Journal articles: 'GRADIENT COMPUTATION'

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Ding, Zhiyan, and Qin Li. "Constrained Ensemble Langevin Monte Carlo." Foundations of Data Science 4, no. 1 (2022): 37. http://dx.doi.org/10.3934/fods.2021034.

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The classical Langevin Monte Carlo method looks for samples from a target distribution by descending the samples along the gradient of the target distribution. The method enjoys a fast convergence rate. However, the numerical cost is sometimes high because each iteration requires the computation of a gradient. One approach to eliminate the gradient computation is to employ the concept of "ensemble." A large number of particles are evolved together so the neighboring particles provide gradient information to each other. In this article, we discuss two algorithms that integrate the ensemble feature into LMC, and the associated properties.In particular, we find that if one directly surrogates the gradient using the ensemble approximation, the algorithm, termed Ensemble Langevin Monte Carlo, is unstable due to a high variance term. If the gradients are replaced by the ensemble approximations only in a constrained manner, to protect from the unstable points, the algorithm, termed Constrained Ensemble Langevin Monte Carlo, resembles the classical LMC up to an ensemble error but removes most of the gradient computation.

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B N, Shobha, Govind R. Kadambi, S. R. Shankapal, and Yuri Vershinim. "Effect of variation in colour gradient information for optic flow computations." International Journal of Engineering & Technology 3, no. 4 (September 17, 2014): 445. http://dx.doi.org/10.14419/ijet.v3i4.2722.

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Optic flow algorithms provide mapping of 3D velocities on 2D image space. Optic flow is computed on the pair of images which are in sequence and is normally gray scale images. Optic flow computation using Horn and Schunck assumes that brightness consistency is maintained. Colour optic flow has the advantage that optic flow vectors are obtained even when there is a variation of brightness in the input images. The use of colour bands for optic flow is investigated by considering gradients of colour bands and component gradients. Results of applying these two types of gradients to three colour models are presented and analyzed. Decision logic is proposed to select the best colour model for colour optic flow computation based on gradient analysis. Keywords: Activity Measure. Colour Bands, Component Gradients, Decision Logic, Optic Flow Computation.

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Sengupta, B., K. J. Friston, and W. D. Penny. "Efficient gradient computation for dynamical models." NeuroImage 98 (September 2014): 521–27. http://dx.doi.org/10.1016/j.neuroimage.2014.04.040.

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Xu, Jingyan, and Frederic Noo. "Efficient gradient computation for optimization of hyperparameters." Physics in Medicine & Biology 67, no. 3 (February 7, 2022): 03NT01. http://dx.doi.org/10.1088/1361-6560/ac4442.

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Abstract We are interested in learning the hyperparameters in a convex objective function in a supervised setting. The complex relationship between the input data to the convex problem and the desirable hyperparameters can be modeled by a neural network; the hyperparameters and the data then drive the convex minimization problem, whose solution is then compared to training labels. In our previous work (Xu and Noo 2021 Phys. Med. Biol. 66 19NT01), we evaluated a prototype of this learning strategy in an optimization-based sinogram smoothing plus FBP reconstruction framework. A question arising in this setting is how to efficiently compute (backpropagate) the gradient from the solution of the optimization problem, to the hyperparameters to enable end-to-end training. In this work, we first develop general formulas for gradient backpropagation for a subset of convex problems, namely the proximal mapping. To illustrate the value of the general formulas and to demonstrate how to use them, we consider the specific instance of 1D quadratic smoothing (denoising) whose solution admits a dynamic programming (DP) algorithm. The general formulas lead to another DP algorithm for exact computation of the gradient of the hyperparameters. Our numerical studies demonstrate a 55%–65% computation time savings by providing a custom gradient instead of relying on automatic differentiation in deep learning libraries. While our discussion focuses on 1D quadratic smoothing, our initial results (not presented) support the statement that the general formulas and the computational strategy apply equally well to TV or Huber smoothing problems on simple graphs whose solutions can be computed exactly via DP.

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Hill, S. "Reduced gradient computation in prediction error identification." IEEE Transactions on Automatic Control 30, no. 8 (August 1985): 776–78. http://dx.doi.org/10.1109/tac.1985.1104062.

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Calugaru, Dan-Gabriel, and Jean-Marie Crolet. "Gradient computation in a nonlinear inverse problem." Comptes Rendus Mathematique 336, no. 8 (April 2003): 691–96. http://dx.doi.org/10.1016/s1631-073x(03)00130-4.

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Berlin, Konstantin, Nail A. Gumerov, David Fushman, and Ramani Duraiswami. "HierarchicalO(N) computation of small-angle scattering profiles and their associated derivatives." Journal of Applied Crystallography 47, no. 2 (March 28, 2014): 755–61. http://dx.doi.org/10.1107/s1600576714004671.

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The need for fast approximate algorithms for Debye summation arises in computations performed in crystallography, small/wide-angle X-ray scattering and small-angle neutron scattering. When integrated into structure refinement protocols these algorithms can provide significant speed up over direct all-atom-to-all-atom computation. However, these protocols often employ an iterative gradient-based optimization procedure, which then requires derivatives of the profile with respect to atomic coordinates. This article presents an accurate,O(N) cost algorithm for the computation of scattering profile derivatives. The results reported here show orders of magnitude improvement in computational efficiency, while maintaining the prescribed accuracy. This opens the possibility to efficiently integrate small-angle scattering data into the structure determination and refinement of macromolecular systems.

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Zhang, Jianfei, and Lei Zhang. "Efficient CUDA Polynomial Preconditioned Conjugate Gradient Solver for Finite Element Computation of Elasticity Problems." Mathematical Problems in Engineering 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/398438.

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Graphics processing unit (GPU) has obtained great success in scientific computations for its tremendous computational horsepower and very high memory bandwidth. This paper discusses the efficient way to implement polynomial preconditioned conjugate gradient solver for the finite element computation of elasticity on NVIDIA GPUs using compute unified device architecture (CUDA). Sliced block ELLPACK (SBELL) format is introduced to store sparse matrix arising from finite element discretization of elasticity with fewer padding zeros than traditional ELLPACK-based formats. Polynomial preconditioning methods have been investigated both in convergence and running time. From the overall performance, the least-squares (L-S) polynomial method is chosen as a preconditioner in PCG solver to finite element equations derived from elasticity for its best results on different example meshes. In the PCG solver, mixed precision algorithm is used not only to reduce the overall computational, storage requirements and bandwidth but to make full use of the capacity of the GPU devices. With SBELL format and mixed precision algorithm, the GPU-based L-S preconditioned CG can get a speedup of about 7–9 to CPU-implementation.

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Yang, Jucheng, Xiaojing Wang, Shujie Han, Jie Wang, Dong Sun Park, and Yuan Wang. "Improved Real-Time Facial Expression Recognition Based on a Novel Balanced and Symmetric Local Gradient Coding." Sensors 19, no. 8 (April 22, 2019): 1899. http://dx.doi.org/10.3390/s19081899.

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In the field of Facial Expression Recognition (FER), traditional local texture coding methods have a low computational complexity, while providing a robust solution with respect to occlusion, illumination, and other factors. However, there is still need for improving the accuracy of these methods while maintaining their real-time nature and low computational complexity. In this paper, we propose a feature-based FER system with a novel local texture coding operator, named central symmetric local gradient coding (CS-LGC), to enhance the performance of real-time systems. It uses four different directional gradients on 5 × 5 grids, and the gradient is computed in the center-symmetric way. The averages of the gradients are used to reduce the sensitivity to noise. These characteristics lead to symmetric of features by the CS-LGC operator, thus providing a better generalization capability in comparison to existing local gradient coding (LGC) variants. The proposed system further transforms the extracted features into an eigen-space using a principal component analysis (PCA) for better representation and less computation; it estimates the intended classes by training an extreme learning machine. The recognition rate for the JAFFE database is 95.24%, whereas that for the CK+ database is 98.33%. The results show that the system has advantages over the existing local texture coding methods.

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Smistad, Erik, and Frank Lindseth. "Multigrid gradient vector flow computation on the GPU." Journal of Real-Time Image Processing 12, no. 3 (October 30, 2014): 593–601. http://dx.doi.org/10.1007/s11554-014-0466-2.

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Davies, E. R. "Optimising computation of hexagonal differential gradient edge detector." Electronics Letters 27, no. 17 (1991): 1526. http://dx.doi.org/10.1049/el:19910959.

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de Moraes, Rafael J., Wessel de Zeeuw, José Roberto P. Rodrigues, Hadi Hajibeygi, and Jan Dirk Jansen. "Iterative multiscale gradient computation for heterogeneous subsurface flow." Advances in Water Resources 129 (July 2019): 210–21. http://dx.doi.org/10.1016/j.advwatres.2019.05.016.

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Gao, Jiaqi, Dongdong Peng, Tian Zhou, Tianhao Wang, and Chao Xu. "Terrain Matching Localization for Underwater Vehicle Based on Gradient Fitting." Journal of Sensors 2018 (December 30, 2018): 1–14. http://dx.doi.org/10.1155/2018/3717430.

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Terrain matching positioning is a promising method to overcome the problem that the inertial navigation error of the underwater vehicle accumulates over time. In the conventional terrain matching method, all measurement points are commonly used for matching and positioning. However, this method fails to be taken into a balanced consideration on both the computation complexity and the positioning accuracy. To reduce the computation and ensure the accuracy at the same time, an improved terrain matching method based on the gradient fitting is proposed in this paper. In the method, the gradient distributions of multiple terrain regions are firstly analyzed. Then, normal distribution is used to fit them, and according to the distribution, points with larger gradient values are selected as matching points. Finally, minimum absolute difference matching is chosen to match for positioning. Simulation results using multibeam sonar show that the improved terrain matching localization method not only reduces the computational complexity but also improves the accuracy of positioning.

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Danner, Gábor, Árpád Berta, István Hegedűs, and Márk Jelasity. "Robust Fully Distributed Minibatch Gradient Descent with Privacy Preservation." Security and Communication Networks 2018 (May 14, 2018): 1–15. http://dx.doi.org/10.1155/2018/6728020.

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Privacy and security are among the highest priorities in data mining approaches over data collected from mobile devices. Fully distributed machine learning is a promising direction in this context. However, it is a hard problem to design protocols that are efficient yet provide sufficient levels of privacy and security. In fully distributed environments, secure multiparty computation (MPC) is often applied to solve these problems. However, in our dynamic and unreliable application domain, known MPC algorithms are not scalable or not robust enough. We propose a light-weight protocol to quickly and securely compute the sum query over a subset of participants assuming a semihonest adversary. During the computation the participants learn no individual values. We apply this protocol to efficiently calculate the sum of gradients as part of a fully distributed minibatch stochastic gradient descent algorithm. The protocol achieves scalability and robustness by exploiting the fact that in this application domain a “quick and dirty” sum computation is acceptable. We utilize the Paillier homomorphic cryptosystem as part of our solution combined with extreme lossy gradient compression to make the cost of the cryptographic algorithms affordable. We demonstrate both theoretically and experimentally, based on churn statistics from a real smartphone trace, that the protocol is indeed practically viable.

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Fan, Xiaomeng, Yuwei Wu, Zhi Gao, Yunde Jia, and Mehrtash Harandi. "Efficient Riemannian Meta-Optimization by Implicit Differentiation." Proceedings of the AAAI Conference on Artificial Intelligence 36, no. 4 (June 28, 2022): 3733–40. http://dx.doi.org/10.1609/aaai.v36i4.20287.

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To solve optimization problems with nonlinear constrains, the recently developed Riemannian meta-optimization methods show promise, which train neural networks as an optimizer to perform optimization on Riemannian manifolds. A key challenge is the heavy computational and memory burdens, because computing the meta-gradient with respect to the optimizer involves a series of time-consuming derivatives, and stores large computation graphs in memory. In this paper, we propose an efficient Riemannian meta-optimization method that decouples the complex computation scheme from the meta-gradient. We derive Riemannian implicit differentiation to compute the meta-gradient by establishing a link between Riemannian optimization and the implicit function theorem. As a result, the updating our optimizer is only related to the final two iterations, which in turn speeds up our method and reduces the memory footprint significantly. We theoretically study the computational load and memory footprint of our method for long optimization trajectories, and conduct an empirical study to demonstrate the benefits of the proposed method. Evaluations of three optimization problems on different Riemannian manifolds show that our method achieves state-of-the-art performance in terms of the convergence speed and the quality of optima.

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Schwabacher, Mark, and Andrew Gelsey. "Intelligent gradient-based search of incompletely defined design spaces." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 11, no. 3 (June 1997): 199–210. http://dx.doi.org/10.1017/s0890060400003127.

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AbstractGradient-based numerical optimization of complex engineering designs offers the promise of rapidly producing better designs. However, such methods generally assume that the objective function and constraint functions are continuous, smooth, and defined everywhere. Unfortunately, realistic simulators tend to violate these assumptions. We present a rule-based technique for intelligently computing gradients in the presence of such pathologies in the simulators, and show how this gradient computation method can be used as part of a gradient-based numerical optimization system. We tested the resulting system in the domain of conceptual design of supersonic transport aircraft, and found that using rule-based gradients can decrease the cost of design space search by one or more orders of magnitude.

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De Keyser, J. "Least-squares multi-spacecraft gradient calculation with automatic error estimation." Annales Geophysicae 26, no. 11 (October 21, 2008): 3295–316. http://dx.doi.org/10.5194/angeo-26-3295-2008.

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Abstract. Multi-spacecraft missions allow the gradient of important physical quantities in the terrestrial environment to be determined. The gradient can be computed from four simultaneous measurements in a straightforward way, but this computation does not produce proper error estimates, making it hard to assess the meaningfulness of the result. Recently developed least-squares gradient computation techniques offer the possibility to obtain more precise results with all-inclusive error estimates, provided that information about the non-linearity of the space and time variations of the observed quantity is given. The present paper describes several heuristics for estimating these variations, thereby enabling a fully automatic computation of the gradient and the associated error estimates. The performance of these heuristics is illustrated with synthetic data corresponding to 4- and 10-spacecraft configurations.

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Skierczynski, B. A., S. Usami, S. Chien, and R. Skalak. "Active Motion of Polymorphonuclear Leukocytes in Response to Chemoattractant in a Micropipette." Journal of Biomechanical Engineering 115, no. 4B (November 1, 1993): 503–9. http://dx.doi.org/10.1115/1.2895531.

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A novel experimental method of producing and observing the active motion of polymorphonuclear leukocytes (PMNs) using a micropipette technique has been recently developed (Usami et al., 1992). The present paper develops a quantitative theory for the chemoattractant gradients and cell locomotion observed in these experiments. In previous experimental methods (e.g., the Boyden chamber, the Zygmond chamber and the Dunn chamber) for study chemotaxis of leukocytes, fibroblasts, and PMNs, the exact nature of the concentration gradient of the chemoattractant is unknown. The cells may themselves modify the local gradient of the chemoattractant. In experiments using the micropipette, an internal source of chemoattractant provides well-defined boundary and initial conditions which allow the computation of the chemoattractant concentration gradient during the active locomotion of the PMNs. Since the cell completely fills the pipette lumen, convection is limited to the motion of the cells themselves. In coordinates moving with cell, it is assumed that diffusion is the only mechanism of mass transport of the chemoattractant (fMLP). Computations of the fMLP concentration during locomotion of the cell were carried out for a range of rates of fMLP binding by the receptors expressed on the front face of the cell membrane. The results show that the front face of the cell is subjected to increasing fMLP concentration during the cell motion. The sequence of events involve receptor binding of fMLP, signal transduction, polymerization of the cell cytoskeleton at the membrane of the front face, spatially dependent adhesion to the pipette wall, and localized contraction of the cytoskeleton. This sequence of events leads to the steady locomotion of the leukocytes in the micropipette. The computation of the distribution of the fMLP concentration during cell locomotion with constant velocity in micropipette experiments shows that the cell is exposed to increasing concentration of fMLP. This suggests that chemotaxis maybe induced by temporal gradient of an attractant.

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Ozfatura, Emre, Sennur Ulukus, and Deniz Gündüz. "Straggler-Aware Distributed Learning: Communication–Computation Latency Trade-Off." Entropy 22, no. 5 (May 13, 2020): 544. http://dx.doi.org/10.3390/e22050544.

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When gradient descent (GD) is scaled to many parallel workers for large-scale machine learning applications, its per-iteration computation time is limited by straggling workers. Straggling workers can be tolerated by assigning redundant computations and/or coding across data and computations, but in most existing schemes, each non-straggling worker transmits one message per iteration to the parameter server (PS) after completing all its computations. Imposing such a limitation results in two drawbacks: over-computation due to inaccurate prediction of the straggling behavior, and under-utilization due to discarding partial computations carried out by stragglers. To overcome these drawbacks, we consider multi-message communication (MMC) by allowing multiple computations to be conveyed from each worker per iteration, and propose novel straggler avoidance techniques for both coded computation and coded communication with MMC. We analyze how the proposed designs can be employed efficiently to seek a balance between the computation and communication latency. Furthermore, we identify the advantages and disadvantages of these designs in different settings through extensive simulations, both model-based and real implementation on Amazon EC2 servers, and demonstrate that proposed schemes with MMC can help improve upon existing straggler avoidance schemes.

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Xiao, Yunchen, Len Thomas, and Mark A. J. Chaplain. "Calibrating models of cancer invasion: parameter estimation using approximate Bayesian computation and gradient matching." Royal Society Open Science 8, no. 6 (June 2021): 202237. http://dx.doi.org/10.1098/rsos.202237.

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We present two different methods to estimate parameters within a partial differential equation model of cancer invasion. The model describes the spatio-temporal evolution of three variables—tumour cell density, extracellular matrix density and matrix degrading enzyme concentration—in a one-dimensional tissue domain. The first method is a likelihood-free approach associated with approximate Bayesian computation; the second is a two-stage gradient matching method based on smoothing the data with a generalized additive model (GAM) and matching gradients from the GAM to those from the model. Both methods performed well on simulated data. To increase realism, additionally we tested the gradient matching scheme with simulated measurement error and found that the ability to estimate some model parameters deteriorated rapidly as measurement error increased.

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Lucka, Felix, Mailyn Pérez-Liva, Bradley E. Treeby, and Ben T. Cox. "High resolution 3D ultrasonic breast imaging by time-domain full waveform inversion." Inverse Problems 38, no. 2 (December 30, 2021): 025008. http://dx.doi.org/10.1088/1361-6420/ac3b64.

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Abstract Ultrasound tomography (UST) scanners allow quantitative images of the human breast’s acoustic properties to be derived with potential applications in screening, diagnosis and therapy planning. Time domain full waveform inversion (TD-FWI) is a promising UST image formation technique that fits the parameter fields of a wave physics model by gradient-based optimization. For high resolution 3D UST, it holds three key challenges: firstly, its central building block, the computation of the gradient for a single US measurement, has a restrictively large memory footprint. Secondly, this building block needs to be computed for each of the 103–104 measurements, resulting in a massive parallel computation usually performed on large computational clusters for days. Lastly, the structure of the underlying optimization problem may result in slow progression of the solver and convergence to a local minimum. In this work, we design and evaluate a comprehensive computational strategy to overcome these challenges: firstly, we exploit a gradient computation based on time reversal that dramatically reduces the memory footprint at the expense of one additional wave simulation per source. Secondly, we break the dependence on the number of measurements by using source encoding (SE) to compute stochastic gradient estimates. Also we describe a more accurate, TD-specific SE technique with a finer variance control and use a state-of-the-art stochastic LBFGS method. Lastly, we design an efficient TD multi-grid scheme together with preconditioning to speed up the convergence while avoiding local minima. All components are evaluated in extensive numerical proof-of-concept studies simulating a bowl-shaped 3D UST breast scanner prototype. Finally, we demonstrate that their combination allows us to obtain an accurate 442 × 442 × 222 voxel image with a resolution of 0.5 mm using Matlab on a single GPU within 24 h.

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Awotunde, A. A. A., and R. N. N. Horne. "An Improved Adjoint-Sensitivity Computation for Multiphase Flow Using Wavelets." SPE Journal 17, no. 02 (February 8, 2012): 402–17. http://dx.doi.org/10.2118/133866-pa.

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Summary In history matching, one of the challenges in the use of gradient-based Newton algorithms (e.g., Gauss-Newton and Leven-berg-Marquardt) in solving the inverse problem is the huge cost associated with the computation of the sensitivity matrix. Although the Newton type of algorithm gives faster convergence than most other gradient-based inverse solution algorithms, its use is limited to small- and medium-scale problems in which the sensitivity coefficients are easily and quickly computed. Modelers often use less-efficient algorithms (e.g., conjugate-gradient and quasi-Newton) to model large-scale problems because these algorithms avoid the direct computation of sensitivity coefficients. To find a direction of descent, such algorithms often use less-precise curvature information that would be contained in the gradient of the objective function. Using a sensitivity matrix gives more-complete information about the curvature of the function; however, this comes with a significant computational cost for large-scale problems. An improved adjoint-sensitivity computation is presented for time-dependent partial-differential equations describing multiphase flow in hydrocarbon reservoirs. The method combines the wavelet parameterization of data space with adjoint-sensitivity formulation to reduce the cost of computing sensitivities. This reduction in cost is achieved by reducing the size of the linear system of equations that are typically solved to obtain the sensitivities. This cost-saving technique makes solving an inverse problem with algorithms (e.g., Levenberg-Marquardt and Gauss-Newton) viable for large multiphase-flow history-matching problems. The effectiveness of this approach is demonstrated for two numerical examples involving multiphase flow in a reservoir with several production and injection wells.

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Gao, Peichao, Samuel A. Cushman, Gang Liu, Sijing Ye, Shi Shen, and Changxiu Cheng. "FracL: A Tool for Characterizing the Fractality of Landscape Gradients from a New Perspective." ISPRS International Journal of Geo-Information 8, no. 10 (October 22, 2019): 466. http://dx.doi.org/10.3390/ijgi8100466.

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The past several years have witnessed much progress in landscape ecology and fractal analysis. In landscape ecology, the gradient model of landscape patterns (i.e., landscape gradient) has emerged as a new operating paradigm, where most landscape metrics do not apply because they were developed for the patch mosaic model. In the fractal analysis, a new definition of fractal has been proposed, and various new fractal metrics have been developed. This technical note aims to provide an intersection of these two lines of advance, which will further present an opportunity to advance geo-informatics by considering the latest progress in both landscape ecology and fractal analysis. We first present an overview of the new definition of fractal and all the fractal metrics developed under this new definition. Since the chief obstacle to geographers and landscape ecologists in applying these metrics is the lack of readily accessible methods for their easy computation, we then develop an integrated tool to compute them on landscape gradients. The developed tool facilitates the computation of these new fractal metrics. A case study was carried out with real-life landscape gradients, namely a digital terrain model. These new fractal metrics and the developed tool can be expected to facilitate the fractal characterization of the patterns of gradient landscapes and the understanding of landscape dynamics from a new perspective.

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Chen, Mengyun, Kaixin Gao, Xiaolei Liu, Zidong Wang, Ningxi Ni, Qian Zhang, Lei Chen, et al. "THOR, Trace-based Hardware-driven Layer-Oriented Natural Gradient Descent Computation." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 8 (May 18, 2021): 7046–54. http://dx.doi.org/10.1609/aaai.v35i8.16867.

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It is well-known that second-order optimizer can accelerate the training of deep neural networks, however, the huge computation cost of second-order optimization makes it impractical to apply in real practice. In order to reduce the cost, many methods have been proposed to approximate a second-order matrix. Inspired by KFAC, we propose a novel Trace-based Hardware-driven layer-ORiented Natural Gradient Descent Computation method, called THOR, to make the second-order optimization applicable in the real application models. Specifically, we gradually increase the update interval and use the matrix trace to determine which blocks of Fisher Information Matrix (FIM) need to be updated. Moreover, by resorting the power of hardware, we have designed a Hardware-driven approximation method for computing FIM to achieve better performance. To demonstrate the effectiveness of THOR, we have conducted extensive experiments. The results show that training ResNet-50 on ImageNet with THOR only takes 66.7 minutes to achieve a top-1 accuracy of 75.9 % under an 8 Ascend 910 environment with MindSpore, a new deep learning computing framework. Moreover, with more computational resources, THOR can only takes 2.7 minutes to 75.9 % with 256 Ascend 910.

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Basnet, Min Bahadur, Mohammad Anas, Zarghaam Haider Rizvi, Asmer Hamid Ali, Mohammad Zain, Giovanni Cascante, and Frank Wuttke. "Enhancement of In-Plane Seismic Full Waveform Inversion with CPU and GPU Parallelization." Applied Sciences 12, no. 17 (September 2, 2022): 8844. http://dx.doi.org/10.3390/app12178844.

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Full waveform inversion is a widely used technique to estimate the subsurface parameters with the help of seismic measurements on the surface. Due to the amount of data, model size and non-linear iterative procedures, the numerical computation of Full Waveform Inversion are computationally intensive and time-consuming. This paper addresses the parallel computation of seismic full waveform inversion with Graphical Processing Units. Seismic full-waveform inversion of in-plane wave propagation in the finite difference method is presented here. The stress velocity formulation of the wave equation in the time domain is used. A four nodded staggered grid finite-difference method is applied to solve the equation, and the perfectly matched layers are considered to satisfy Sommerfeld’s radiation condition at infinity. The gradient descent method with conjugate gradient method is used for adjoined modelling in full-waveform inversion. The host code is written in C++, and parallel computation codes are written in CUDA C. The computational time and performance gained from CUDA C and OpenMP parallel computation in different hardware are compared to the serial code. The performance improvement is enhanced with increased model dimensions and remains almost constant after a certain threshold. A GPU performance gain of up to 90 times is obtained compared to the serial code.

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Zhu, Tiefeng, Zaizai Yan, and Xiuyun Peng. "A Modified Nonlinear Conjugate Gradient Method for Engineering Computation." Mathematical Problems in Engineering 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/1425857.

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A general criterion for the global convergence of the nonlinear conjugate gradient method is established, based on which the global convergence of a new modified three-parameter nonlinear conjugate gradient method is proved under some mild conditions. A large amount of numerical experiments is executed and reported, which show that the proposed method is competitive and alternative. Finally, one engineering example has been analyzed for illustrative purposes.

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Cong, Jason, Guojie Luo, and Eric Radke. "Highly Efficient Gradient Computation for Density-Constrained Analytical Placement." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 27, no. 12 (December 2008): 2133–44. http://dx.doi.org/10.1109/tcad.2008.2006158.

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Zhou, J. X., Q. Zhou, T. Chen, Y. J. Yin, and X. Shen. "Computation of feed-path based on temperature gradient method." Materials Research Innovations 19, sup5 (May 2015): S5–811—S5–816. http://dx.doi.org/10.1179/1432891714z.0000000001198.

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Carlson, J. R., N. Duquesne, C. L. Rumsey, and T. B. Gatski. "Computation of turbulent wake flows in variable pressure gradient." Computers & Fluids 30, no. 2 (February 2001): 161–87. http://dx.doi.org/10.1016/s0045-7930(00)00007-4.

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Moraes, Rafael J. de, José R. P. Rodrigues, Hadi Hajibeygi, and Jan Dirk Jansen. "Multiscale gradient computation for flow in heterogeneous porous media." Journal of Computational Physics 336 (May 2017): 644–63. http://dx.doi.org/10.1016/j.jcp.2017.02.024.

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Bozma, H. I. "Computation of Nash equilibria: admissibility of parallel gradient descent." Journal of Optimization Theory and Applications 90, no. 1 (July 1996): 45–61. http://dx.doi.org/10.1007/bf02192245.

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Steinmann, Paul. "Formulation and computation of geometrically non-linear gradient damage." International Journal for Numerical Methods in Engineering 46, no. 5 (October 20, 1999): 757–79. http://dx.doi.org/10.1002/(sici)1097-0207(19991020)46:5<757::aid-nme731>3.0.co;2-n.

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Zmigrod, Ran, Tim Vieira, and Ryan Cotterell. "Efficient Computation of Expectations under Spanning Tree Distributions." Transactions of the Association for Computational Linguistics 9 (2021): 675–90. http://dx.doi.org/10.1162/tacl_a_00391.

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Abstract We give a general framework for inference in spanning tree models. We propose unified algorithms for the important cases of first-order expectations and second-order expectations in edge-factored, non-projective spanning-tree models. Our algorithms exploit a fundamental connection between gradients and expectations, which allows us to derive efficient algorithms. These algorithms are easy to implement with or without automatic differentiation software. We motivate the development of our framework with several cautionary tales of previous research, which has developed numerous inefficient algorithms for computing expectations and their gradients. We demonstrate how our framework efficiently computes several quantities with known algorithms, including the expected attachment score, entropy, and generalized expectation criteria. As a bonus, we give algorithms for quantities that are missing in the literature, including the KL divergence. In all cases, our approach matches the efficiency of existing algorithms and, in several cases, reduces the runtime complexity by a factor of the sentence length. We validate the implementation of our framework through runtime experiments. We find our algorithms are up to 15 and 9 times faster than previous algorithms for computing the Shannon entropy and the gradient of the generalized expectation objective, respectively.

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Dalla, Carlos Eduardo Rambalducci, Wellington Betencurte da Silva, Júlio Cesar Sampaio Dutra, and Marcelo José Colaço. "A comparative study of gradient-based and meta-heuristic optimization methods using Griewank benchmark function/ Um estudo comparativo de métodos de otimização baseados em gradientes e meta-heurísticos usando a função de benchmark do Griewank." Brazilian Journal of Development 7, no. 6 (June 7, 2021): 55341–50. http://dx.doi.org/10.34117/bjdv7n6-102.

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Optimization methods are frequently applied to solve real-world problems such, engineering design, computer science, and computational chemistry. This paper aims to compare gradient-based algorithms and the meta-heuristic particle swarm optimization to minimize the multidimensional benchmark Griewank function, a multimodal function with widespread local minima. Several approaches of gradient-based methods such as steepest descent, conjugate gradient with Fletcher-Reeves and Polak-Ribiere formulations, and quasi-Newton Davidon-Fletcher-Powell approach were compared. The results presented showed that the meta-heuristic method is recommended for function with this behavior because is no needed prior information of the search space. The performance comparison includes computation time and convergence of global and local optimum.

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Fikl, Alexandru, Vincent Le Chenadec, and Taraneh Sayadi. "Control and Optimization of Interfacial Flows Using Adjoint-Based Techniques." Fluids 5, no. 3 (September 10, 2020): 156. http://dx.doi.org/10.3390/fluids5030156.

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The applicability of adjoint-based gradient computation is investigated in the context of interfacial flows. Emphasis is set on the approximation of the transport of a characteristic function in a potential flow by means of an algebraic volume-of-fluid method. A class of optimisation problems with tracking-type functionals is proposed. Continuous (differentiate-then-discretize) and discrete (discretize-then-differentiate) adjoint-based gradient computations are formulated and compared in a one-dimensional configuration, the latter being ultimately used to perform optimisation in two dimensions. The gradient is used in truncated Newton and steepest descent optimisers, and the algorithms are shown to recover optimal solutions. These validations raise a number of open questions, which are finally discussed with directions for future work.

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Fattah, Esmail Abdul, Janet Van Niekerk, and Håvard Rue. "Smart Gradient - An adaptive technique for improving gradient estimation." Foundations of Data Science 4, no. 1 (2022): 123. http://dx.doi.org/10.3934/fods.2021037.

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Computing the gradient of a function provides fundamental information about its behavior. This information is essential for several applications and algorithms across various fields. One common application that requires gradients are optimization techniques such as stochastic gradient descent, Newton's method and trust region methods. However, these methods usually require a numerical computation of the gradient at every iteration of the method which is prone to numerical errors. We propose a simple limited-memory technique for improving the accuracy of a numerically computed gradient in this gradient-based optimization framework by exploiting (1) a coordinate transformation of the gradient and (2) the history of previously taken descent directions. The method is verified empirically by extensive experimentation on both test functions and on real data applications. The proposed method is implemented in the <inline-formula><tex-math id="M1">\begin{document}$\texttt{R} $\end{document}</tex-math></inline-formula> package <inline-formula><tex-math id="M2">\begin{document}$ \texttt{smartGrad}$\end{document}</tex-math></inline-formula> and in C<inline-formula><tex-math id="M3">\begin{document}$ \texttt{++} $\end{document}</tex-math></inline-formula>.

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Sun, Zong Hai. "Constraint Projection Adaptive Natural Gradient Online Algorithm for SVM." Advanced Materials Research 139-141 (October 2010): 1692–96. http://dx.doi.org/10.4028/www.scientific.net/amr.139-141.1692.

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The training of Support Vector Machine (SVM) is an optimization problem of quadratic programming which can not be applied to the online training in real time applications or time-variant data source. The online algorithms proposed by other researchers have high computational complexity and slow training speed, which can not be well applied to the time-variant problems as well. In this paper the projection gradient and adaptive natural gradient is combined. The constraint projection adaptive natural gradient online algorithm for SVM is proposed. The computation complexity of the constraint projection adaptive natural gradient algorithm is . The learning performance is compared via prediction of the concentration of component A of Continuous Stirred Tank Reactor. The results of simulation demonstrate that the time taken by the constraint projection adaptive natural gradient online algorithm for SVM is far less than that of incremental algorithm, while keep higher prediction precision.

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Moldovanu, Simona, Lenuta Pană Toporaș, Anjan Biswas, and Luminita Moraru. "Combining Sparse and Dense Features to Improve Multi-Modal Registration for Brain DTI Images." Entropy 22, no. 11 (November 14, 2020): 1299. http://dx.doi.org/10.3390/e22111299.

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A new solution to overcome the constraints of multimodality medical intra-subject image registration is proposed, using the mutual information (MI) of image histogram-oriented gradients as a new matching criterion. We present a rigid, multi-modal image registration algorithm based on linear transformation and oriented gradients for the alignment of T2-weighted (T2w) images (as a fixed reference) and diffusion tensor imaging (DTI) (b-values of 500 and 1250 s/mm2) as floating images of three patients to compensate for the motion during the acquisition process. Diffusion MRI is very sensitive to motion, especially when the intensity and duration of the gradient pulses (characterized by the b-value) increases. The proposed method relies on the whole brain surface and addresses the variability of anatomical features into an image stack. The sparse features refer to corners detected using the Harris corner detector operator, while dense features use all image pixels through the image histogram of oriented gradients (HOG) as a measure of the degree of statistical dependence between a pair of registered images. HOG as a dense feature is focused on the structure and extracts the oriented gradient image in the x and y directions. MI is used as an objective function for the optimization process. The entropy functions and joint entropy function are determined using the HOGs data. To determine the best image transformation, the fiducial registration error (FRE) measure is used. We compare the results against the MI-based intensities results computed using a statistical intensity relationship between corresponding pixels in source and target images. Our approach, which is devoted to the whole brain, shows improved registration accuracy, robustness, and computational cost compared with the registration algorithms, which use anatomical features or regions of interest areas with specific neuroanatomy. Despite the supplementary HOG computation task, the computation time is comparable for MI-based intensities and MI-based HOG methods.

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Cao, Shaohua, Shu Chen, Hui Chen, Hanqing Zhang, Zijun Zhan, and Weishan Zhang. "HCOME: Research on Hybrid Computation Offloading Strategy for MEC Based on DDPG." Electronics 12, no. 3 (January 21, 2023): 562. http://dx.doi.org/10.3390/electronics12030562.

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With the growth of the Internet of Things, smart devices are subsequently generating a large number of computation-intensive and latency-sensitive tasks. Mobile edge computing can provide resources in close proximity, greatly reducing service latency and alleviating congestion in mobile core networks. Due to the instability of the mobile edge computing environment, it was difficult to guarantee the quality of service for users. To address this problem, a hybrid computation offloading framework based on Deep Deterministic Policy Gradient (DDPG) in IoT is proposed. The framework is a system consisting of edge servers and user devices. It is used to acquire the environment state through Software Defined Network technologies and generate the offloading strategy by Deep Deterministic Policy Gradient. The optimization objectives in this paper include the total system overhead of the mobile edge computing system, and considering both network load and computational load, an optimal offloading strategy can be obtained to enable users to obtain a better quality of service. Finally, the experimental results show that the algorithm outperforms the comparison algorithm and can reduce the system latency by 20%, while the network load and computational load are also more stable.

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Iwahana, Kazuki, Naoto Yanai, Jason Paul Cruz, and Toru Fujiwara. "SPGC: Integration of Secure Multiparty Computation and Differential Privacy for Gradient Computation on Collaborative Learning." Journal of Information Processing 30 (2022): 209–25. http://dx.doi.org/10.2197/ipsjjip.30.209.

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Fu, Ping, Hui Liu, Xihua Chu, and Yuanjie Xu. "A Multiscale Computational Formulation for Gradient Elasticity Problems of Heterogeneous Structures." International Journal of Computational Methods 13, no. 05 (August 31, 2016): 1650030. http://dx.doi.org/10.1142/s0219876216500304.

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In this paper, a multiscale computational formulation is developed for modeling two- and three-dimensional gradient elasticity behaviors of heterogeneous structures. To capture the microscopic properties at the macroscopic level effectively, a numerical multiscale interpolation function of coarse element is constructed by employing the oversampling element technique based on the staggered gradient elasticity scheme. By virtue of these functions, the equivalent quantities of the coarse element could be obtained easily, resulting in that the material microscopic characteristics are reflected to the macroscopic scale. Consequently, the displacement field of the original boundary value problem could be calculated at the macroscopic level, and the corresponding microscopic gradient-enriched solutions could also be evaluated by adopting the downscaling computation on the sub-grids of each coarse element domain, which will reduce the computational cost significantly. Furthermore, several representative numerical experiments are performed to demonstrate the validity and efficiency of the proposed multiscale formulation.

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Brzoska, Jean-Bruno, Frédéric Flin, and Jean Barckicke. "Explicit iterative computation of diffusive vapour field in the 3-D snow matrix: preliminary results for low flux metamorphism." Annals of Glaciology 48 (2008): 13–18. http://dx.doi.org/10.3189/172756408784700798.

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AbstractThe metamorphism of seasonal snow is classically considered as limited by vapour diffusion in the pore phase. To account for the lack of knowledge of the ice–vapour reaction coefficient near 0°C, the assumption of a reaction-limited metamorphism was first tested in three-dimensional simulations at low and very low temperature gradients; however, the validity of such results is difficult to verify experimentally. By a reasoned use of traditional iterative schemes, vapour diffusion is now simulated in three dimensions on tomographic snow data, mapping the gradient of vapour pressure near the grains. Repeating this process may provide a way to simulate the isothermal metamorphism without grain packing at a reasonable expense of computation time. Preliminary results are compared with existing computations made within the reaction-limited hypothesis.

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Nicolet, Baptiste, Fabrice Rousselle, Jan Novak, Alexander Keller, Wenzel Jakob, and Thomas Müller. "Recursive Control Variates for Inverse Rendering." ACM Transactions on Graphics 42, no. 4 (July 26, 2023): 1–13. http://dx.doi.org/10.1145/3592139.

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We present a method for reducing errors---variance and bias---in physically based differentiable rendering (PBDR). Typical applications of PBDR repeatedly render a scene as part of an optimization loop involving gradient descent. The actual change introduced by each gradient descent step is often relatively small, causing a significant degree of redundancy in this computation. We exploit this redundancy by formulating a gradient estimator that employs a recursive control variate , which leverages information from previous optimization steps. The control variate reduces variance in gradients, and, perhaps more importantly, alleviates issues that arise from differentiating loss functions with respect to noisy inputs, a common cause of drift to bad local minima or divergent optimizations. We experimentally evaluate our approach on a variety of path-traced scenes containing surfaces and volumes and observe that primal rendering efficiency improves by a factor of up to 10.

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Tan, Yanli, and Yongqiang Zhao. "A Fast Otsu Thresholding Method Based on an Improved 2D Histogram." International Journal of Circuits, Systems and Signal Processing 15 (August 12, 2021): 953–59. http://dx.doi.org/10.46300/9106.2021.15.102.

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The regional division of a traditional 2D histogram is difficult to obtain satisfactory image segmentation results. Based on the gray level-gradient 2D histogram, we proposed a fast 2D Otsu method based on integral image. In this method, the average gray level is replaced by the gray level gradient in the neighborhood of pixels, and the edge features of the image are extracted according to the gray level difference between adjacent pixels to improve the segmentation effect. Calculating the integral image from the two-dimensional histogram reduces the computational complexity of searching the optimal threshold, thus reducing the amount of computation. The simulation results demonstrate that the proposed algorithm has better performance in image segmentation, with the increased computational speed and improved real-time capability.

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Thurston, Jeffrey B., and R. James Brown. "Automated source‐edge location with a new variable pass‐band horizontal‐gradient operator." GEOPHYSICS 59, no. 4 (April 1994): 546–54. http://dx.doi.org/10.1190/1.1443615.

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A new class of space‐domain convolution operators permits computation of the components of the horizontal gradient of gridded potential‐field data. These so‐called gradient‐component operators allow one to vary the passband and thus control the frequency content of the resulting horizontal‐gradient map. This facilitates computation of gradient maps that accommodate data of widely varying frequency content. Examination of the transfer functions of these operators suggests that this method of numerical differentiation is well suited to potential‐field data: in particular, the operators suppress long wavelengths and highfrequency noise bands and amplify signal. Maps of the horizontal gradient of certain potential fields (e.g., gravity, pseudogravity) may be combined with algorithms that locate relative maxima, so‐called thresholding, to automate the procedure of source‐body edge detection, which is a useful tool in mapping, for example, basement grain, fault patterns, and igneous intrusive bodies. We apply this new operator, together with an existing thresholding algorithm, to a field example from western Canada and demonstrate its potential for improved imaging of the horizontal‐gradient magnitude and thus improved edge detection.

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Cao, Zuohao, Jianmin Ma, and Wayne R. Rouse. "Improving Computation of Sensible Heat Flux over a Water Surface Using the Variational Method." Journal of Hydrometeorology 7, no. 4 (August 1, 2006): 678–86. http://dx.doi.org/10.1175/jhm513.1.

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Abstract In this study, the authors have performed the variational computations for surface sensible heat fluxes over a large northern lake using observed wind, temperature gradient, and moisture gradient. In contrast with the conventional (Monin–Obukhov similarity theory) MOST-based flux-gradient method, the variational approach sufficiently utilizes observational meteorological conditions over the lake, where the conventional flux-gradient method performs poorly. Verifications using direct eddy-correlation measurements over Great Slave Lake, the fifth largest lake in North America in terms of surface area, during the open water period of 1999 demonstrate that the variational method yields good agreements between the computed and the measured sensible heat fluxes. It is also demonstrated that the variational method is more accurate than the flux-gradient method in computations of sensible heat flux across the air–water interface.

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Lachenmaier, Nicolas, Daniel Baumgärtner, Heinz-Peter Schiffer, and Johannes Kech. "Gradient-Free and Gradient-Based Optimization of a Radial Turbine." International Journal of Turbomachinery, Propulsion and Power 5, no. 3 (July 6, 2020): 14. http://dx.doi.org/10.3390/ijtpp5030014.

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A turbocharger’s radial turbine has a strong impact on the fuel consumption and transient response of internal combustion engines. This paper summarizes the efforts to design a new radial turbine aiming at high efficiency and low inertia by applying two different optimization techniques to a parametrized CAD model. The first workflow wraps 3D fluid and solid simulations within a meta-model assisted genetic algorithm to find an efficient turbine subjected to several constraints. In the next step, the chosen turbine is re-parametrized and fed into the second workflow which makes use of a gradient projection algorithm to further fine-tune the design. This requires the computation of gradients with respect to the CAD parametrization, which is done by calculating and combining surface sensitivities and design velocities. Both methods are applied successfully, i.e., the first delivers a well-performing turbine, which, by the second method, is further improved by 0.34% in efficiency.

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Zhao, Kang, Sida Huang, Pan Pan, Yinghan Li, Yingya Zhang, Zhenyu Gu, and Yinghui Xu. "Distribution Adaptive INT8 Quantization for Training CNNs." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 4 (May 18, 2021): 3483–91. http://dx.doi.org/10.1609/aaai.v35i4.16462.

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Researches have demonstrated that low bit-width (e.g., INT8) quantization can be employed to accelerate the inference process. It makes the gradient quantization very promising since the backward propagation requires approximately twice more computation than forward one. Due to the variability and uncertainty of gradient distribution, a lot of methods have been proposed to attain training stability. However, most of them ignore the channel-wise gradient distributions and the impact of gradients with different magnitudes, resulting in the degradation of final accuracy. In this paper, we propose a novel INT8 quantization training framework for convolutional neural network to address the above issues. Specifically, we adopt Gradient Vectorized Quantization to quantize the gradient, based on the observation that layer-wise gradients contain multiple distributions along the channel dimension. Then, Magnitude-aware Clipping Strategy is introduced by taking the magnitudes of gradients into consideration when minimizing the quantization error, and we present a theoretical derivation to solve the quantization parameters of different distributions. Experimental results on broad range of computer vision tasks, such as image classification, object detection and video classification, demonstrate that the proposed Distribution Adaptive INT8 Quantization training method has achieved almost lossless training accuracy for different backbones, including ResNet, MobileNetV2, InceptionV3, VGG and AlexNet, which is superior to the state-of-the-art techniques. Moreover, we further implement the INT8 kernel that can accelerate the training iteration more than 200% under the latest Turing architecture, i.e., our method excels on both training accuracy and speed.

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De Henau, V., G. D. Raithby, and B. E. Thompson. "Prediction of Flows With Strong Curvature and Pressure Gradient Using the k–ε Turbulence Model." Journal of Fluids Engineering 112, no. 1 (March 1, 1990): 40–47. http://dx.doi.org/10.1115/1.2909366.

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The k–ε turbulence model of Launder and Spalding has found widespread application in the computation of fluid flows. Under conditions of strong adverse pressure gradient or strong streamline curvature, the accuracy of the model is known to decrease. The present paper adds to the available case studies involving these types of flows by providing predictions for two problems. In the first problem, Simpson’s flow, an adverse pressure gradient induces the flow along a planar surface to separate. The second problem is unseparated flow over an airfoil. In addition to predictions using the standard k–ε model, results are reported from another k–ε model by Hanjalic and Launder, that has been modified to account for adverse pressure gradients.

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Lakrisenko, Polina, Paul Stapor, Stephan Grein, Łukasz Paszkowski, Dilan Pathirana, Fabian Fröhlich, Glenn Terje Lines, Daniel Weindl, and Jan Hasenauer. "Efficient computation of adjoint sensitivities at steady-state in ODE models of biochemical reaction networks." PLOS Computational Biology 19, no. 1 (January 3, 2023): e1010783. http://dx.doi.org/10.1371/journal.pcbi.1010783.

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Dynamical models in the form of systems of ordinary differential equations have become a standard tool in systems biology. Many parameters of such models are usually unknown and have to be inferred from experimental data. Gradient-based optimization has proven to be effective for parameter estimation. However, computing gradients becomes increasingly costly for larger models, which are required for capturing the complex interactions of multiple biochemical pathways. Adjoint sensitivity analysis has been pivotal for working with such large models, but methods tailored for steady-state data are currently not available. We propose a new adjoint method for computing gradients, which is applicable if the experimental data include steady-state measurements. The method is based on a reformulation of the backward integration problem to a system of linear algebraic equations. The evaluation of the proposed method using real-world problems shows a speedup of total simulation time by a factor of up to 4.4. Our results demonstrate that the proposed approach can achieve a substantial improvement in computation time, in particular for large-scale models, where computational efficiency is critical.

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Journal articles on the topic 'GRADIENT COMPUTATION'

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