Hierarchical Algorithms for Training Kolmogorov-Arnold Networks
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Cyr, Eric Christopher [Sandia National Laboratorie↗
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Parallel Implementation of the Recursive Approximation of an Unsupervised Hierarchical Segmentation Algorithm
The hierarchical image segmentation algorithm (referred to as HSEG) is a hybrid of hierarchical step-wise optimization (HSWO) and constrained spectral clustering that produces a hierarchical set of image segmentations. HSWO is an iterative approach to region grooving segmentation in which the optimal image segmentation is found at N(sub R) regions, given a segmentation at N(sub R+1) regions. HSEG's addition of constrained spectral clustering makes it a computationally intensive algorithm, for all but, the smallest of images. To counteract this, a computationally efficient recursive approximation of HSEG (called RHSEG) has been devised. Further improvements in processing speed are obtained through a parallel implementation of RHSEG. This chapter describes this parallel implementation and demonstrates its computational efficiency on a Landsat Thematic Mapper test scene.
Tilton, James C.↗
A Hierarchical OPF Algorithm with Improved Gradient Evaluation in Three-Phase Networks
Linear approximation commonly used in solving alternating-current optimal power flow (AC-OPF) simplifies the system models but incurs accumulated voltage errors in large power networks. Such errors will make the primal-dual type gradient algorithms converge to solutions with voltage violation. In this paper, we improve a recent hierarchical OPF algorithm that rested on primal-dual gradients evaluated with a linearized distribution power flow model. Specifically, we propose a more accurate gradient evaluation method based on an unbalanced three-phase nonlinear distribution power flow model to mitigate the errors arising from linearization. The resultant gradients feature a blocked structure that enables our development of an improved hierarchical primal-dual algorithm to solve the OPF problem. Numerical results on the IEEE 123-bus test feeder and a 4,518-node test feeder show that the proposed method can enhance voltage safety at comparable computational efficiency with the linearized algorithm.
approximation algorithms↗
A multiscale design method using interpretable machine learning for phononic materials with closely interacting scales
Manipulating the dispersive characteristics of vibrational waves is beneficial for many applications, e.g., high-precision instruments. architected hierarchical phononic materials have sparked promise tunability of elastodynamic waves and vibrations over multiple frequency ranges. In this article, hierarchical unit-cells are obtained, where features at each length scale result in a band gap within a targeted frequency range. Our novel approach, the ‘‘hierarchical unit-cell template method,’’ is an interpretable machine-learning approach that uncovers global unit-cell shape/topology patterns corresponding to predefined band-gap objectives. A scale-separation effect is observed where the coarse-scale band-gap objective is mostly unaffected by the fine-scale features despite the closeness of their length scales, thus enabling an efficient hierarchical algorithm. Moreover, the hierarchical patterns revealed are not predefined or self-similar hierarchies as common in current hierarchical phononic materials. Furthermore, our approach offers a flexible and efficient method for the exploration of new regions in the hierarchical design space, extracting minimal effective patterns for inverse design in applications targeting multiple frequency ranges.
Architected materials↗
A parallel algorithm for global routing
A Parallel Hierarchical algorithm for Global Routing (PHIGURE) is presented. The router is based on the work of Burstein and Pelavin, but has many extensions for general global routing and parallel execution. Main features of the algorithm include structured hierarchical decomposition into separate independent tasks which are suitable for parallel execution and adaptive simplex solution for adding feedthroughs and adjusting channel heights for row-based layout. Alternative decomposition methods and the various levels of parallelism available in the algorithm are examined closely. The algorithm is described and results are presented for a shared-memory multiprocessor implementation.
Brouwer, Randall J.↗
International Satellite Cloud Climatology Project (ISCCP) Stage D1 3-Hourly Cloud Product - Revised Algorithm in Hierarchical Data Format (ISCCP_D1)
Since 1983 an international group of institutions has collected and analyzed satellite radiance measurements from up to five geostationary and two polar orbiting satellites to infer the global distribution of cloud properties and their diurnal, seasonal and interannual variations. The primary focus of the first phase of the project (1983-1995) was the elucidation of the role of clouds in the radiation budget (top of the atmosphere and surface). In the second phase of the project (1995 onwards) the analysis also concerns improving understanding of clouds in the global hydrological cycle. [Location=TROPOSPHERE] [Temporal_Coverage: Start_Date=1983-07-01; Stop_Date=] [Spatial_Coverage: Southernmost_Latitude=-90; Northernmost_Latitude=90; Westernmost_Longitude=-180; Easternmost_Longitude=180] [Data_Resolution: Latitude_Resolution=280 Km; Longitude_Resolution=280 Km; Temporal_Resolution=3 Hourly].
CLOUD LIQUID WATER PATH↗
International Satellite Cloud Climatology Project (ISCCP) Stage D2 Monthly Cloud Product - Revised Algorithm in Hierarchical Data Format (ISCCP_D2)
Since 1983 an international group of institutions has collected and analyzed satellite radiance measurements from up to five geostationary and two polar orbiting satellites to infer the global distribution of cloud properties and their diurnal, seasonal and interannual variations. The primary focus of the first phase of the project (1983-1995) was the elucidation of the role of clouds in the radiation budget (top of the atmosphere and surface). In the second phase of the project (1995 onwards) the analysis also concerns improving understanding of clouds in the global hydrological cycle. [Location=TROPOSPHERE] [Temporal_Coverage: Start_Date=1983-07-01; Stop_Date=] [Spatial_Coverage: Southernmost_Latitude=-90; Northernmost_Latitude=90; Westernmost_Longitude=-180; Easternmost_Longitude=180] [Data_Resolution: Latitude_Resolution=280 Km; Longitude_Resolution=280 Km; Temporal_Resolution=Monthly].
CLOUD TOP PRESSURE↗
A family of solution algorithms for nonlinear structural analysis based on relaxation equations
A family of hierarchical algorithms for nonlinear structural equations are presented. The algorithms are based on the Davidenko-Branin type homotopy and shown to yield consistent hierarchical perturbation equations. The algorithms appear to be particularly suitable to problems involving bifurcation and limit point calculations. An important by-product of the algorithms is that it provides a systematic and economical means for computing the stepsize at each iteration stage when a Newton-like method is employed to solve the systems of equations. Some sample problems are provided to illustrate the characteristics of the algorithms.
Park, K. C.↗
Development of practical multiband algorithms for estimating land-surface temperature from EOS/MODIS data
A practical multiband, hierarchical algorithm for estimating land-surface temperature from NASA's future Earth Observing System (EOS) instruments Moderate Resolution Imaging Spectroradiometer (MODIS) and Advance Spaceborne Thermal Emission and Reflection Radiometer (ASTER) is developed through comprehensive, accurate, radiative transfer simulations at moderate spectral steps of 1-5/cm for wide ranges of atmospheric and surface conditions. The algorithm will accept empirical or estimated information about the surface emissivity and reflectivity and the atmospheric temperature and water-vapor profiles. Ground-based and aircraft measurements are necessary to validate and improve the algorithm and to establish its quality. Its accuracy depends on the calibration accuracy of thermal infrared data, uncertainties in surface heterogeneity, and temperature-dependent atmospheric absorption coefficients. Better knowledge of land-surface spectral emissivities and more accurate coefficients for atmospheric molecular band absorption and water vapor continuum absorption are needed to develop global land-surface temperature algorithms accurate to 1-2 K.
Dozier, J.↗
Approximation, abstraction and decomposition in search and optimization
In this paper, I discuss four different areas of my research. One portion of my research has focused on automatic synthesis of search control heuristics for constraint satisfaction problems (CSPs). I have developed techniques for automatically synthesizing two types of heuristics for CSPs: Filtering functions are used to remove portions of a search space from consideration. Another portion of my research is focused on automatic synthesis of hierarchic algorithms for solving constraint satisfaction problems (CSPs). I have developed a technique for constructing hierarchic problem solvers based on numeric interval algebra. Another portion of my research is focused on automatic decomposition of design optimization problems. We are using the design of racing yacht hulls as a testbed domain for this research. Decomposition is especially important in the design of complex physical shapes such as yacht hulls. Another portion of my research is focused on intelligent model selection in design optimization. The model selection problem results from the difficulty of using exact models to analyze the performance of candidate designs.
Ellman, Thomas↗
A Hierarchical Optimization Method for Electric Vertical Takeoff and Landing Aircraft Network Design
Electric vertical takeoff and landing aircraft (eVTOLs) are expected to serve urban air mobility in a station-to-station configuration, which makes the optimal network design of eVTOL stations a critical question to explore. Existing approaches often face limitations, such as the inability to interact station locations with demand or difficulty in finding the optimal solution for large study regions. Here, this paper first proposes a mathematical model to generate optimal eVTOL station locations while considering associated potential eVTOL demand, and then proposes a heuristic algorithm, Hierarchical Optimization MEthod (HOME), to efficiently solve the model. With a case study of Southern California, HOME was compared to 1) directly solving the original integer linear programming-based network design problem, and 2) employing the widely used genetic algorithm. Results suggest that HOME can find optimal solutions with limited computational resources. The proposed framework powered by HOME provides a computationally efficient way to support urban air mobility planning.
97 MATHEMATICS AND COMPUTING↗
Multi-frequency and polarimetric radar backscatter signatures for discrimination between agricultural crops at the Flevoland experimental test site
We describe the calibration and analysis of multi-frequency, multi-polarization radar backscatter signatures over an agriculture test site in the Netherlands. The calibration procedure involved two stages: in the first stage, polarimetric and radiometric calibrations (ignoring noise) were carried out using square-base trihedral corner reflector signatures and some properties of the clutter background. In the second stage, a novel algorithm was used to estimate the noise level in the polarimetric data channels by using the measured signature of an idealized rough surface with Bragg scattering (the ocean in this case). This estimated noise level was then used to correct the measured backscatter signatures from the agriculture fields. We examine the significance of several key parameters extracted from the calibrated and noise-corrected backscatter signatures. The significance is assessed in terms of the ability to uniquely separate among classes from 13 different backscatter types selected from the test site data, including eleven different crops, one forest and one ocean area. Using the parameters with the highest separation for a given class, we use a hierarchical algorithm to classify the entire image. We find that many classes, including ocean, forest, potato, and beet, can be identified with high reliability, while the classes for which no single parameter exhibits sufficient separation have higher rates of misclassification. We expect that modified decision criteria involving simultaneous consideration of several parameters increase performance for these classes.
Freeman, A.↗
Optimizing Multi-Robot Placements for Wire Arc Additive Manufacturing
Wire arc additive manufacturing is a metal additive manufacturing process in which the material is deposited using arc welding technology. It is gaining popularity due to high material deposition rates and faster build time. It is en-abled using robotic manipulators and can build relatively large-scale parts faster when compared with other metal additive manufacturing processes. However, the size of the large-scale parts is limited by the size of the industrial manipulator being used for the process. This limitation is overcome by using a fixed configuration multi-robot cell in which manipulators work cooperatively to build large-scale parts quickly. A fixed multi-robot cell with closely spaced industrial manipulators has high flexibility, but it restricts the part size that can be built. If the manipulators are spread out, the cell loses its flexibility but can build relatively larger parts. This issue can be avoided by using larger size manipulators, which are expensive, or by moving the modest size manipulators based on the part geometries. This paper presents a novel algorithm to generate multi-robot placements for different part geometries to be built using wire arc additive manufacturing. Furthermore, the algorithm hierarchically optimizes the build time and the inverse kinematics consistency in robot paths to improve the process efficiency and part quality. We compare the results with fixed multi-robot cells and provide insights to users to make an informed decision on whether to use a fixed or a flexible multi-robot cell for wire arc additive manufacturing.
Bhatt, Prahar↗
CONGO²: Scalable Online Anomaly Detection and Localization in Power Electronics Networks
Rapid and accurate detection and localization of electronic disturbances simultaneously are important for preventing its potential damages and determining potential remedies. Existing anomaly detection methods are severely limited by the low accuracy, the expensive computational cost and the need for highly trained personnel. There is an urgent need for a scalable online algorithm for in-field analysis of large-scale power electronics networks. Here in this paper, we propose a fast and accurate algorithm for anomaly detection and localization of power electronics networks: stratified colored-node graph (CONGO2). This algorithm hierarchically models the change of correlated waveforms and then correlated sensors using the colored-node graph. By aggregating the change of each sensor with its neighbors’ inputs, we can spontaneously identify and localize the anomaly that cannot be detected by data collected from a single sensor. As our proposed method only focuses on the changes within a short time frame, it is highly computational efficient and only needs small data storage. Thus, our method is ideal for online and reliable anomaly detection and localization of large-scale power electronic networks. Compared to existing anomaly detection methods, our method is entirely data-driven without training data, highly accurate and reliable for wide-spectrum anomalies detection, and more importantly, capable of both detection and localization. Thus, it is ideal for infield deployment for large-scale power electronic networks. As illustrated by a distributed energy resources (DERs) power grid with 37-node, our method can effectively detect and localize various cyber and physical attacks.
42 ENGINEERING↗
A hierarchical, automated target recognition algorithm for a parallel analog processor
A hierarchical approach is described for an automated target recognition (ATR) system, VIGILANTE, that uses a massively parallel, analog processor (3DANN). The 3DANN processor is capable of performing 64 concurrent inner products of size 1x4096 every 250 nanoseconds.
algorithm analog processor eigenvector↗
Reinforcement Learning for Weakly-Coupled MDPs and an Application to Planetary Rover Control
Weakly-coupled Markov decision processes can be decomposed into subprocesses that interact only through a small set of bottleneck states. We study a hierarchical reinforcement learning algorithm designed to take advantage of this particular type of decomposability. To test our algorithm, we use a decision-making problem faced by autonomous planetary rovers. In this problem, a Mars rover must decide which activities to perform and when to traverse between science sites in order to make the best use of its limited resources. In our experiments, the hierarchical algorithm performs better than Q-learning in the early stages of learning, but unlike Q-learning it converges to a suboptimal policy. This suggests that it may be advantageous to use the hierarchical algorithm when training time is limited.
Bernstein, Daniel S.↗
Recursive Hierarchical Image Segmentation by Region Growing and Constrained Spectral Clustering
This paper describes an algorithm for hierarchical image segmentation (referred to as HSEG) and its recursive formulation (referred to as RHSEG). The HSEG algorithm is a hybrid of region growing and constrained spectral clustering that produces a hierarchical set of image segmentations based on detected convergence points. In the main, HSEG employs the hierarchical stepwise optimization (HS WO) approach to region growing, which seeks to produce segmentations that are more optimized than those produced by more classic approaches to region growing. In addition, HSEG optionally interjects between HSWO region growing iterations merges between spatially non-adjacent regions (i.e., spectrally based merging or clustering) constrained by a threshold derived from the previous HSWO region growing iteration. While the addition of constrained spectral clustering improves the segmentation results, especially for larger images, it also significantly increases HSEG's computational requirements. To counteract this, a computationally efficient recursive, divide-and-conquer, implementation of HSEG (RHSEG) has been devised and is described herein. Included in this description is special code that is required to avoid processing artifacts caused by RHSEG s recursive subdivision of the image data. Implementations for single processor and for multiple processor computer systems are described. Results with Landsat TM data are included comparing HSEG with classic region growing. Finally, an application to image information mining and knowledge discovery is discussed.
Tilton, James C.↗
Haar-Like Wavelets on Hierarchical Trees
Here, discrete wavelet methods, originally formulated in the setting of regularly sampled signals, can be adapted to data defined on a point cloud if some multiresolution structure is imposed on the cloud. A wide variety of hierarchical clustering algorithms can be used for this purpose, and the multiresolution structure obtained can be encoded by a hierarchical tree of subsets of the cloud. Prior work introduced the use of Haar-like bases defined with respect to such trees for approximation and learning tasks on unstructured data. This paper builds on that work in two directions. First, we present an algorithm for constructing Haar-like bases on general discrete hierarchical trees. Second, with an eye towards data compression, we present thresholding techniques for data defined on a point cloud with error controlled in the $L$ $\infty$ norm and in a Hölder-type norm. In a concluding trio of numerical examples, we apply our methods to compress a point cloud dataset, study the tightness of the $L$ $\infty$ error bound, and use thresholding to identify MNIST classifiers with good generalizability.
97 MATHEMATICS AND COMPUTING↗