Abstract: \begin{document} We have developed fast implementations of B-spline/NURBS based finite element solvers, written using PETSc. PETSc is frequently used in software packages to leverage its optimized and parallel implementation of solvers, however we also are using PETSc data structures to assemble the linear systems. These structures in PETSC (called DA’s) were originally intended for the parallel assembly of linear systems resulting from finite differences. We have reworked this structure for linear systems resulting from isogeometric analysis based on tensor product spline spaces. The result of which is the PetIGA framework for solving problems using isogeometric analysis which is scalable and greatly simplified over previous solvers. Our infrastructure has also allowed us to develop scalable solvers for a variety of problems. We have chosen to pursue nonlinear time dependent problems~\cite{PetIGAp, PetIGAc}, such as: \begin{itemize} \item Cahn-Hilliard \item Navier-Stokes-Korteweg \item Variational Multiscale for Navier-Stokes \item Diffusive Wave Approximation to Shallow Water Equations \item Phase-Field Crystal (PFC) equation and its time integration \item Divergence-conforming B-spline modeling of nanoparticle suspensions \end{itemize} We also have solvers for an assortment of linear problems: Poisson, elasticity, Helmholtz, thin shells, advection-diffusion, and diffusion-reaction. All solvers are written to be inherently parallel and run on anything from a laptop to a supercomputer such as Shaheen, KAUST’s IBM-BlueGeneP supercomputer. In this presentation we will focus on new time integration techniques for phase-field modeling which are energy stable and allow for stable linearizations of the underlying non-linear model~\cite{PFC}. \begin{thebibliography}{99} \setlength{\parskip}{0pt} \bibitem{PetIGAp} N. Collier, L. Dalcin, and V.M. Calo, ``PetIGA: High-Performance Isogeometric Analysis,'' submitted, 2013. \bibitem{PetIGAc} L. Dalcin and N. Collier, ``PetIGA: A framework for high performance Isogeometric Analysis,'' https://bitbucket.org/dalcinl/petiga/, 2013 \bibitem{PFC} P. Vignal, L. Dalcin, D.L. Brown, N. Collier, and V.M. Calo, ``Energy-stable time-discretizations for the phase-field crystal equation,'' in preparation, 2014. \end{thebibliography}

Abstract: In this paper, we present a multi-frontal solver algorithm for the adaptive finite element method expressed by graph grammar productions. The graph grammar productions construct first the binary elimination tree, and then process frontal matrices stored in distributed manner in nodes of the elimination tree. The solver is specialized for a class of one, two and three dimensional h refined meshes whose elimination tree has a regular structure. In particular, this class contains all one dimensional grids, two and three dimensional grids refined towards point singularities, two dimensional grids refined in an anisotropic way towards edge singularity as well as three dimensional grids refined in an anisotropic way towards edge or face singularities. In all these cases, the structure of the elimination tree and the structure of the frontal matrices are similar. The solver is implemented within the Galois environment, which allows parallel execution of graph grammar productions. We also compare the performance of the Galois implementation of our graph grammar based solver with the MUMPS solver

Abstract: For the simulation of wave propagation problems, it is necessary to truncate the computational domain. Perfectly Matched Layers are often employed for that purpose, especially in high contrast layered materials where absorbing boundary conditions are difficult to design. In here, we define a Perfectly Matched Layer that automatically adjusts its parameters without any user interaction. The user only has to indicate the desired decay in the surrounding layer. With this Perfectly Matched Layer, we show that even in the most complex scenarios where the material contrast properties are as high as sixteen orders of magnitude, we do not introduce numerical reflections when truncating the domain, thus, obtaining accurate solutions.

Abstract: In this paper we present a theoretical proof of linear computational cost and complexity for a recently developed direct solver driven by hypergraph grammar productions. The solver is specialized for computational meshes with point singularities in two and three dimensions. Linear complexity is achieved due to utilizing the special structure of such grids. We describe the algorithm and estimate the exact computational cost on an example of a two-dimensional mesh containing a point singularity. We extend this reasoning to the three dimensional meshes. Numerical results fully support our theoretical estimates.

Abstract: In a dynamic world, the process of prioritizing where to invest limited conservation resources is extremely complex. It needs to incorporate information on features (species, or landforms), planning units, ongoing or predicted future threats, and the costs and effectiveness of potential conservation actions. Extended research has been conducted on the spatial and temporal conservation prioritization using software tools such as Marxan, C-Plan, and Zonation to aid managers in their decision-making process. However, these tools are limited in various ways in addressing the full complexity of day-to-day management decisions. Some tools fail to consider variation in: land values in space and time; multiple threats and their spatio-temporal variations; multiple conservation actions applied to individual areas; the feasibility, effectiveness, and varying costs of actions; and the dynamic nature of biodiversity responses in space and time. Optimizing such a multi-dimensional system is a large challenge in complexity mathematics. What is needed is a new software tool that builds on current approaches, but allows for more realistic scenarios as described above, developed and parameterised in close collaboration with managers. This includes the modification of existing tools and the creation of new algorithms. The new software will be trialled in conservation planning exercises for islands in north-western Western Australia and the Great Barrier Reef. The current approaches mostly exploit simulated annealing as it was proven the fastest and sufficiently efficient for problems which do not need the best solution. The new software, however, intends to include sub-models on threats, costs, and contribution of action on individual islands. We are examining the option of use constraint programming to incorporate these sub-models into the decision process, with desirable time resolution.

Abstract: In this paper, we present a high-performance framework for solving partial differential equations using Isogeometric Analysis. It is called PetIGA, and in this work we show how it can be used to solve phase-field problems. We specifically chose the Cahn-Hilliard equation, and the phase-field crystal equation as study-problems. These two models allow us to highlight some of the main advantages that we have access to while using PetIGA for scientific computing.

Abstract: We discretized the two-dimensional linear momentum, microrotation, energy and mass conservation equations from the microrotational theory, with the finite element method, using B-spline basis to create divergence conforming spaces to obtain pointwise divergence free solutions [8]. Weak boundary conditions impositions was handled using Niche’s method for tangential conditions, while normal conditions were imposed strongly.We solved the heat driven cavity problem as a test case, including a variation of the parameters that differentiate micropolar fluids from conventional fluids under different Rayleigh numbers, for a better understanding of the system.

Abstract: In this paper we present a hypergraph grammar model for transformations of two and three dimensional grids. The hypergraph grammar describes the proces for generating uniform grids with two or three dimensional rectangular or hexahedral elements, followed by the proces of h refinements, which involves breaking selected elements into four or eight son elements, in two or three dimensions, respectively. We also provide graph grammar productions for two projection algorithms we use to pre-process material data. The first one is the projection based interpolation solver algorithm used for computing H1 or L2 projections of MRI scan of human head, in two and three dimensions. The second one is utilized for solving the non-stationary problem modeling the three dimensional heat transport in the human head generated by the cellphone usage.

Abstract: After applying the Finite Element Method (FEM) to the diffusion-type and wave-type Partial Differential Equations (PDEs), a first order and a second order Ordinary Differential Equation (ODE) systems are obtained respectively. These ODE systems usually present high stiffness, so numerical methods with good stability properties are required in their resolution. MATLAB offers a set of open source adaptive step functions for solving ODEs. One of these functions is the ode15s recommended for stiff problems and which is based on the Backward Differentiation Formulae (BDF). We describe the error estimation and the step size control implemented in this function. The ode15s is a variable order algorithm, and even though it has an adaptive step size implementation, the advancing formula and the local error estimation that uses correspond to the constant step size formula. We have focused on the second order accurate and unconditionally stable BDF (BDF2) and we have implemented a real adaptive step size BDF2 algorithm using the same strategy as the BDF2 implemented in the ode15s, resulting the new algorithm more efficient than the one implemented in MATLAB.

Abstract: This paper presents a graph-transformation-based multi-frontal direct solver with an optimization technique that allows for a significant decrease of time complexity in some multi-scale simulations of the Step and Flash Imprint Lithography (SFIL). The multi-scale simulation consists of a macro-scale linear elasticity model with thermal expansion coefficient and a nano-scale molecular statics model. The algorithm is exemplified with a photopolimerization simulation that involves densification of a polymer inside a feature followed by shrinkage of the feature after removal of the template. The solver is optimized thanks to a mechanism of reusing sub-domains with similar geometries and similar material properties. The graph transformation formalism is used to describe the algorithm - such an approach helps automatically localize sub-domains that can be reused.

Abstract: Hybridizing agent-based paradigm with evolutionary computation can enhance the field of meta-heuristics in a significant way, giving to usually passive individuals autonomy and capabilities of perception and interaction with other ones, treating them as agents. In the paper as a follow-up to the previous research, an evolutionary multi-agent system (EMAS) is examined in difficult discrete benchmark problems. As a means for comparison, classical evolutionary algorithm (constructed along with Michalewicz model) implemented in island-model is used. The results encourage for further research regarding application of EMAS in discrete problem domain.

Abstract: Agents provide a feasible mean for maintaining and manipulating large scale data. This paper deals with the problem of information exchange between different agents. It uses graph based formalism for the representation of knowledge maintained by an agent and graph transformations as a mean of knowledge exchange. Such a rigorous formalism ensures the cohesion of graph-based knowledge held by agents after each modification and exchange action. The approach presented in this paper is illustrated by a case study dealing with the problem of personal data held in different places (maintained by different agents) and the process of transmitting such information

Abstract: In this paper we propose an agent-based system for Service-Oriented Architecture self-adaptation. Services are supervised by autonomous agents which are responsible for deciding which service should be chosen for interoperation. Agents learn the choice strategy autonomously using supervised learning. In experiments we show that supervised learning (Naive Bayes, C4.5 and Ripper) allows to achieve much better efficiency than simple strategies such as random choice or round robin. What is also important, supervised learning generates a knowledge in a readable form, which may be analyzed by experts.

Abstract: Metaheuristics resulting from the hybridization of multi-agent systems with evolutionary computing are efficient in many optimization problems. Evolutionary multi-agent systems (EMAS) are more similar to biological evolution than classical evolutionary algorithms. However, technological limitations prevented the use of fully asynchronous agents in previous EMAS implementations. In this paper we present a new algorithm for agent-based evolutionary computations. The individuals are represented as fully autonomous and asynchronous agents. Evolutionary operations are performed continuously and no artificial generations need to be distinguished. Our results show that such asynchronous evolutionary operators and the resulting absence of explicit generations lead to significantly better results. An efficient implementation of this algorithm was possible through the use of Erlang technology, which natively supports lightweight processes and asynchronous communication.

Abstract: In this paper we present a hypergraph grammar based multi-frontal solver for three dimensional grids with point singularities. We show experimentally that the computational cost of the resulting solver algorithm is linear with respect to the number of degrees of freedom. We also propose a reutilization algorithm that enables to reuse LU factorizations over unrefined parts of the mesh when new local refinements are executed by the hypergraph grammar productions.