In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.

Conservation laws ae solved by a local Gaerkin finite element procedure with adapfive space-time mesh refinement ad explicit time integration. The Courat stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to methods having a single global time step. Processor load imbalaces, introduced at adaptive enrichment steps, are corrected by using traversals of an octtee representing a spatial decomposition of the domain. To accommodate the variable time steps, octtee partitioning is extended to use weights derived from element size. Partition boundary smoothing reduces the communications volume of partitioning procedures for a modest cost. Computational results comparing parallel octtee ad inertial partitioning procedures ae presented for the three-dimensional Euler equations of compressible flow solved on an IBM SP2 computer.

We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independently, and a nearly load-balanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection. Only the initial fan-out of the mesh decomposition to the processors requires communication. Two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution, namely, the construction of a global conforming mesh from the independent subproblems, followed by a final smoothing phase using the subdomain solutions as an initial guess. We present a series of convincing numerical experiments that illustrate the e#ectiveness of this approach. The justification of the initial refinement prediction step, as well as the justification of skipping the two communication-intensive steps, ...

Adaptive scientific simulations require that periodic repartitioning occur dynamically throughout the course of the computation. The repartitionings should be computed so as to minimize both the inter-processor communications incurred during the iterative mesh-based computation and the data redistribution costs required to balance the load. Recently developed schemes for computing repartitionings provide the user with only a limited control of the tradeoffs among these objectives. This paper describes a new Unified Repartitioning Algorithm that can tradeoff one objective for the other dependent upon a user-defined parameter describing the relative costs of these objectives. We show that the Unified Repartitioning Algorithm is able to reduce the precise overheads associated with repartitioning as well as or better than other repartitioning schemes for a variety of problems, regardless of the relative costs of performing inter-processor communication and data redistribution. Our experimental results show that this scheme is extremely fast and scalable to large problems.

Introduction The finite element method (FEM) has become a standard analysis tool for solving partial differential equations (PDEs). Computationally demanding threedimensional problems make adaptive methods and parallel computation essential. Adaptive FEMs provide reliability, robustness, and time and space efficiency. In such a method, the computational domain is discretized into a mesh. During the adaptive solution process, portions of the mesh may be refined or coarsened (h-refinement) or moved to follow evolving phenomena (r-refinement). The method order may also be varied (p-refinement). Each adaptive process concentrates the computational effort in areas where the solution resolution would otherwise be inadequate [7]. Conventional array-based data representations, which work well for fixed-mesh solutions, are not wellsuited to solutions involving mesh adaptivity [1]. Traversal of the data must be efficient in all cases, but w

The Zoltan dynamic load balancing library provides applications with a reusable object oriented interface to several load balancing techniques, including coordinate bisection, octree/space filling curve methods, and multilevel graph partitioners. We describe enhancements to Zoltan’s octree load balancing procedure and its distributed structures that improve performance of the space filling curve (SFC) traversals by

Large-scale, dynamic, and heterogeneous networks of computational resources (a.k.a. grids) promise to provide high performance and scalability to computationally intensive applications. To fulfill this promise, grid environments require complex resource management. We propose decentralized middlewaretriggered dynamic reconfiguration strategies to enable application adaptation to the constantly changing resource availability of Internet-scale shared computational grids. As a proof of concept, we present a software framework for dynamically reconfigurable distributed applications. The Internet Operating System (IOS) is a middleware infrastructure which aims at freeing application developers from dealing with non-functional concerns while seeking to optimize application performance and global resource utilization. IOS consists of distributed middleware agents that are capable of interconnecting themselves in various virtual peer-to-peer topologies. IOS middleware agents: 1) profile application communication patterns, 2) evaluate the dynamics of the underlying physical resources, and 3) reconfigure application components by changing their mappings to physical resources through migration and by changing their granularity through a split and merge mechanism. A key characteristic of IOS is its decentralized coordination, thereby avoiding the use of global knowledge and thus enabling scalable reconfiguration. The IOS middleware is programming model-independent: we have implemented an actor programming model interface for SALSA programs and also a process programming model interface for MPI programs. Experimental results show that adaptive middleware can be an effective approach to reconfiguring distributed applications with various ratios of communication to computation in order to improve their performance, and more effectively utilize grid resources. 1

Abstract. With the proliferation of large scale dynamic execution environments such as grids, the need for providing efficient and scalable application adaptation strategies for long running parallel and distributed applications has emerged. Message passing interfaces have been initially designed with a traditional machine model in mind which assumes homogeneous and static environments. It is inevitable that long running message passing applications will require support for dynamic reconfiguration to maintain high performance under varying load conditions. In this paper we describe a framework that provides iterative MPI applications with reconfiguration capabilities. Our approach is based on integrating MPI applications with a middleware that supports process migration and large scale distributed application reconfiguration. We present our architecture for reconfiguring MPI applications, and verify our design with a heat diffusion application in a dynamic setting. 1

We consider the problem of redistributing data on homogeneous and heterogeneous ring of processors. The problem arises in several applications, each time after that a load-balancing mechanism is invoked (but we do not discuss the load-balancing mechanism itself). We provide algorithms that aim at optimizing the data redistribution, both for unidirectional and bi-directional rings, and we give complete proofs of correctness. One major contribution of the paper is that we are able to prove the optimality of the proposed algorithms in all cases except that of a bi-directional heterogeneous ring, for which the problem remains open.

This paper describes two predictive load balancing schemes designed for use with parallel adaptive finite element methods. We also provide an overview of data structures suitable for distributed storage of finite element mesh data as well as software designed for mesh adaptation and load balancing. During the course of a parallel computation, processor load irabalances are introduced at adaptive enrichment steps. The predictive load balancing methods introduced here use a priori estimates of work load for adaptive refinement and subsequent computation to improve enrichment ciency and reduce total balancing time. These components have been used to build a system for solving compressible flow problems. Computational results on an IBM SP2 computer are presented for transient solutions of the three-dimensional Euler equations of compressible flow.