples of application of ATPs and SMT-solvers in the mathematical setup of an open-closed stringy topological quantum computer will be provided. © 2015 SPIE.

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