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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.
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In this talk, we will present interface-aware methods for diffusion equation. The focus is on the necessity of interface-aware, mixed cells, interface reconstruction, discontinuity of material diffusion coefficients, order of accuracy in time, property for large time steps, and simpler interface-aware methods. Numerical examples are provided to demonstrate the features on these aspects of the methods
Hydrodynamic codes (hydrocodes) are common tools for modeling hypervelocity impacts to provide insight into the physical phenomenon. Hydrocodes can simulate impacts from micrometer to kilometer spatial scales and reach impact velocities difficult to achieve in experimental settings. However, numerical models are approximations, and demonstrating that a numerical method is capable of providing physical results for these models is essential. In this work, we employ a hydrocode verification technique that leverages hydrodynamic similarity, a mathematical property of the conservation equations of fluid mechanics that form the basis for hydrocode models. Using the FLAG hydrocode, we simulate aluminum (Al) and basalt projectiles and targets at spatial scales spanning 7 orders of magnitude (hundreds of micrometers to kilometers). These materials were chosen because Al-6061 is a common material in spacecraft and satellites and basalt is a useful approximation of rocky astronomical bodies. Our results show that hydrodynamic similarity holds for each material model used and across spatial scales. We show that under certain conditions hydrodynamic similarity can apply in the presence of gravity and that similarity does not hold in the presence of strength models. We conclude that the FLAG hydrocode preserves important mathematical properties of fluid dynamics in hypervelocity impacts of Al-6061 and basalt.
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This paper describes efficient algorithms for computing rank-revealing factorizations of matrices that are too large to fit in main memory (RAM), and must instead be stored on slow external memory devices such as disks (out-of-core or out-of-memory). Traditional algorithms for computing rank-revealing factorizations (such as the column pivoted QR factorization and the singular value decomposition) are very communication intensive as they require many vector-vector and matrix-vector operations, which become prohibitively expensive when data is not in RAM. Randomization allows to reformulate new methods so that large contiguous blocks of the matrix are processed in bulk. The paper describes two distinct methods. The first is a blocked version of column pivoted Householder QR, organized as a “left-looking” method to minimize the number of the expensive write operations. The second method results employs a UTV factorization. It is organized as an algorithm-by-blocks to overlap computations and I/O operations. As it incorporates power iterations, it is much better at revealing the numerical rank. Numerical experiments on several computers demonstrate that the new algorithms are almost as fast when processing data stored on slow memory devices as traditional algorithms are for data stored in RAM.
Fluid flow in the transonic regime finds relevance in aerospace engineering, particularly in the design of commercial air transportation vehicles. Computational fluid dynamics models of transonic flow for aerospace applications are computationally expensive to solve because of the high degrees of freedom as well as the coupled nature of the conservation laws. While these issues pose a bottleneck for the use of such models in aerospace design, computational costs can be significantly minimized by constructing special, structure-preserving surrogate models called reduced-order models. In this work, we propose a machine learning method to construct reduced-order models via deep neural networks and we demonstrate its ability to preserve accuracy with a significantly lower computational cost. In addition, our machine learning methodology is physics-informed and constrained through the utilization of an interpretable encoding by way of proper orthogonal decomposition. Application to the inviscid transonic flow past the RAE2822 airfoil under varying freestream Mach numbers and angles of attack, as well as airfoil shape parameters with a deforming mesh, shows that the proposed approach adapts to high-dimensional parameter variation well. Notably, the proposed framework precludes the knowledge of numerical operators utilized in the data generation phase, thereby demonstrating its potential utility in the fast exploration of design space for diverse engineering applications. Comparison against a projection-based nonintrusive model order reduction method demonstrates that the proposed approach produces comparable accuracy and yet is orders of magnitude computationally cheap to evaluate, despite being agnostic to the physics of the problem.
Forthcoming cosmic microwave background (CMB) polarized anisotropy experiments have the potential to revolutionize our understanding of the Universe and fundamental physics. The sought-after, tale-telling signatures will be however distributed over voluminous data sets which these experiments will collect. These data sets will need to be efficiently processed and unwanted contributions due to astrophysical, environmental, and instrumental effects characterized and efficiently mitigated in order to uncover the signatures. This poses a significant challenge to data analysis methods, techniques, and software tools which will not only have to be able to cope with huge volumes of data but to do so with unprecedented precision driven by the demanding science goals posed for the new experiments. A keystone of efficient CMB data analysis is solvers of very large linear systems of equations. Such systems appear in very diverse contexts throughout CMB data analysis pipelines, however they typically display similar algebraic structures and can therefore be solved using similar numerical techniques. Linear systems arising in the so-called map-making problem are one of the most prominent and common ones. In this work we present a massively parallel, flexible and extensible framework, comprised of a numerical library, MIDAPACK, and a high level code, MAPPRAISER, which provide tools for solving efficiently such systems. Here, the framework implements iterative solvers based on conjugate gradient techniques: enlarged and preconditioned using different preconditioners. We demonstrate the framework on simulated examples reflecting basic characteristics of the forthcoming data sets issued by ground-based and satellite-borne instruments, executing it on as many as 16,384 compute cores. The software is developed as an open source project freely available to the community at: https://github.com/B3Dcmb/midapack.
Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a combination of deterministic and stochastic terms. However, these models usually lack interpretability and have difficulty in generalizing beyond their training domain. Here, this paper introduces the Finite Expression Method (FEX), a symbolic learning approach designed to derive interpretable mathematical representations of the deterministic component of SDEs. For the stochastic component, we integrate FEX with advanced generative modeling techniques to provide a comprehensive representation of SDEs. The numerical experiments on linear, nonlinear, and multidimensional SDEs demonstrate that FEX generalizes well beyond the training domain and delivers more accurate long-term predictions compared to neural network-based methods. The symbolic expressions identified by FEX not only improve prediction accuracy but also offer valuable scientific insights into the underlying dynamics of the systems.
The Hartree–Fock problem provides the conceptual and mathematical underpinning of a large portion of quantum chemistry. As efforts in quantum technology aim to enhance computational chemistry algorithms, the Hartree–Fock method, central to many other numerical approaches, is a natural target for quantum enhanced algorithms. While quantum computers and quantum simulation offer many prospects for the future of modern chemistry, the non-deterministic polynomial-complete Hartree–Fock problem is not a likely candidate. Here, we highlight this fact from a number of perspectives including computational complexity, practical examples, and the full characterization of energy landscapes for simple systems.
This paper presents an adaptive randomized algorithm for computing the butterfly factorization of an m × n matrix with m ≈ n provided that both the matrix and its transpose can be rapidly applied to arbitrary vectors. The resulting factorization is composed of O(log n) sparse factors, each containing O(n) nonzero entries. The factorization can be attained using O(n 3/2 log n) computation and O(n log n) memory resources. Furthermore, the proposed algorithm can be implemented in parallel and can apply to matrices with strong or weak admissibility conditions arising from surface integral equation solvers as well as multi-frontal-based finite-difference, finite-element, or finite-volume solvers. A distributed-memory parallel implementation of the algorithm demonstrates excellent scaling behavior.
Using the Löwdin orthonormalization of tall-skinny matrices as a proxy-app for wavefunction-based Density Functional Theory solvers, we investigate a distributed memory parallel strategy focusing on Graphics Processing Unit (GPU)-accelerated nodes as available on some of the top ranked supercomputers at the present time. Here we present numerical results in the strong limit regime, as it is particularly relevant for First-Principles Molecular Dynamics. We also examine how matrix product-based iterative solvers provide a competitive alternative to dense eigensolvers on GPUs, allowing to push the strong scaling limit of these computations to a larger number of distributed tasks. Our strategy, which relies on replicated Gram matrices and efficient collective communications using the NCCL library, leads to a time-to-solution under 0.5 s for the Löwdin orthonormalization of a tall-skinny matrix of 3000 columns on Summit at Oak Ridge Leadership Facility (OLCF). Given the similarity in computational operations between one iteration of a DFT solver and this proxy-app, this shows the possibility of solving accurately the DFT equations well under a minute for 3000 electronic wave functions, and thus perform First-Principles molecular dynamics of physical systems much larger than traditionally solved on CPU systems.
Herein we propose a multi-resolution convolutional autoencoder (MrCAE) architecture that integrates and leverages three highly successful mathematical architectures: (i) multigrid methods, (ii) convolutional autoencoders and (iii) transfer learning. The method provides an adaptive, hierarchical architecture that capitalizes on a progressive training approach for multiscale spatio-temporal data. This framework allows for inputs across multiple scales: starting from a compact (small number of weights) network architecture and low-resolution data, our network progressively deepens and widens itself in a principled manner to encode new information in the higher resolution data based on its current performance of reconstruction. Basic transfer learning techniques are applied to ensure information learned from previous training steps can be rapidly transferred to the larger network. As a result, the network can dynamically capture different scaled features at different depths of the network. The performance gains of this adaptive multiscale architecture are illustrated through a sequence of numerical experiments on synthetic examples and real-world spatial-temporal data.
The purpose of this project is to examine the energy impact of urban-scale traffic for the Los Angeles Basin by developing and implementing a scalable traffic assignment model. An energy optimization function will be posed and when integrated into the optimization code for travel assignment it can be mathematically proven to converge. The energy optimization function can then be compared to the typical travel time optimization that is traditionally used in traffic assignment models. The analysis will begin with static traffic assignment models with the routing for all origin and destinations computed in parallel on high performance computing facilities. Convergence of the numerical methods rely on the solution of convex programs (or extensions of these). This step will mostly consist of demonstrating the ability to parallelize the Frank Wolfe algorithm on various platforms.
Yttrium hydride (YH x ) is one of the materials being considered for moderating thermal and epithermal nuclear microreactors. One potential issue with YH x use is that the hydrogen redistributes in the hydride when thermal and concentration gradients are present. This hydrogen redistribution leads to spatial gradients in the hydrogen concentration, thus affecting neutron transport in the reactor. Here, by building upon observations in prior works, this paper aims to gain a better understanding of the reactivity feedback associated with such hydrogen redistributions. In particular, we wish to understand the sign (+/–) of the hydrogen redistribution neutronic feedback, its order of magnitude, and its underlying physical causes. To achieve this goal, the DireWolf multiphysics software driver was used to solve the coupled radiation transport, heat transfer, heat pipe two-phase flow, and hydrogen redistribution equations for the Simplified Microreactor Benchmark Assessment (SiMBA) problem, a full-core microreactor numerical benchmark developed at Idaho National Laboratory.
In this study, we propose a novel approach to model viscoelasticity materials, where rate-dependent and non-linear constitutive relationships are approximated with deep neural networks. We assume that inputs and outputs of the neural networks are not directly observable, and therefore common training techniques with input–output pairs for the neural networks are inapplicable. To that end, we develop a novel computational approach to both calibrate parametric and learn neural-network-based constitutive relations of viscoelasticity materials from indirect displacement data in the context of multiple-physics systems. We show that limited displacement data holds sufficient information to quantify the viscoelasticity behavior. We formulate the inverse computation – modeling viscoelasticity properties from observed displacement data – as a PDE-constrained optimization problem and minimize the error functional using a gradient-based optimization method. The gradients are computed by a combination of automatic differentiation and implicit function differentiation rules. The effectiveness of our method is demonstrated through numerous benchmark problems in geomechanics and porous media transport.
Here, we present a novel mass-momentum consistent coupling between a geometric volume-of-fluid scheme and an incompressible flow solver with differing directional-splitting approaches. The advection of the volume fraction is performed using a direction-split algorithm, whereas the momentum advection algorithm uses a traditional unsplit, fractional-step approach. Both algorithms employ finite-volume discretizations based on Cartesian meshes. In solving the mass-momentum consistency problem, momentum fluxes at the cell faces are weighted by the density fluxes based on the already advected volume fraction. The success of our approach lies on introducing a Favre-averaged velocity interpolation at the liquid/gas interface along with a minmod slope limiter. Mesh-convergence studies show that when the minmod slope limiter is used, the two-phase solver retains an accuracy between first and second order, but when a purely upwind scheme is considered, its accuracy drops to first order. Finally, after considering several validation problems, the solver is shown to agree well with reference numerical and experimental data while retaining its robustness and efficiency.
Here, the paper presents data-driven algorithms to estimate in a distributed manner the eigenvalues, right and left eigenvectors of an unknown linear (or linearized) interconnected dynamic system. In particular, the proposed algorithms do not require the identification of the system model in advance before performing the estimation. As a first step, we consider interconnected dynamical system with distinct eigenvalues. The proposed strategy first estimates the eigenvalues using the well-known Prony method. The right and left eigenvectors are then estimated by solving distributively a set of linear equations. One important feature of the proposed algorithms is that the topology of communication network used to perform the distributed estimation can be chosen arbitrarily, given that it is connected, and is also independent of the structure or sparsity of the system (state) matrix. The proposed distributed algorithms are demonstrated via a numerical example.
Bayesian approaches have been successfully integrated into training deep neural networks. One popular family is stochastic gradient Markov chain Monte Carlo methods (SG-MCMC), which have gained increasing interest due to their ability to handle large datasets and the potential to avoid overfitting. Although standard SG-MCMC methods have shown great performance in a variety of problems, they may be inefficient when the random variables in the target posterior densities have scale differences or are highly correlated. Here, we present an adaptive Hessian approximated stochastic gradient MCMC method to incorporate local geometric information while sampling from the posterior. The idea is to apply stochastic approximation (SA) to sequentially update a preconditioning matrix at each iteration. The preconditioner possesses second-order information and can guide the random walk of a sampler efficiently. Instead of computing and saving the full Hessian of the log posterior, we use limited memory of the samples and their stochastic gradients to approximate the inverse Hessian-vector multiplication in the updating formula. Moreover, by smoothly optimizing the preconditioning matrix via SA, our proposed algorithm can asymptotically converge to the target distribution with a controllable bias under mild conditions. To reduce the training and testing computational burden, we adopt a magnitude-based weight pruning method to enforce the sparsity of the network. Our method is user-friendly and demonstrates better learning results compared to standard SG-MCMC updating rules. The approximation of inverse Hessian alleviates storage and computational complexities for large dimensional models. Numerical experiments are performed on several problems, including sampling from 2D correlated distribution, synthetic regression problems, and learning the numerical solutions of heterogeneous elliptic PDE. The numerical results demonstrate great improvement in both the convergence rate and accuracy.
In this work, we present trapped solitary wave solutions of a coupled nonlinear Schrödinger (NLS) system in 1 + 1 dimensions in the presence of an external, supersymmetric and complex $\mathcal{PT}$-symmetric potential. The Schrödinger system this work focuses on possesses exact solutions whose existence, stability, and spatio-temporal dynamics are investigated by means of analytical and numerical methods. Two different variational approximations are considered where the stability and dynamics of the solitary waves are explored in terms of eight and twelve time-dependent collective coordinates (CCs). We find regions of stability for specific potential choices as well as analytic expressions for the small oscillation frequencies in the CC approximation. Our findings are further supported by performing systematic numerical simulations of the NLS system.