On the physical consistency of data assimilation and a computationally effective approximate smoother
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data assimilation numerical modeling ocean circula↗
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Accelerated solution of discrete ordinates approximation to the Boltzmann transport equation via model reduction
Radiation heat transfer is an important phenomenon in many physical systems of practical interest. When participating media is important, the radiative transfer equation (RTE) must be solved for the radiative intensity as a function of location, time, direction, and wavelength. In many heat-transfer applications, a quasi-steady assumption is valid, thereby removing time dependence. The dependence on wavelength is often treated through a weighted sum of gray gases (WSGG) approach. The discrete ordinates method (DOM) is one of the most common methods for approximating the angular (i.e., directional) dependence. The DOM exactly solves for the radiative intensity for a finite number of discrete ordinate directions and computes approximations to integrals over the angular space using a quadrature rule; the chosen ordinate directions correspond to the nodes of this quadrature rule. This paper applies a projection-based model-reduction approach to make high-order quadrature computationally feasible for the DOM for purely absorbing applications. First, the proposed approach constructs a reduced basis from (high-fidelity) solutions of the radiative intensity computed at a relatively small number of ordinate directions. Then, the method computes inexpensive approximations of the radiative intensity at the (remaining) quadrature points of a high-order quadrature using a reduced-order model constructed from the reduced basis. Finally, this results in a much more accurate solution than might have been achieved using only the ordinate directions used to compute the reduced basis. One- and three-dimensional test problems highlight the efficiency of the proposed method.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
NISQ+: Boosting quantum computing power by approximating quantum error correction
Quantum computers are growing in size, and design decisions are being made now that attempt to squeeze more computation out of these machines. In this spirit, we design a method to boost the computational power of near-term quantum computers by adapting protocols used in quantum error correction to implement "Approximate Quantum Error Correction (AQEC)." By approximating fully-fledged error correction mechanisms, we can increase the compute volume (qubits × gates, or "Simple Quantum Volume (SQV)") of near-term machines. The crux of our design is a fast hardware decoder that can approximately decode detected error syndromes rapidly. Specifically, we demonstrate a proof-of-concept that approximate error decoding can be accomplished online in near-term quantum systems by designing and implementing a novel algorithm in Single-Flux Quantum (SFQ) superconducting logic technology. This avoids a critical decoding backlog, hidden in all offline decoding schemes, that leads to idle time exponential in the number of T gates in a program. Our design utilizes one SFQ processing module per physical qubit. Employing state-of-the-art SFQ synthesis tools, we show that the circuit area, power, and latency are within the constraints of contemporary quantum system designs. Under pure dephasing error models, the proposed accelerator and AQEC solution is able to expand SQV by factors between 3,402 and 11,163 on expected near-term machines. The decoder achieves a 5% accuracy-threshold and pseudo-thresholds of ~ 5%,4.75%,4.5%, and 3.5% physical error-rates for code distances 3,5,7, and 9. Decoding solutions are achieved in a maximum of ~20 nanoseconds on the largest code distances studied. By avoiding the exponential idle time in offline decoders, we achieve a 10x reduction in required code distances to achieve the same logical performance as alternative designs.
97 MATHEMATICS AND COMPUTING↗
SFU-Driven Transparent Approximation Acceleration on GPUs
Approximate computing, the technique that sacrifices certain amount of accuracy in exchange for substantial performance boost or power reduction, is one of the most promising solutions to enable power control and performance scaling towards exascale. Although most existing approximation designs target the emerging data-intensive applications that are comparatively more error-tolerable, there is still high demand for the acceleration of traditional scientific applications (e.g., weather and nuclear simulation), which often comprise intensive transcendental function calls and are very sensitive to accuracy loss. To address this challenge, we focus on a very important but often ignored approximation unit on GPUs.
Approximate computing↗
The effect of direct current bias in the computation of power spectra
The effect of dc bias in the approximate computation of spectra of Gaussian processes is determined by hard limiting.
Rodemich, E. R.↗
An approximate solution to improve computational efficiency of impedance-type payload load prediction
The computational efficiency of the impedance type loads prediction method was studied. Three goals were addressed: devise a method to make the impedance method operate more efficiently in the computer; assess the accuracy and convenience of the method for determining the effect of design changes; and investigate the use of the method to identify design changes for reduction of payload loads. The method is suitable for calculation of dynamic response in either the frequency or time domain. It is concluded that: the choice of an orthogonal coordinate system will allow the impedance method to operate more efficiently in the computer; the approximate mode impedance technique is adequate for determining the effect of design changes, and is applicable for both statically determinate and statically indeterminate payload attachments; and beneficial design changes to reduce payload loads can be identified by the combined application of impedance techniques and energy distribution review techniques.
White, C. W.↗
A computational procedure to analyze metal matrix laminates with nonlinear lamination residual strains
An approximate computational procedure is described for the analysis of angleplied laminates with residual nonlinear strains. The procedure consists of a combination of linear composite mechanics and incremental linear laminate theory. The procedure accounts for initial nonlinear strains, unloading, and in-situ matrix orthotropic nonlinear behavior. The results obtained in applying the procedure to boron/aluminum angleplied laminates show that this is a convenient means to accurately predict the initial tangent properties of angleplied laminates in which the matrix has been strained nonlinearly by the lamination residual stresses. The procedure predicted initial tangent properties results which were in good agreement with measured data obtained from boron/aluminum angleplied laminates.
Chamis, C. C.↗
A computational procedure to analyze metal matrix laminates with nonlinear lamination residual strains
An approximate computational procedure is described for the analysis of angleplied laminates with residual nonlinear strains. The procedure consists of a combination of linear composite mechanics and incremental linear laminate theory. The procedure accounts for initial nonlinear strains, unloading, and in-situ matrix orthrotropic nonlinear behavior. The results obtained in applying the procedure to boron/aluminum angleplied laminates show that this is a convenient means to accurately predict the initial tangent properties of angleplied laminates in which the matrix has been strained nonlinearly by the lamination residual stresses. The procedure predicted initial tangent properties results which were in good agreement with measured data obtained from boron/aluminum angleplied laminates.
Chamis, C. C.↗
On least squares approximations to indefinite problems of the mixed type
A least squares method is presented for computing approximate solutions of indefinite partial differential equations of the mixed type such as those that arise in connection with transonic flutter analysis. The method retains the advantages of finite difference schemes namely simplicity and sparsity of the resulting matrix system. However, it offers some great advantages over finite difference schemes. First, the method is insensitive to the value of the forcing frequency, i.e., the resulting matrix system is always symmetric and positive definite. As a result, iterative methods may be successfully employed to solve the matrix system, thus taking full advantage of the sparsity. Furthermore, the method is insensitive to the type of the partial differential equation, i.e., the computational algorithm is the same in elliptic and hyperbolic regions. In this work the method is formulated and numerical results for model problems are presented. Some theoretical aspects of least squares approximations are also discussed.
Fix, G. J.↗
APPRENTICE
SF-19-096 In this software, there are multiple approaches to compute approximations to multivariate functions. This includes: (a) an approach that builds on Stieltjes process to efficiently compute the coefficients of the rational approximation, (b) an approach that is based on an optimization formulation to get pole-free coefficients of the rational approximation and (c) approaches that finds the coefficients of rational and polynomial approximations by finding the linear least-squares solution using singular value decomposition.
KRISNAMOORTHY, MOHAN↗
Approximate Cartesian state transition matrix
Based on a Taylor series expansion, an easily-computed approximation to the Cartesian state transition matrix is presented for a general velocity-independent force field. Suitable for the short time intervals encountered in onboard navigation applications of the extended Kalman filter, it provides approximately five decimal digits of accuracy for earth orbiting spacecraft with update intervals of one minute, and better accuracy for shorter intervals.
Markley, F. L.↗
First-principles calculations and experimental studies of: XYZ 2 thermoelectric compounds: Detailed analysis of van der Waals interactions
First-principles calculations can accelerate the search for novel high-performance thermoelectric materials. However, the prediction of the thermoelectric properties is strongly dependent on the approximations used for the calculations. Here, thermoelectric properties were calculated with different computational approximations ( i.e. , PBE-GGA, HSE06, spin–orbit coupling and DFT-D3) for three layered XYZ 2 compounds (TmAgTe 2 , YAgTe 2 , and YCuTe 2 ). In addition to the computations, the structural, electrical and thermal properties of these compounds were measured experimentally and compared to the computations. Furthermore, an enhanced prediction of the crystal structure and heat capacity was achieved with the inclusion of van der Waals interactions due to more accurate modeling of the interatomic forces. In particular, a large shift of the acoustic phonons and low-frequency optical phonons to lower frequencies was observed from the dispersion-optimized structure. From the phonon dispersion curves of these compounds, the ultralow thermal conductivity in the investigated XYZ 2 compounds could be described by a recent developed minimum thermal conductivity model. Additionally, for the prediction of the electrical conductivity, a temperature-dependent relaxation time was used, and it was limited by acoustic phonons. While HSE06 has only a small influence on the electrical properties due to a computed band gap energy of >0.25 eV, the inclusion of both van der Waals interactions and spin–orbit coupling leads to a more accurate band structure, resulting in better prediction of electrical properties. Furthermore, the experimental thermoelectric properties of YAgTe 2 , TmAg 0.95 Zn 0.05 Te 2 and TmAg 0.95 Mg 0.05 Te 2 were measured, showing an increase in zT of TmAg 0.95 Zn 0.05 Te 2 by more than 35% ( zT = 0.47 ± 0.12) compared to TmAgTe 2 .
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Assimilation of TOPEX Sea Level Measurements with a Reduced-Gravity, Shallow Water Model of the Tropical Pacific Ocean
Sea surface height variability measured by TOPEX is analyzed in the tropical Pacific Ocean by way of assimilation into a wind-driven, reduced-gravity, shallow water model using an approximate Kalman filter and smoother. The analysis results in an optimal fit of the dynamic model to the observations, providing it dynamically consistent interpolation of sea level and estimation of the circulation. Nearly 80% of the expected signal variance is accounted for by the model within 20 deg of the equator, and estimation uncertainty is substantially reduced by the voluminous observation. Notable features resolved by the analysis include seasonal changes associated with the North Equatorial Countercurrent and equatorial Kelvin and Rossby waves. Significant discrepancies are also found between the estimate and TOPEX measurements, especially near the eastern boundary. Improvements in the estimate made by the assimilation are validated by comparisons with independent tide gauge and current meter observations. The employed filter and smoother are based on approximately computed estimation error covariance matrices, utilizing a spatial transformation and an symptotic approximation. The analysis demonstrates the practical utility of a quasi-optimal filter and smoother.
Fukumori, Ichiro↗
Transport of infrared radiation in cuboidal clouds
The transport of infrared radiation in a single cuboidal cloud is modeled using a variable azimuth two-stream approximation. Computations are made at 10 microns for a Deirmendjian (1969) C-1 water cloud where the single scattering albedo is equal to 0.638 and the asymmetry parameter is 0.865. The results indicate that the emittance of the top face of the model cloud is always less than that for a plane parallel cloud of the same optical depth. The hemispheric flux escaping from the cloud top possesses a gradient from the center to the edges which are warmer when the cloud is over warmer ground. Cooling rate calculations in the 8-13.6 micron region demonstrate that there is cooling out of the sides of the cloud at all levels even when there is heating of the core from the ground below. The radiances exiting from model cuboidal clouds are computed by path integration over the source function obtained with the two-stream approximation. Results indicate that the brightness temperature measured from finite clouds will overestimate the cloud-top temperature.
Harshvardhan, MR.↗
Initialization and Restart in Stochastic Local Search: Computing a Most Probable Explanation in Bayesian Networks
For hard computational problems, stochastic local search has proven to be a competitive approach to finding optimal or approximately optimal problem solutions. Two key research questions for stochastic local search algorithms are: Which algorithms are effective for initialization? When should the search process be restarted? In the present work we investigate these research questions in the context of approximate computation of most probable explanations (MPEs) in Bayesian networks (BNs). We introduce a novel approach, based on the Viterbi algorithm, to explanation initialization in BNs. While the Viterbi algorithm works on sequences and trees, our approach works on BNs with arbitrary topologies. We also give a novel formalization of stochastic local search, with focus on initialization and restart, using probability theory and mixture models. Experimentally, we apply our methods to the problem of MPE computation, using a stochastic local search algorithm known as Stochastic Greedy Search. By carefully optimizing both initialization and restart, we reduce the MPE search time for application BNs by several orders of magnitude compared to using uniform at random initialization without restart. On several BNs from applications, the performance of Stochastic Greedy Search is competitive with clique tree clustering, a state-of-the-art exact algorithm used for MPE computation in BNs.
Mengshoel, Ole J.↗
Verifying quantitative reliability for programs that execute on unreliable hardware
Emerging high-performance architectures are expected to contain unreliable components that may exhibit soft errors, which silently corrupt the results of computations. Full detection and masking of soft errors is challenging, expensive, and, for some applications, unnecessary. For example, approximate computing applications (such as multimedia processing, machine learning, and big data analytics) can often naturally tolerate soft errors. We introduce Rely, a programming language that enables developers to reason about the quantitative reliability of an application—namely, the probability that it produces the correct result when executed on unreliable hardware. Rely allows developers to specify the reliability requirements for each value that a function produces. We perform a static quantitative reliability analysis that verifies quantitative requirements on the reliability of an application, enabling a developer to perform sound and verified reliability engineering. The analysis takes a Rely program with a reliability specification and a hardware specification that characterizes the reliability of the underlying hardware components and verifies that the program satisfies its reliability specification when executed on the underlying unreliable hardware platform. We demonstrate the application of quantitative reliability analysis on six computations implemented in Rely.
97 MATHEMATICS AND COMPUTING↗
Projection techniques for iterative solution of A(bar x) equals (bar b) with successive right-hand sides
Two projection techniques for computing approximate solutions to linear systems of the form A(bar x)(sup n) = (bar b)(sup n), for a sequence n = 1, 2, ..., e.g., such as arises from time discretization of a partial differential equation, are presented. The inexpensive approximate solutions can be used as initial guesses for iterative solution of the system, resulting in significantly reduced computational expense. Examples of two- and three-dimensional incompressible Navier-Stokes calculations are presented in which x represents the pressure, and A is a discrete Poisson operator. In flows containing significant dynamic activity, these projection techniques lead to as much as a two-fold reduction in solution time.
Fischer, Paul F.↗
The second-generation Shifted Boundary Method and its numerical analysis
Recently, the Shifted Boundary Method (SBM) was proposed within the class of unfitted (or immersed, or embedded) finite element methods. By reformulating the original boundary value problem over a surrogate (approximate) computational domain, the SBM avoids integration over cut cells and the associated problematic issues regarding numerical stability and matrix conditioning. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. In this article, we present enhanced variational SBM formulations for the Poisson and Stokes problems with improved flexibility and robustness. These simplified variational forms allow to relax some of the assumptions required by the mathematical proofs of stability and convergence of earlier implementations. First, we show that these new SBM implementations can be proved asymptotically stable and convergent even without the rather restrictive assumption that the inner product between the normals to the true and surrogate boundaries is positive. Second, we show that it is not necessary to introduce a stabilization term involving the tangential derivatives of the solution at Dirichlet boundaries, therefore avoiding the calibration of an additional stabilization parameter. Finally, we prove enhanced L 2 -estimates without the cumbersome assumption – of earlier proofs – that the surrogate domain is convex. Instead we rely on a conventional assumption that the boundary of the true domain is smooth, which can also be replaced by requiring convexity of the true domain. The aforementioned improvements open the way to a more general and efficient implementation of the Shifted Boundary Method, particularly in complex three-dimensional geometries. We complement these theoretical developments with numerical experiments in two and three dimensions.
42 ENGINEERING↗