Proceedings of the 6th European Conference on Computational Mechanics: Solids, Structures and Coupled Problems, ECCM 2018 and 7th European Conference on Computational Fluid Dynamics, ECFD 2018
SNOWPAC (Stochastic Nonlinear Optimization With Path-Augmented Constraints) is a method for stochastic nonlinear constrained derivative-free optimization. For such problems, it extends the path-augmented constraints framework introduced by the deterministic optimization method NOWPAC and uses a noise-adapted trust region approach and Gaussian processes for noise reduction. In recent developments, SNOWPAC is available in the DAKOTA framework which offers a highly flexible interface to couple the optimizer with different sampling strategies or surrogate models. In this paper we discuss details of SNOWPAC and demonstrate the coupling with DAKOTA. We showcase the approach by presenting design optimization results of a shape in a 2D supersonic duct. This simulation is supposed to imitate the behavior of the flow in a SCRAMJET simulation but at a much lower computational cost. Additionally different mesh or model fidelities can be tested. Thus, it serves as a convenient test case before moving to costly SCRAMJET computations. Here, we study deterministic results and results obtained by introducing uncertainty on inflow parameters. As sampling strategies we compare classical Monte Carlo sampling with multilevel Monte Carlo approaches for which we developed new error estimators. All approaches show a reasonable optimization of the design over the objective while maintaining or seeking feasibility. Furthermore, we achieve significant reductions in computational cost by using multilevel approaches that combine solutions from different grid resolutions.
This study explores the use of a Krylov iterative method (GMRES) as a smoother for an algebraic multigrid (AMG) preconditioned Newton–Krylov iterative solution approach for a fully-implicit variational multiscale (VMS) finite element (FE) resistive magnetohydrodynamics (MHD) formulation. The efficiency of this approach is critically dependent on the scalability and performance of the AMG preconditioner for the linear solutions and the performance of the smoothers play an essential role. Krylov smoothers are considered an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition (DD) with incomplete LU factorization solves on each subdomain. This brief study presents three time dependent resistive MHD test cases to evaluate the method. The results demonstrate that the GMRES smoother can be faster due to a decrease in the preconditioner setup time and a reduction in outer GMRESR solver iterations, and requires less memory (typically 35% less memory for global GMRES smoother) than the DD ILU smoother.
Wear prediction is important in designing reliable machinery for slurry industry. It usually relies on multi-phase computational fluid dynamics, which is accurate but computationally expensive. Each run of the simulations can take hours or days even on a high-performance computing platform. The high computational cost prohibits a large number of simulations in the process of design optimization. In contrast to physics-based simulations, data-driven approaches such as machine learning are capable of providing accurate wear predictions at a small fraction of computational costs, if the models are trained properly. In this paper, a recently developed WearGP framework [1] is extended to predict the global wear quantities of interest by constructing Gaussian process surrogates. The effects of different operating conditions are investigated. The advantages of the WearGP framework are demonstrated by its high accuracy and low computational cost in predicting wear rates.
This article describes a parallel implementation of a two-level overlapping Schwarz preconditioner with the GDSW (Generalized Dryja–Smith–Widlund) coarse space described in previous work [12, 10, 15] into the Trilinos framework; cf. [16]. The software is a significant improvement of a previous implementation [12]; see Sec. 4 for results on the improved performance.
Recent work has demonstrated that block preconditioning can scalably accelerate the performance of iterative solvers applied to linear systems arising in implicit multiphysics PDE simulations. The idea of block preconditioning is to decompose the system matrix into physical sub-blocks and apply individual specialized scalable solvers to each sub-block. It can be advantageous to block into simpler segregated physics systems or to block by discretization type. This strategy is particularly amenable to multiphysics systems in which existing solvers, such as multilevel methods, can be leveraged for component physics and to problems with disparate discretizations in which scalable monolithic solvers are rare. This work extends our recent work on scalable block preconditioning methods for structure-preserving discretizatons of the Maxwell equations and our previous work in MHD system solvers to the context of multifluid electromagnetic plasma systems. We argue how a block preconditioner can address both the disparate discretization, as well as strongly-coupled off-diagonal physics that produces fast time-scales (e.g. plasma and cyclotron frequencies). We propose a block preconditioner for plasma systems that allows reuse of existing multigrid solvers for different degrees of freedom while capturing important couplings, and demonstrate the algorithmic scalability of this approach at time-scales of interest.
We propose a very large family of benchmarks for probing the performance of quantum computers. We call them volumetric benchmarks (VBs) because they generalize IBM's benchmark for measuring quantum volume [1]. The quantum volume benchmark defines a family of square circuits whose depth d and width w are the same. A volumetric benchmark defines a family of rectangular quantum circuits, for which d and w are uncoupled to allow the study of time/space performance trade-offs. Each VB defines a mapping from circuit shapes - (w, d) pairs - to test suites C(w, d). A test suite is an ensemble of test circuits that share a common structure. The test suite C for a given circuit shape may be a single circuit C, a specific list of circuits {C1... CN} that must all be run, or a large set of possible circuits equipped with a distribution Pr(C). The circuits in a given VB share a structure, which is limited only by designers' creativity. We list some known benchmarks, and other circuit families, that fit into the VB framework: several families of random circuits, periodic circuits, and algorithm-inspired circuits. The last ingredient defining a benchmark is a success criterion that defines when a processor is judged to have “passed” a given test circuit. We discuss several options. Benchmark data can be analyzed in many ways to extract many properties, but we propose a simple, universal graphical summary of results that illustrates the Pareto frontier of the d vs w trade-off for the processor being benchmarked.
We propose a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND (sparsified Nested Dissection). It is based on nested dissection, sparsification, and low-rank compression. After eliminating all interiors at a given level of the elimination tree, the algorithm sparsifies all separators corresponding to the interiors. This operation reduces the size of the separators by eliminating some degrees of freedom but without introducing any fill-in. This is done at the expense of a small and controllable approximation error. The result is an approximate factorization that can be used as an efficient preconditioner. We then perform several numerical experiments to evaluate this algorithm. We demonstrate that a version using orthogonal factorization and block-diagonal scaling takes fewer CG iterations to converge than previous similar algorithms on various kinds of problems. Furthermore, this algorithm is provably guaranteed to never break down and the matrix stays symmetric positive-definite throughout the process. We evaluate the algorithm on some large problems show it exhibits near-linear scaling. The factorization time is roughly \scrO (N), and the number of iterations grows slowly with N.
We present a numerical method for synchronous and concurrent solution of transient elastodynamics problem where the computational domain is divided into subdomains that may reside on separate computational platforms. This work employs the variational multiscale discontinuous Galerkin (VMDG) method to develop interdomain transmission conditions for transient problems. The fine-scale modeling concept leads to variationally consistent coupling terms at the common interfaces. The method admits a large class of time discretization schemes, and decoupling of the solution for each subdomain is achieved by selecting any explicit algorithm. Numerical tests with a manufactured solution problem show optimal convergence rates. The energy history in a free vibration problem is in agreement with that of the solution from a monolithic computational domain.
Gunther, Stefanie; Ruthotto, Lars; Schroder, Jacob B.; Cyr, Eric C.; Gauger, Nicolas R.
Residual neural networks (ResNets) are a promising class of deep neural networks that have shown excellent performance for a number of learning tasks, e.g., image classification and recognition. Mathematically, ResNet architectures can be interpreted as forward Euler discretizations of a nonlinear initial value problem whose time-dependent control variables represent the weights of the neural network. Hence, training a ResNet can be cast as an optimal control problem of the associated dynamical system. For similar time-dependent optimal control problems arising in engineering applications, parallel-in-time methods have shown notable improvements in scalability. This paper demonstrates the use of those techniques for efficient and effective training of ResNets. The proposed algorithms replace the classical (sequential) forward and backward propagation through the network layers with a parallel nonlinear multigrid iteration applied to the layer domain. This adds a new dimension of parallelism across layers that is attractive when training very deep networks. From this basic idea, we derive multiple layer-parallel methods. The most efficient version employs a simultaneous optimization approach where updates to the network parameters are based on inexact gradient information in order to speed up the training process. Using numerical examples from supervised classification, we demonstrate that the new approach achieves a training performance similar to that of traditional methods, but enables layer-parallelism and thus provides speedup over layer-serial methods through greater concurrency.
We derive a formulation of the nonhydrostatic equations in spherical geometry with a Lorenz staggered vertical discretization. The combination conserves a discrete energy in exact time integration when coupled with a mimetic horizontal discretization. The formulation is a version of Dubos and Tort (2014, https://doi.org/10.1175/MWR-D-14-00069.1) rewritten in terms of primitive variables. It is valid for terrain following mass or height coordinates and for both Eulerian or vertically Lagrangian discretizations. The discretization relies on an extension to Simmons and Burridge (1981, https://doi.org/10.1175/1520-0493(1981)109<0758:AEAAMC>2.0.CO;2) vertical differencing, which we show obeys a discrete derivative product rule. This product rule allows us to simplify the treatment of the vertical transport terms. Energy conservation is obtained via a term-by-term balance in the kinetic, internal, and potential energy budgets, ensuring an energy-consistent discretization up to time truncation error with no spurious sources of energy. We demonstrate convergence with respect to time truncation error in a spectral element code with a horizontal explicit vertically implicit implicit-explicit time stepping algorithm.
Concern about memory errors has been widespread in high-performance computing (HPC) for decades. These concerns have led to significant research on detecting and correcting memory errors to improve performance and provide strong guarantees about the correctness of the memory contents of scientific simulations. However, power concerns and changes in memory architectures threaten the viability of current approaches to protecting memory (e.g., Chipkill). Returning to less protective error-correcting codes (ECC), e.g., single-error correction, double-error detection (SECDED), may increase the frequency of memory errors, including silent data corruption (SDC). SDC has the potential to silently cause applications to produce incorrect results and mislead domain scientists. We propose an approach for exploiting unnecessary bits in pointer values to support encoding the pointer with a Reed-Solomon code. Encoding the pointer allows us to provides strong capabilities for correcting and detecting corruption of pointer values. In this paper, we provide a detailed description of how we can exploit unnecessary pointer bits to store Reed-Solomon parity symbols. We evaluate the performance impacts of this approach and examine the effectiveness of the approach against corruption. Our results demonstrate that encoding and decoding is fast (less than 45 per event) and that the protection it provides is robust (the rate of miscorrection is less than 5% even for significant corruption). The data and analysis presented in this paper demonstrates the power of our approach. It is fast, tunable, requires no additional per-pointer storage resources, and provides robust protection against pointer corruption.
There is a wealth of psychological theory regarding the drive for individuals to congregate and form social groups, positing that people may organize out of fear, social pressure, or even to manage their self-esteem. We evaluate three such theories for multi-scale validity by studying them not only at the individual scale for which they were originally developed, but also for applicability to group interactions and behavior. We implement this multi-scale analysis using a dataset of communications and group membership derived from a long-running online game, matching the intent behind the theories to quantitative measures that describe players’ behavior. Once we establish that the theories hold for the dataset, we increase the scope to test the theories at the higher scale of group interactions. Despite being formulated to describe individual cognition and motivation, we show that some group dynamics theories hold at the higher level of group cognition and can effectively describe the behavior of joint decision making and higher-level interactions.
