Time Parallel Algorithms for Solution of time-Dependent PDE
Parallel algorithms for solution of time-dependent partial differential equations (PDEs) are developed.
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Parallel algorithms for solution of time-dependent partial differential equations (PDEs) are developed.
The use of hybrid interconnection structures that combine link connections and bus connections for real-time parallel processing is discussed. Idealistic parallel computation models for two real-time computing applications are described with attention given to a tightly coupled network model for object tracking and a network model for image processing. Consideration is given to the following different interconnection structures: the crossbar, the hypercube, the circular linked array, and the bus array.
Here, a major challenge in simulating the thermal behavior in additive manufacturing processes is the disparate length and time scales between transport phenomena occurring in the melt pool and the component. A common simulation approach relies on spatial decomposition for parallel computing, but due to the nature of heat transfer in AM, where most of the computational expenditure is localized near the melt pool, the computational speedup from spatial parallelization saturates quickly. Therefore, additional parallelism by means of time-domain decomposition is needed to fully take advantage of high-performance computing (HPC) resources. This work introduces a time-parallel method to improve the computational scalability of additive manufacturing simulations on HPC systems, while maintaining high temporal resolution of heat transfer near the melt pool. The method, inspired by the nonlinear paraexp formalism, performs an iterative superposition of nonlinear solutions to the initial value problem, integrating the heat equation across overlapping time-parallel intervals. For a single layer of the NIST AMB2018–01 L7 benchmark problem, the method achieves a 38.51x speedup in wall-clock time with a maximum error in the global temperature solution of 0.99%. This reduces the total solution time from 196.72 min to 5.11 min on 128 nodes of the ORNL Frontier supercomputer. The tradeoff between accuracy and total wall-clock time is investigated and recommendations for time-parallel deployment for AM problems are made.
Run time methods are studied to automatically parallelize and schedule iterations of a do loop in certain cases, where compile-time information is inadequate. The methods presented involve execution time preprocessing of the loop. At compile-time, these methods set up the framework for performing a loop dependency analysis. At run time, wave fronts of concurrently executable loop iterations are identified. Using this wavefront information, loop iterations are reordered for increased parallelism. Symbolic transformation rules are used to produce: inspector procedures that perform execution time preprocessing and executors or transformed versions of source code loop structures. These transformed loop structures carry out the calculations planned in the inspector procedures. Performance results are presented from experiments conducted on the Encore Multimax. These results illustrate that run time reordering of loop indices can have a significant impact on performance. Furthermore, the overheads associated with this type of reordering are amortized when the loop is executed several times with the same dependency structure.
Run-time methods are studied to automatically parallelize and schedule iterations of a do loop in certain cases where compile-time information is inadequate. The methods presented involve execution time preprocessing of the loop. At compile-time, these methods set up the framework for performing a loop dependency analysis. At run-time, wavefronts of concurrently executable loop iterations are identified. Using this wavefront information, loop iterations are reordered for increased parallelism. Symbolic transformation rules are used to produce: inspector procedures that perform execution time preprocessing, and executors or transformed versions of source code loop structures. These transformed loop structures carry out the calculations planned in the inspector procedures. Performance results are presented from experiments conducted on the Encore Multimax. These results illustrate that run-time reordering of loop indexes can have a significant impact on performance.
The possibility of using parallel processing to accelerate the simulation of Timed Petri Nets (TPN's) was studied. It was recognized that complex system development tools often transform system descriptions into TPN's or TPN-like models, which are then simulated to obtain information about system behavior. Viewed this way, it was important that the parallelization of TPN's be as automatic as possible, to admit the possibility of the parallelization being embedded in the system design tool. Later years of the grant were devoted to examining the problem of joint performance and reliability analysis, to explore whether both types of analysis could be accomplished within a single framework. In this final report, the results of our studies are summarized. We believe that the problem of parallelizing TPN's automatically for MIMD architectures has been almost completely solved for a large and important class of problems. Our initial investigations into joint performance/reliability analysis are two-fold; it was shown that Monte Carlo simulation, with importance sampling, offers promise of joint analysis in the context of a single tool, and methods for the parallel simulation of general Continuous Time Markov Chains, a model framework within which joint performance/reliability models can be cast, were developed. However, very much more work is needed to determine the scope and generality of these approaches. The results obtained in our two studies, future directions for this type of work, and a list of publications are included.
This paper presents the implementation of a new class of massively parallel algorithms for solving certain time-dependent partial differential equations (PDEs) on massively parallel-supercomputers.
Evolutionary partial differential equations are usually solved by decretization in time and space, and by applying a marching in time procedure to data and algorithms potentially parallelized in the spatial domain.
The class of problems that can be effectively compiled by parallelizing compilers is discussed. This is accomplished with the doconsider construct which would allow these compilers to parallelize many problems in which substantial loop-level parallelism is available but cannot be detected by standard compile-time analysis. We describe and experimentally analyze mechanisms used to parallelize the work required for these types of loops. In each of these methods, a new loop structure is produced by modifying the loop to be parallelized. We also present the rules by which these loop transformations may be automated in order that they be included in language compilers. The main application area of the research involves problems in scientific computations and engineering. The workload used in our experiment includes a mixture of real problems as well as synthetically generated inputs. From our extensive tests on the Encore Multimax/320, we have reached the conclusion that for the types of workloads we have investigated, self-execution almost always performs better than pre-scheduling. Further, the improvement in performance that accrues as a result of global topological sorting of indices as opposed to the less expensive local sorting, is not very significant in the case of self-execution.
Quantum optimal control plays a crucial role in quantum computing by providing the interface between compiler and hardware. Solving the optimal control problem is particularly challenging for multi-qubit gates, due to the exponential growth in computational complexity with the system's dimensionality and the deterioration of optimization convergence. To ameliorate the computational complexity of time-integration, this paper introduces a multiple-shooting approach in which the time domain is divided into multiple windows and the intermediate states at window boundaries are treated as additional optimization variables. Further, this enables parallel computation of state evolution across time-windows, significantly accelerating objective function and gradient evaluations. Since the initial state matrix in each window is only guaranteed to be unitary upon convergence of the optimization algorithm, the conventional gate trace infidelity is replaced by a generalized infidelity that is convex for non-unitary state matrices. Continuity of the state across window boundaries is enforced by equality constraints. A quadratic penalty optimization method is used to solve the constrained optimal control problem, and an efficient adjoint technique is employed to calculate the gradients in each iteration. We demonstrate the effectiveness of the proposed method through numerical experiments on quantum Fourier transform gates in systems with 2, 3, and 4 qubits, noting a speedup of 80x for evaluating the gradient in the 4-qubit case, highlighting the method's potential for optimizing control pulses in multi-qubit quantum systems.
The objective of the current effort is to provide a computational framework for design and analysis of the entire fuel supply system of a liquid rocket engine, including high-fidelity unsteady turbopump flow analysis. This capability is needed to support the design of pump sub-systems for advanced space transportation vehicles that are likely to involve liquid propulsion systems. To date, computational tools for design/analysis of turbopump flows are based on relatively lower fidelity methods. An unsteady, three-dimensional viscous flow analysis tool involving stationary and rotational components for the entire turbopump assembly has not been available for real-world engineering applications. The present effort provides developers with information such as transient flow phenomena at start up, and non-uniform inflows, and will eventually impact on system vibration and structures. In the proposed paper, the progress toward the capability of complete simulation of the turbo-pump for a liquid rocket engine is reported. The Space Shuttle Main Engine (SSME) turbo-pump is used as a test case for evaluation of the hybrid MPI/Open-MP and MLP versions of the INS3D code. CAD to solution auto-scripting capability is being developed for turbopump applications. The relative motion of the grid systems for the rotor-stator interaction was obtained using overset grid techniques. Unsteady computations for the SSME turbo-pump, which contains 114 zones with 34.5 million grid points, are carried out on Origin 3000 systems at NASA Ames Research Center. Results from these time-accurate simulations with moving boundary capability will be presented along with the performance of parallel versions of the code.
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The utility of cluster computers (Beowulfs) for efficient parallel image processing of large-scale mosaics and stereo image correlations is presented.
Here, we consider the parallel-in-time solution of scalar nonlinear conservation laws in one spatial dimension. The equations are discretized in space with a conservative finite-volume method using weighted essentially nonoscillatory (WENO) reconstructions, and in time with high-order explicit Runge–Kutta methods. The solution of the global, discretized space-time problem is sought via a nonlinear iteration that uses a novel linearization strategy in cases of nondifferentiable equations. Under certain choices of discretization and algorithmic parameters, the nonlinear iteration coincides with Newton’s method, although, more generally, it is a preconditioned residual correction scheme. At each nonlinear iteration, the linearized problem takes the form of a certain discretization of a linear conservation law over the space-time domain in question. An approximate parallel-in-time solution of the linearized problem is computed with a single multigrid reduction-in-time (MGRIT) iteration; however, any other effective parallel-in-time method could be used in its place. The MGRIT iteration employs a novel coarse-grid operator that is a modified conservative semi-Lagrangian discretization and generalizes those we have developed previously for nonconservative scalar linear hyperbolic problems. Numerical tests are performed for the inviscid Burgers and Buckley–Leverett equations. For many test problems, the solver converges in just a handful of iterations with a convergence rate independent of mesh resolution, including problems with (interacting) shocks and rarefactions.
We consider the parallel-in-time solution of both linear and nonlinear hyperbolic partial differential equation (PDE) systems in one spatial dimension. In the nonlinear setting, the discretized equations are solved with a preconditioned residual iteration based on a global linearization. The linear(ized) equation systems are approximately solved parallel-in-time using a block preconditioner applied in the characteristic variables of the underlying linear(ized) hyperbolic PDE. This change of variables is motivated by the observation that intervariable coupling between characteristic variables is weak, at least locally where spatio-temporal variations in the eigenvectors of the associated flux Jacobian are sufficiently small, while that between the original variables is not. For an ℓ-dimensional system of PDEs, applying the preconditioner consists of solving a sequence of ℓ scalar linear(ized)-advection-like problems, each associated with a different characteristic wave-speed in the underlying linear(ized) PDE. Furthermore, we approximately solve these linear advection problems using multigrid reduction-in-time (MGRIT); however, any other suitable parallel-in-time method could be used. Numerical examples are shown for the (linear) acoustics equations in heterogeneous media and for the (nonlinear) shallow water equations and Euler equations of gas dynamics with shocks and rarefactions. For many test problems, the solver converges in just a handful of iterations and with mesh-independent convergence rates.
Extensions to an architecture for real-time, distributed (parallel) knowledge-based systems called the Parallel Real-time Artificial Intelligence System (PRAIS) are discussed. PRAIS strives for transparently parallelizing production (rule-based) systems, even under real-time constraints. PRAIS accomplished these goals (presented at the first annual C Language Integrated Production System (CLIPS) conference) by incorporating a dynamic task scheduler, operating system extensions for fact handling, and message-passing among multiple copies of CLIPS executing on a virtual blackboard. This distributed knowledge-based system tool uses the portability of CLIPS and common message-passing protocols to operate over a heterogeneous network of processors. Results using the original PRAIS architecture over a network of Sun 3's, Sun 4's and VAX's are presented. Mechanisms using the producer-consumer model to extend the architecture for fault-tolerance and distributed truth maintenance initiation are also discussed.
In this paper, we present a novel strategy for incorporating massive parallelism into the solution of Maxwell's equations using finite-difference time-domain methods. In a departure from previous techniques wherein spatial parallelism is used, our approach exploits massive temporal parallelism by computing all of the time steps in parallel.
In the past few decades, researchers have created a veritable zoo of quantum algorithms by drawing inspiration from classical computing, information theory, and even from physical phenomena. Here, we present quantum algorithms for parallel-in-time simulations that are inspired by the Page and Wootters formalism. In this framework, and thus in our algorithms, the classical time variable of quantum mechanics is promoted to the quantum realm by introducing a Hilbert space of “clock” qubits that are then entangled with the “system” qubits. We show that our algorithms can compute temporal properties over 𝑁 different times of many-body systems by only using log(𝑁) clock qubits. As such, we achieve an exponential trade-off between time and spatial complexities. In addition, we rigorously prove that the entanglement created between the system qubits and the clock qubits has operational meaning, as it encodes valuable information about the system’s dynamics. We also provide a circuit depth estimation of all the protocols, showing a running time advantage in computation times over traditional sequential-in-time algorithms. In particular, for the case when the dynamics are determined by the Aubry-Andre model, we present a hybrid method for which our algorithms have a depth that only scales as 𝒪(log(𝑁)𝑛). As a by-product, we can relate the previous schemes to the problem of equilibration of an isolated quantum system, thus indicating that our framework enables a new dimension for studying dynamical properties of many-body systems.