Containing epidemic outbreaks by message-passing techniques

The problem of targeted network immunization can be defined as the one of finding a subset of nodes in a network to immunize or vaccinate in order to minimize a tradeoff between the cost of vaccination and the final (stationary) expected infection under a given epidemic model. Although computing the expected infection is a hard computational problem, simple and efficient mean-field approximations have been put forward in the literature in recent years. The optimization problem can be recast into a constrained one in which the constraints enforce local mean-field equations describing the average stationary state of the epidemic process. For a wide class of epidemic models, including the susceptible-infected-removed and the susceptible-infected-susceptible models, we define a message-passing approach to network immunization that allows us to study the statistical properties of epidemic outbreaks in the presence of immunized nodes as well as to find (nearly) optimal immunization sets for a given choice of parameters and costs. The algorithm scales linearly with the size of the graph and it can be made efficient even on large networks. We compare its performance with topologically based heuristics, greedy methods, and simulated annealing

A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementationComment: 25 page

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

High order resolution and parallel implementation on unstructured grids

The numerical solution of the two-dimensional inviscid Euler flow equations is given. The unstructured mesh is generated by the advancing front technique. A cell-centred upwind finite volume method has been adopted to discretize the Euler equations. Both explicit and point implicit time stepping algorithms are derived. The flux calculation using Roe's and Osher's approximate Riemann solvers are studied. It is shown that both the Roe and Osher's schemes produce an accurate representation of discontinuities (e.g. shock wave). It is also shown that better convergence performance has been achieved by the point implicit scheme than that by the explicit scheme. Validations have been done for subsonic and transonic flow over airfoils, supersonic flow past a compression corner and hypersonic flow past cylinder and blunt body geometries. An adaptive remeshing procedure is also applied to the numerical solution with the objective of getting improved results. The issue of high order reconstruction on unstructured grids has been discussed. The methodology of the Taylor series expansion is adopted. The calculation of the gradient at a reference point is carried out by the use of either the Green-Gauss integral formula or the least-square methods. Some recently developed limiter construction methods have been used and their performance has been demonstrated using the test example of the transonic flow over a RAE 2822 airfoil. It has been shown that similar pressure distributions are obtained by all limiters except for shock wave regions where the limiter is active. The convergence problem is illustrated by the mid-mod type limiter. It seems only the Venkatakrishnan limiter provides improved convergence. Other limiters do not appear to work as well as that shown in their original publications. Also the convergence history given by the least-square method appears better than that by the Green-Gauss method in the test

A scalable H-matrix approach for the solution of boundary integral equations on multi-GPU clusters

In this work, we consider the solution of boundary integral equations by means of a scalable hierarchical matrix approach on clusters equipped with graphics hardware, i.e. graphics processing units (GPUs). To this end, we extend our existing single-GPU hierarchical matrix library hmglib such that it is able to scale on many GPUs and such that it can be coupled to arbitrary application codes. Using a model GPU implementation of a boundary element method (BEM) solver, we are able to achieve more than 67 percent relative parallel speed-up going from 128 to 1024 GPUs for a model geometry test case with 1.5 million unknowns and a real-world geometry test case with almost 1.2 million unknowns. On 1024 GPUs of the cluster Titan, it takes less than 6 minutes to solve the 1.5 million unknowns problem, with 5.7 minutes for the setup phase and 20 seconds for the iterative solver. To the best of the authors' knowledge, we here discuss the first fully GPU-based distributed-memory parallel hierarchical matrix Open Source library using the traditional H-matrix format and adaptive cross approximation with an application to BEM problems

Containing Epidemic Outbreaks by Message-Passing Techniques

The problem of targeted network immunization can be defined as the one of finding a subset of nodes in a network to immunize or vaccinate in order to minimize a tradeoff between the cost of vaccination and the final (stationary) expected infection under a given epidemic model. Although computing the expected infection is a hard computational problem, simple and efficient mean-field approximations have been put forward in the literature in recent years. The optimization problem can be recast into a constrained one in which the constraints enforce local mean-field equations describing the average stationary state of the epidemic process. For a wide class of epidemic models, including the susceptible-infected-removed and the susceptible-infected-susceptible models, we define a message-passing approach to network immunization that allows us to study the statistical properties of epidemic outbreaks in the presence of immunized nodes as well as to find (nearly) optimal immunization sets for a given choice of parameters and costs. The algorithm scales linearly with the size of the graph, and it can be made efficient even on large networks. We compare its performance with topologically based heuristics, greedy methods, and simulated annealing on both random graphs and real-world networks

ParMooN - a modernized program package based on mapped finite elements

{\sc ParMooN} is a program package for the numerical solution of elliptic and parabolic partial differential equations. It inherits the distinct features of its predecessor {\sc MooNMD} \cite{JM04}: strict decoupling of geometry and finite element spaces, implementation of mapped finite elements as their definition can be found in textbooks, and a geometric multigrid preconditioner with the option to use different finite element spaces on different levels of the multigrid hierarchy. After having presented some thoughts about in-house research codes, this paper focuses on aspects of the parallelization for a distributed memory environment, which is the main novelty of {\sc ParMooN}. Numerical studies, performed on compute servers, assess the efficiency of the parallelized geometric multigrid preconditioner in comparison with some parallel solvers that are available in the library {\sc PETSc}. The results of these studies give a first indication whether the cumbersome implementation of the parallelized geometric multigrid method was worthwhile or not.Comment: partly supported by European Union (EU), Horizon 2020, Marie Sk{\l}odowska-Curie Innovative Training Networks (ITN-EID), MIMESIS, grant number 67571

A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementationbackward stochastic differential equations, parallel computing, Monte- Carlo methods, non linear PDE, American options, local volatility model.