Lecture Notes in Computational Science and Engineering
Maljaars, Jakob M.; Labeur, Robert J.; Trask, Nathaniel A.; Sulsky, Deborah L.
A particle-mesh strategy is presented for scalar transport problems which provides diffusion-free advection, conserves mass locally (i.e. cellwise) and exhibits optimal convergence on arbitrary polyhedral meshes. This is achieved by expressing the convective field naturally located on the Lagrangian particles as a mesh quantity by formulating a dedicated particle-mesh projection based via a PDE-constrained optimization problem. Optimal convergence and local conservation are demonstrated for a benchmark test, and the application of the scheme to mass conservative density tracking is illustrated for the Rayleigh–Taylor instability.
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.
Truly predictive numerical simulations can only be obtained by performing Uncertainty Quantification. However, many realistic engineering applications require extremely complex and computationally expensive high-fidelity numerical simulations for their accurate performance characterization. Very often the combination of complex physical models and extreme operative conditions can easily lead to hundreds of uncertain parameters that need to be propagated through high-fidelity codes. Under these circumstances, a single fidelity uncertainty quantification approach, i.e. a workflow that only uses high-fidelity simulations, is unfeasible due to its prohibitive overall computational cost. To overcome this difficulty, in recent years multifidelity strategies emerged and gained popularity. Their core idea is to combine simulations with varying levels of fidelity/accuracy in order to obtain estimators or surrogates that can yield the same accuracy of their single fidelity counterparts at a much lower computational cost. This goal is usually accomplished by defining a priori a sequence of discretization levels or physical modeling assumptions that can be used to decrease the complexity of a numerical model realization and thus its computational cost. Less attention has been dedicated to low-fidelity models that can be built directly from a small number of available high-fidelity simulations. In this work we focus our attention on reduced order models (ROMs). Our main goal in this work is to investigate the combination of multifidelity uncertainty quantification and ROMs in order to evaluate the possibility to obtain an efficient framework for propagating uncertainties through expensive numerical codes. We focus our attention on sampling-based multifidelity approaches, like the multifidelity control variate, and we consider several scenarios for a numerical test problem, namely the Kuramoto-Sivashinsky equation, for which the efficiency of the multifidelity-ROM estimator is compared to the standard (single-fidelity) Monte Carlo approach.
As computer architectures are rapidly evolving (e.g. those designed for exascale), multiple portability frameworks have been developed to avoid new architecture-specific development and tuning. However, portability frameworks depend on compilers for auto-vectorization and may lack support for explicit vectorization on heterogeneous platforms. Alternatively, programmers can use intrinsics-based primitives to achieve more efficient vectorization, but the lack of a gpu back-end for these primitives makes such code non-portable. A unified, portable, Single Instruction Multiple Data (simd) primitive proposed in this work, allows intrinsics-based vectorization on cpus and many-core architectures such as Intel Knights Landing (knl), and also facilitates Single Instruction Multiple Threads (simt) based execution on gpus. This unified primitive, coupled with the Kokkos portability ecosystem, makes it possible to develop explicitly vectorized code, which is portable across heterogeneous platforms. The new simd primitive is used on different architectures to test the performance boost against hard-to-auto-vectorize baseline, to measure the overhead against efficiently vectroized baseline, and to evaluate the new feature called the “logical vector length” (lvl). The simd primitive provides portability across cpus and gpus without any performance degradation being observed experimentally.
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 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.
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.
Particle-in-cell (PIC) simulation methods are attractive for representing species distribution functions in plasmas. However, as a model, they introduce uncertain parameters, and for quantifying their prediction uncertainty it is useful to be able to assess the sensitivity of a quantity-of-interest (QoI) to these parameters. Such sensitivity information is likewise useful for optimization. However, computing sensitivity for PIC methods is challenging due to the chaotic particle dynamics, and sensitivity techniques remain underdeveloped compared to those for Eulerian discretizations. This challenge is examined from a dual particle–continuum perspective that motivates a new sensitivity discretization. Two routes to sensitivity computation are presented and compared: a direct fully-Lagrangian particle-exact approach provides sensitivities of each particle trajectory, and a new particle-pdf discretization, which is formulated from a continuum perspective but discretized by particles to take the advantages of the same type of Lagrangian particle description leveraged by PIC methods. Since the sensitivity particles in this approach are only indirectly linked to the plasma-PIC particles, they can be positioned and weighted independently for efficiency and accuracy. The corresponding numerical algorithms are presented in mathematical detail. The advantage of the particle-pdf approach in avoiding the spurious chaotic sensitivity of the particle-exact approach is demonstrated for Debye shielding and sheath configurations. In essence, the continuum perspective makes implicit the distinctness of the particles, which circumvents the Lyapunov instability of the N-body PIC system. The cost of the particle-pdf approach is comparable to the baseline PIC simulation.
Bayesian optimization (BO) is an efficient and flexible global optimization framework that is applicable to a very wide range of engineering applications. To leverage the capability of the classical BO, many extensions, including multi-objective, multi-fidelity, parallelization, and latent-variable modeling, have been proposed to address the limitations of the classical BO framework. In this work, we propose a novel multi-objective (MO) extension, called srMOBO-3GP, to solve the MO optimization problems in a sequential setting. Three different Gaussian processes (GPs) are stacked together, where each of the GP is assigned with a different task: the first GP is used to approximate a single-objective computed from the MO definition, the second GP is used to learn the unknown constraints, and the third GP is used to learn the uncertain Pareto frontier. At each iteration, a MO augmented Tchebycheff function converting MO to single-objective is adopted and extended with a regularized ridge term, where the regularization is introduced to smooth the single-objective function. Finally, we couple the third GP along with the classical BO framework to explore the richness and diversity of the Pareto frontier by the exploitation and exploration acquisition function. The proposed framework is demonstrated using several numerical benchmark functions, as well as a thermomechanical finite element model for flip-chip package design optimization.
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.
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.
Porosity in additively manufactured metals can reduce material strength and is generally undesirable. Although studies have shown relationships between process parameters and porosity, monitoring strategies for defect detection and pore formation are still needed. In this paper, instantaneous anomalous conditions are detected in-situ via pyrometry during laser powder bed fusion additive manufacturing and correlated with voids observed using post-build micro-computed tomography. Large two-color pyrometry data sets were used to estimate instantaneous temperatures, melt pool orientations and aspect ratios. Machine learning algorithms were then applied to processed pyrometry data to detect outlier images and conditions. It is shown that melt pool outliers are good predictors of voids observed post-build. With this approach, real time process monitoring can be incorporated into systems to detect defect and void formation. Alternatively, using the methodology presented here, pyrometry data can be post processed for porosity assessment.
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.
Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional complexity, namely, uncertain parameters in the PDEs. Uncertainty quantification (UQ) is necessary to characterize, prioritize, and study the influence of these uncertain parameters. Sensitivity analysis, a classical tool in UQ, is frequently used to study the sensitivity of a model to uncertain parameters. In this article, we introduce "hyper-differential sensitivity analysis" which considers the sensitivity of the solution of a PDE-constrained optimization problem to uncertain parameters. Our approach is a goal-oriented analysis which may be viewed as a tool to complement other UQ methods in the service of decision making and robust design. We formally define hyper-differential sensitivity indices and highlight their relationship to the existing optimization and sensitivity analysis literatures. Assuming the presence of low rank structure in the parameter space, computational efficiency is achieved by leveraging a generalized singular value decomposition in conjunction with a randomized solver which converts the computational bottleneck of the algorithm into an embarrassingly parallel loop. Two multi-physics examples, consisting of nonlinear steady state control and transient linear inversion, demonstrate efficient identification of the uncertain parameters which have the greatest influence on the optimal solution.
