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Transfer-function approximations made optimal in special least-squares sense. Algorithm computes reduced-order rational-fraction approximates to single-input/single-output transfer functions. Reduces amount of computation needed for such purposes as numerical simulation of dynamics and design of control subsystems.
Observations regarding the use of advanced computational fluid dynamics (CFD) analysis, sensitivity analysis (SA), and design codes in gradient-based multidisciplinary design optimization (MDO) reflect our perception of the interactions required of CFD and our experience in recent aerodynamic design optimization studies using CFD. Sample results from these latter studies are summarized for conventional optimization (analysis - SA codes) and simultaneous analysis and design optimization (design code) using both Euler and Navier-Stokes flow approximations. The amount of computational resources required for aerodynamic design using CFD via analysis - SA codes is greater than that required for design codes. Thus, an MDO formulation that utilizes the more efficient design codes where possible is desired. However, in the aerovehicle MDO problem, the various disciplines that are involved have different design points in the flight envelope; therefore, CFD analysis - SA codes are required at the aerodynamic 'off design' points. The suggested MDO formulation is a hybrid multilevel optimization procedure that consists of both multipoint CFD analysis - SA codes and multipoint CFD design codes that perform suboptimizations.
Two-level least squares acceleration approaches are applied to the Chebyshev acceleration method and the restarted conjugate residual method in solving systems of linear algebraic equations with sparse unsymmetric coefficient matrices arising from finite volume or finite element approximations of boundary-value problems on irregular grids. Application of the proposed idea to other iterative restarted processes also is considered. The efficiency of the algorithms suggested is investigated numerically on a set of model Dirichlet problems for the convection-diffusion equation.
Here we propose a new type of neural networks, Kronecker neural networks (KNNs), that form a general framework for neural networks with adaptive activation functions. KNNs employ the Kronecker product, which provides an efficient way of constructing a very wide network while keeping the number of parameters low. Our theoretical analysis reveals that under suitable conditions, KNNs induce a faster decay of the loss than that by the feed-forward networks. This is also empirically verified through a set of computational examples. Furthermore, under certain technical assumptions, we establish global convergence of gradient descent for KNNs. As a specific case, we propose the Rowdy activation function that is designed to get rid of any saturation region by injecting sinusoidal fluctuations, which include trainable parameters. The proposed Rowdy activation function can be employed in any neural network architecture like feed-forward neural networks, Recurrent neural networks, Convolutional neural networks etc. The effectiveness of KNNs with Rowdy activation is demonstrated through various computational experiments including function approximation using feed-forward neural networks, solution inference of partial differential equations using the physics-informed neural networks, and standard deep learning benchmark problems using convolutional and fully-connected neural networks.
Observational tests of a model for the formation of the Local Group are presented and analyzed in which the mass concentration grows by gravitational accretion of local-pressure matter onto two seed masses in an otherwise homogeneous initial mass distribution. The evolution of the mass distribution is studied in an analytic approximation and a numerical computation. The initial seed mass and separation are adjusted to produce the observed present separation and relative velocity of the Andromeda Nebula and the Galaxy. If H(0) is adjusted to about 80 km/s/Mpc with density parameter Omega = 1, then the model gives a good fit to the motions of the outer members of the Local Group. The same model gives particle orbits at radius of about 100 kpc that reasonably approximate the observed distribution of redshifts of the Galactic satellites.
Among the well-known methods to approximate derivatives of expectancies computed by Monte-Carlo simulations, averages of pathwise derivatives are often the easiest one to apply. Computing them via algorithmic differentiation typically does not require major manual analysis and rewriting of the code, even for very complex programs like simulations of particle-detector interactions in high-energy physics. However, the pathwise derivative estimator can be biased if there are discontinuities in the program, which may diminish its value for applications. This work integrates algorithmic differentiation into the electromagnetic shower simulation code HepEmShow based on G4HepEm, allowing us to study how well pathwise derivatives approximate derivatives of energy depositions in a sampling calorimeter with respect to parameters of the beam and geometry. We found that when multiple scattering is disabled in the simulation, means of pathwise derivatives converge quickly to their expected values, and these are close to the actual derivatives of the energy deposition. Additionally, we demonstrate the applicability of this novel gradient estimator for stochastic gradient-based optimization in a model example.
Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this study we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.
A method is presented for calculating the shape sensitivity of a wing aeroelastic response with respect to changes in geometric shape. Yates' modified strip method is used in conjunction with Giles' equivalent plate analysis to predict the flutter speed, frequency, and reduced frequency of the wing. Three methods are used to calculate the sensitivity of the eigenvalue. The first method is purely a finite difference calculation of the eigenvalue derivative directly from the solution of the flutter problem corresponding to the two different values of the shape parameters. The second method uses an analytic expression for the eigenvalue sensitivities of a general complex matrix, where the derivatives of the aerodynamic, mass, and stiffness matrices are computed using a finite difference approximation. The third method also uses an analytic expression for the eigenvalue sensitivities, but the aerodynamic matrix is computed analytically. All three methods are found to be in good agreement with each other. The sensitivities of the eigenvalues were used to predict the flutter speed, frequency, and reduced frequency. These approximations were found to be in good agreement with those obtained using a complete reanalysis.
Nonlocal operators of fractional type are a popular modeling choice for applications that do not adhere to classical diffusive behavior; however, one major challenge in nonlocal simulations is the selection of model parameters. In this work we propose an optimization-based approach to parameter identification for fractional models with an optional truncation radius. We formulate the inference problem as an optimal control problem where the objective is to minimize the discrepancy between observed data and an approximate solution of the model, and the control variables are the fractional order and the truncation length. For the numerical solution of the minimization problem we propose a gradient-based approach, where we enhance the numerical performance by an approximation of the bilinear form of the state equation and its derivative with respect to the fractional order. Several numerical tests in one and two dimensions illustrate the theoretical results and show the robustness and applicability of our method.
A Cartesian, cell-based scheme for solving the Euler and Navier-Stokes equations in two dimensions is developed and tested. Grids about geometrically complicated bodies are generated automatically, by recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal 'cut' cells are created. The geometry of the cut cells is computed using polygon-clipping algorithms. The grid is stored in a binary-tree data structure which provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite-volume formulation. The convective terms are upwinded, with a limited linear reconstruction of the primitive variables used to provide input states to an approximate Riemann solver for computing the fluxes between neighboring cells. A multi-stage time-stepping scheme is used to reach a steady-state solution. Validation of the Euler solver with benchmark numerical and exact solutions is presented. An assessment of the accuracy of the approach is made by uniform and adaptive grid refinements for a steady, transonic, exact solution to the Euler equations. The error of the approach is directly compared to a structured solver formulation. A non smooth flow is also assessed for grid convergence, comparing uniform and adaptively refined results. Several formulations of the viscous terms are assessed analytically, both for accuracy and positivity. The two best formulations are used to compute adaptively refined solutions of the Navier-Stokes equations. These solutions are compared to each other, to experimental results and/or theory for a series of low and moderate Reynolds numbers flow fields. The most suitable viscous discretization is demonstrated for geometrically-complicated internal flows. For flows at high Reynolds numbers, both an altered grid-generation procedure and a different formulation of the viscous terms are shown to be necessary. A hybrid Cartesian/body-fitted grid generation approach is demonstrated. In addition, a grid-generation procedure based on body-aligned cell cutting coupled with a viscous stensil-construction procedure based on quadratic programming is presented.
We present a computationally efficient method to approximate the probability distribution of seismic Green's functions given the uncertainty of an Earth model. The method is based on the Karhunen-Loève (KL) theorem and an approximation of the Green's function (or seismogram) covariance. Using Monte Carlo (MC) simulations as a control case, we demonstrate that our KL-based method can accurately reproduce a probability distribution of seismograms that results from an uncertain Earth model for a MC-derived seismogram covariance. We then describe a method to estimate the covariance of the seismograms resulting from those Earth models that is not based on MC simulations. We use the estimated Green's function covariance in conjunction with our KL-based method to produce a Green's function probability distribution, and compare that distribution to a Green's function probability distribution produced using a MC finite difference method. We find that the Green's function probability distribution approximated using our KL-based method generally mimics that produced using the MC simulations, especially for direct-arriving body waves. However the accuracy of the KL-based method generally decreases for later times in the simulated Green's function distribution.
Ab initio and semiempirical electronic structure methods are usually implemented in separate software packages or use entirely different code paths. As a result, it can be time-consuming to transfer an established ab initio electronic structure scheme to a semiempirical Hamiltonian. Here we present an approach to unify ab initio and semiempirical electronic structure code paths based on a separation of the wavefunction ansatz and the needed matrix representations of operators. With this separation, the Hamiltonian can refer to either an ab initio or semiempirical treatment of the resulting integrals. We built a semiempirical integral library and interfaced it to the GPU-accelerated electronic structure code TeraChem. Equivalency between ab initio and semiempirical tight-binding Hamiltonian terms is assigned according to their dependence on the one-electron density matrix. The new library provides semiempirical equivalents of the Hamiltonian matrix and gradient intermediates, corresponding to those provided by the ab initio integral library. This enables the straightforward combination of semiempirical Hamiltonians with the full pre-existing ground and excited state functionality of the ab initio electronic structure code. We demonstrate the capability of this approach by combining the extended tight-binding method GFN1-xTB with both spin-restricted ensemble-referenced Kohn–Sham and complete active space methods. We also present a highly efficient GPU implementation of the semiempirical Mulliken-approximated Fock exchange. The additional computational cost for this term becomes negligible even on consumer-grade GPUs, enabling Mulliken-approximated exchange in tight-binding methods for essentially no additional cost.
We describe a cloud-based medical image reconstruction service designed to meet a real-time and daily demand to reconstruct thousands of images from proton cancer treatment facilities worldwide. Rapid reconstruction of a three-dimensional Proton Computed Tomography (pCT) image can require the transfer of 100 GB of data and use of approximately 120 GPU-enabled compute nodes. The nature of proton therapy means that demand for such a service is sporadic and comes from potentially hundreds of clients worldwide. We thus explore the use of a commercial cloud as a scalable and cost-efficient platform for pCT reconstruction. To address the high performance requirements of this application we leverage Amazon Web Services' GPU-enabled cluster resources that are provisioned with high performance networks between nodes. To support episodic demand, we develop an on-demand multi-user provisioning service that can dynamically provision and resize clusters based on image reconstruction requirements, priorities, and wait times. We compare the performance of our pCT reconstruction service running on commercial cloud resources with that of the same application on dedicated local high performance computing resources. We show that we can achieve scalable and on-demand reconstruction of large scale pCT images for simultaneous multi-client requests, processing images in less than 10 minutes for less than $10 per image.
The problem of obtaining accurate estimates of suction requirements on swept laminar flow control wings was discussed. A fast accurate computer code developed to predict suction requirements by integrating disturbance amplification rates was described. Assumptions and approximations used in the present computer code are examined in light of flow conditions on the swept wing which may limit their validity.
This paper presents an adaptive algorithm for motion compensated color image coding. The algorithm can be used for video teleconferencing or broadcast signals. Activity segmentation is used to reduce the bit rate and a variable stage search is conducted to save computations. The adaptive algorithm is compared with the nonadaptive algorithm and it is shown that with approximately 60 percent savings in computing the motion vector and 33 percent additional compression, the performance of the adaptive algorithm is similar to the nonadaptive algorithm. The adaptive algorithm results also show improvement of up to 1 bit/pel over interframe DPCM coding with nonuniform quantization. The test pictures used for this study were recorded directly from broadcast video in color.
Modern Computational Fluid Dynamics (CFD) techniques were used to compute aerodynamic forces and moments of the Space Shuttle Orbiter in specific portions of contingency abort trajectory space. The trajectory space covers a Mach number range of 3.5-15, an angle-of-attack range of 20deg-60deg, an altitude range of 100-190 kft, and several different settings of the control surfaces (elevons, body flap, and speed brake). Presented here are details of the methodology and comparisons of computed aerodynamic coefficients against the values in the current Orbiter Operational Aerodynamic Data Book (OADB). While approximately 40 cases have been computed, only a sampling of the results is provided here. The computed results, in general, are in good agreement with the OADB data (i.e., within the uncertainty bands) for almost all the cases. However, in a limited number of high angle-of-attack cases (at Mach 15), there are significant differences between the computed results, especially the vehicle pitching moment, and the OADB data. A preliminary analysis of the data from the CFD simulations at Mach 15 shows that these differences can be attributed to real-gas/Mach number effects. The aerodynamic coefficients and detailed surface pressure distributions of the present simulations are being used by the Shuttle Program in the evaluation of the capabilities of the Orbiter in contingency abort scenarios.
We propose a novel partitioning method for block-structured adaptive meshes utilizing the meshless Lagrangian particle concept. With the observation that an optimum partitioning has high analogy to the relaxation of a multi-phase fluid to steady state, physically motivated model equations are developed to characterize the background mesh topology and are solved by multi-phase smoothed-particle hydrodynamics. In contrast to well established partitioning approaches, all optimization objectives are implicitly incorporated and achieved during the particle relaxation to stationary state. Distinct partitioning sub-domains are represented by colored particles and separated by a sharp interface with a surface tension model. In order to obtain the particle relaxation, special viscous and skin friction models, coupled with a tailored time integration algorithm are proposed. Numerical experiments show that the present method has several important properties: generation of approximately equal-sized partitions without dependence on the mesh-element type, optimized interface communication between distinct partitioning sub-domains, continuous domain decomposition which is physically localized and implicitly incremental. Therefore it is particularly suitable for load-balancing of high-performance CFD simulations.
We report in regimes of high strain rate, the strength of materials often cannot be measured directly in experiments. Instead, the strength is inferred based on an experimental observable, such as a change in shape, that is matched by simulations supported by a known strength model. In hole closure experiments, the rate and degree to which a central hole in a plate of material closes during a dynamic loading event are used to infer material strength parameters. Due to the complexity of the experiment, many computationally expensive, three-dimensional simulations are necessary to train an emulator for calibration or other analyses. These simulations can be run at multiple grid resolutions, where dense grids are slower but more accurate. In an effort to reduce the computational cost, a combination of simulations with different resolutions can be combined to develop an accurate emulator within a limited training time. We explore the novel design and construction of an appropriate functional recursive multi-fidelity emulator of a strength model for tantalum in hole closure experiments that can be applied to arbitrarily large training data. Hence, by formulating a multi-fidelity model to employ low-fidelity simulations, we were able to reduce the error of our emulator by approximately 81% with only an approximately 1.6% increase in computing resource utilization.