Matrix Diagonalization as a Board Game: Teaching an Eigensolver the Fastest Path to Solution

Matrix diagonalization is at the cornerstone of numerous fields of scientific computing. Diagonalizing a matrix to solve an eigenvalue problem requires a sequential path of iterations that eventually reaches a sufficiently converged and accurate solution for all the eigenvalues and eigenvectors. This typically translates into a high computational cost. Here we demonstrate how reinforcement learning, using the AlphaZero framework, can accelerate Jacobi matrix diagonalizations by viewing the selection of the fastest path to solution as a board game. To demonstrate the viability of our approach we apply the Jacobi diagonalization algorithm to symmetric Hamiltonian matrices that appear in quantum chemistry calculations. We find that a significant acceleration can often be achieved. Our findings highlight the opportunity to use machine learning as a promising tool to improve the performance of numerical linear algebra.Comment: 14 page

Quantum Algorithm Implementations for Beginners

As quantum computers become available to the general public, the need has arisen to train a cohort of quantum programmers, many of whom have been developing classical computer programs for most of their careers. While currently available quantum computers have less than 100 qubits, quantum computing hardware is widely expected to grow in terms of qubit count, quality, and connectivity. This review aims to explain the principles of quantum programming, which are quite different from classical programming, with straightforward algebra that makes understanding of the underlying fascinating quantum mechanical principles optional. We give an introduction to quantum computing algorithms and their implementation on real quantum hardware. We survey 20 different quantum algorithms, attempting to describe each in a succinct and self-contained fashion. We show how these algorithms can be implemented on IBM's quantum computer, and in each case, we discuss the results of the implementation with respect to differences between the simulator and the actual hardware runs. This article introduces computer scientists, physicists, and engineers to quantum algorithms and provides a blueprint for their implementations

A Repository Dedicated To Enhancing Verification and Validation in Metal Plasticity Using Material Point Simulators

Tools for Enhancing Verification and Validation in Metal Plasticity Using Material Point Simulator

Determination of best practice guidelines for performing large eddy simulation of flows in configurations of engineering interest

Large eddy simulation (LES) suffers from two primary sources of error: the numerical discretization scheme and the subgrid stress model (SGS). An attempt has been made to determine optimum combinations of SGS models and numerical schemes for use in performing practical LES for engineering-relevant problems. A formal quantification of numerical error present in finite-volume/finite-difference simulations was conducted. The effect of this error was explicitly added to a pseudospectral LES solver, and the modified pseudospectral solver was used to compute LES of decaying turbulence. In this way SGS modeling error and numerical error could be separately assessed. Verification of results was carried out using a commercially available finite-volume solver (FLUENT). Results showed that some combinations of SGS model and discretization scheme are more suitable for performing LES than others. Favorable combinations from the above findings were tested for an axisymmetric jet at Mach number 0.2. Results indicate good agreement with prior findings

A Continuum Mechanics Approach to Modeling and Simulating Engineering Materials Undergoing Phase Transformation using the Evolving Micro-Structural Model of Inelasticity

Heat treatment for the purpose of material strengthening is accompanied by residual stresses and distortion. During these processing steps, steel alloys experience a phase change that in turn modify their overall mechanical response. To properly account for the cumulative composite behavior, the mechanical response, transformation kinetics and subsequent interaction of each phase have to be properly accounted for. Of interest to material designers and fabricators is modeling and simulating the evolutionary process a part undergoes for the sake of capturing the observable residual stress states and geometric distortion accumulated after processing. In an attempt to capture the aforementioned physical phenomena, this investigation is premised upon a consistent thermodynamic framework. Following this, the single phase Evolving Microstructural Model of Inelasticity state variable model is extended to accommodate the occurrence of multiphases, affirming that the interaction between coexisting phases is through an interfacial stress. Since the efficacy of a multiphase model is dependent on its ability to capture the behavior of constituents phases and their subsequent interaction, we introduce a physically based self-consistent strain partitioning algorithm. With synthesis of the aforementioned ideas, the additional transformation induced plasticity is numerically accounted for by modifying each phase’s flowrule to accommodate an interfacial stress. In addition, for simulating the cohabitation of two phases, the mechanical multiphase model equations is coupled with a previously developed non-diffusional phase transformation kinetics model. A qualitative assessment of the material response based on a Taylor, Sachs and self-consistent polycrystalline approximation is carried out. Further analysis of the multiphase model and its interaction with transformation kinetics is evaluated

High Performance Computing

With energy-efficient architectures, including accelerators and many-core processors, gaining traction, application developers face the challenge of optimizing their applications for multiple hardware features including many-core parallelism, wide processing vector-units and on-chip high-bandwidth memory. In this paper, we discuss the development and utilization of a new application performance tool based on an extension of the classical roofline-model for simultaneously profiling multiple levels in the cache-memory hierarchy. This tool presents a powerful visual aid for the developer and can be used to frame the many-dimensional optimization problem in a tractable way. We show case studies of real scientific applications that have gained insights from the Integrated Roofline Model.<br /