Hierarchical Matrix Research Articles

Boundary-element analysis of the noise scattering for urban aerial mobility vehicles: Solver development and assessment

Abstract Urban aerial mobility (UAM) vehicles with multi-propeller configurations have attracted considerable attention in recent years. However, previous investigations have mainly focused on the aerodynamic noise from individual propellers, with limited focus on the fuselage's sound scattering effects which can alter the far-field noise directivity. In this work, an efficient boundary element solver for sound scattering was developed to fill this gap. The solver employs hierarchical matrices to save computational cost. The benchmark examples showed high accuracy and good scalability. A representative vehicle model was then chosen, the propeller noise was simulated using rotating sources. Results show that the shielding effect in the fuselage/propeller configuration can produce an apparent noise reduction and redirect sound-energy distribution, suggesting the importance of considering the fuselage in low-noise UAM development.

Deep bidirectional hierarchical matrix factorization model for hyperspectral unmixing

Existing deep nonnegative matrix factorization-based approaches treat shallow and deep layers equally or with similar strategies, neglecting the heterogeneous physical structures between the shallow and progressively deeper layers, thus failing to explore the latent different sub-manifold structures in the hyperspectral image. In this paper, we propose a deep nonnegative matrix factorization model with bidirectional constraints to achieve hyperspectral unmixing. The sub-manifold structures in hyperspectral image are fully exploited by filtering and penalizing the shallow abundance layer with a denoised regularizer and a manifold regularizer. In contrast to the shallow abundance layer, the remaining layers are constrained by an extremely common regularizer to avoid over-denoising and maintain fidelity. In this way, the fine cues between different substances are exploited to a large extent. Additionally, the overall reconstruction error can be well controlled because the performance of the designed feedback mechanism can be fine-tuned by the inverse hierarchical constraints. Finally, we employ Nesterov's optimal gradient method to solve the optimization problem effectively. Experiment results are conducted on both synthetic datasets and real datasets, and all results show that the proposed method is superior to recent canonical unmixing methods.

Hierarchical Stability Conditions for Two Types of Time-Varying Delay Generalized Neural Networks.

In this article, the stability analysis for generalized neural networks (GNNs) with a time-varying delay is investigated. About the delay, the differential has only an upper boundary or cannot be obtained. For the both two types of delayed GNNs, up to now, the second-order integral inequalities have been the highest-order integral inequalities utilized to derive the stability conditions. To establish the stability conditions on the basis of the high-order integral inequalities, two challenging issues are required to be resolved. One is the formulation of the Lyapunov-Krasovskii functional (LKF), the other is the high-degree polynomial negative conditions (NCs). By transforming the integrals in N-order generalized free-matrix-based integral inequalities (GFIIs) into the multiple integrals, the hierarchical LKFs are constructed by adopting these multiple integrals. Then, the novel modified matrix polynomial NCs are presented for the 2N-1 degree delay polynomials in the LKF differentials. Thus, the hierarchical linear matrix inequalities (LMIs) are set up and the nonlinear problems caused by the GFIIs are solved at the same time. Eventually, the superiority of the provided hierarchical stability criteria is demonstrated by several numeric examples.

A graphics processing unit accelerated sparse direct solver and preconditioner with block low rank compression

We present the GPU implementation efforts and challenges of the sparse solver package STRUMPACK. The code is made publicly available on github with a permissive BSD license. STRUMPACK implements an approximate multifrontal solver, a sparse LU factorization which makes use of compression methods to accelerate time to solution and reduce memory usage. Multiple compression schemes based on rank-structured and hierarchical matrix approximations are supported, including hierarchically semi-separable, hierarchically off-diagonal butterfly, and block low rank. In this paper, we present the GPU implementation of the block low rank (BLR) compression method within a multifrontal solver. Our GPU implementation relies on highly optimized vendor libraries such as cuBLAS and cuSOLVER for NVIDIA GPUs, rocBLAS and rocSOLVER for AMD GPUs and the Intel oneAPI Math Kernel Library (oneMKL) for Intel GPUs. Additionally, we rely on external open source libraries such as SLATE (Software for Linear Algebra Targeting Exascale), MAGMA (Matrix Algebra on GPU and Multi-core Architectures), and KBLAS (KAUST BLAS). SLATE is used as a GPU-capable ScaLAPACK replacement. From MAGMA we use variable sized batched dense linear algebra operations such as GEMM, TRSM and LU with partial pivoting. KBLAS provides efficient (batched) low rank matrix compression for NVIDIA GPUs using an adaptive randomized sampling scheme. The resulting sparse solver and preconditioner runs on NVIDIA, AMD and Intel GPUs. Interfaces are available from PETSc, Trilinos and MFEM, or the solver can be used directly in user code. We report results for a range of benchmark applications, using the Perlmutter system from NERSC, Frontier from ORNL, and Aurora from ALCF. For a high frequency wave equation on a regular mesh, using 32 Perlmutter compute nodes, the factorization phase of the exact GPU solver is about 6.5× faster compared to the CPU-only solver. The BLR-enabled GPU solver is about 13.8× faster than the CPU exact solver. For a collection of SuiteSparse matrices, the STRUMPACK exact factorization on a single GPU is on average 1.9× faster than NVIDIA’s cuDSS solver.

High-Order Models for Gas Transport in Multiscale Coal Rock.

The coal reservoirs exhibit great heterogeneity and strong anisotropy in multiscale pore/fracture structures. Developing highly accurate multiscale models for real-time prediction of microgaseous flow in complicated porous media with pronounced contrast in transport coefficients is crucial but not yet available, which is time-consuming, expensive, and even computationally impossible. In this study, a multiscale approximate solution of the gas flow and pressure field is derived in this paper for predicting coalbed methane (CBM) transport in macro-microscopic two-scale porous media of typical coal rocks in Guizhou Province, China, and the detailed finite element algorithm is established. The multiscale approximate solutions are constructed from the first- and second-order auxiliary cell functions and low- or high-order derivatives of the homogenized pressure field, which can accurately approximate the solution of the original problem to a certain extent. The numerical accuracy and validity of the multiscale approximate solutions under various working conditions are analyzed in detail by comparing and verifying the results with the direct numerical simulation results of fine-mesh high-precision finite elements. The results show that the homogenized approximate solution can only roughly capture the characteristics of the macroscopic pressure evolution, the first-order approximate solution can partially reproduce the macro-microscopic pressure oscillation phenomenon, and the second-order approximate solution can accurately predict the pressure profile and its evolution law with time in porous media with macro-micro configuration under various complex conditions. The second- and high-order two-scale approximate solutions constructed in this paper exhibit high numerical accuracy and excellent computational efficiency. We also develop multiscale approximate solutions for predicting microgaseous flow at a low reservoir pressure where the slippage effects are very pronounced. Potential applications of our high-order models to the coal reservoirs with a hierarchical matrix and fractures are discussed. The developed multiscale models exhibit excellent applicability to investigating the crossflow effects and coupling mechanisms between the matrix and cleats or fractures with multiple configurations in the CBM and shale gas reservoirs, which is crucial for the effective implementation of hydraulic fracturing technology. This study provides a fundamental understanding of gas transport in multiscale matrix-fracture networks, which helps to improve the optimization design of hydraulic fracturing in tight reservoirs.

Reduced order modeling for real-time monitoring of structural displacements due to electromagnetic forces in large scale tokamaks

Abstract The real-time monitoring of the structural displacement of the vacuum vessel of thermonuclear fusion devices caused by electromagnetic loads is of great interest. In this paper, model order reduction is applied to the integral equation methods and the finite elements method to develop electromagnetic and structural reduced order models (ROMs) compatible with real-time execution which allows for the real-time monitoring of strain and displacement in critical positions of Tokamaks machines. Low-rank compression techniques based on hierarchical matrices are applied to reduce the computational cost during the offline stage when the ROMs are constructed. Numerical results show the accuracy of the approach and demonstrate the compatibility with real-time execution in standard hardware.

Nonlinear Hierarchical Matrix Factorization-Based Tensor Ring Approximation for Multi-dimensional Image Recovery

Nonlinear Hierarchical Matrix Factorization-Based Tensor Ring Approximation for Multi-dimensional Image Recovery

Can GPT-3.5 generate and code discharge summaries?

The aim of this study was to investigate GPT-3.5 in generating and coding medical documents with International Classification of Diseases (ICD)-10 codes for data augmentation on low-resource labels. Employing GPT-3.5 we generated and coded 9606 discharge summaries based on lists of ICD-10 code descriptions of patients with infrequent (or generation) codes within the MIMIC-IV dataset. Combined with the baseline training set, this formed an augmented training set. Neural coding models were trained on baseline and augmented data and evaluated on an MIMIC-IV test set. We report micro- and macro-F1 scores on the full codeset, generation codes, and their families. Weak Hierarchical Confusion Matrices determined within-family and outside-of-family coding errors in the latter codesets. The coding performance of GPT-3.5 was evaluated on prompt-guided self-generated data and real MIMIC-IV data. Clinicians evaluated the clinical acceptability of the generated documents. Data augmentation results in slightly lower overall model performance but improves performance for the generation candidate codes and their families, including 1 absent from the baseline training data. Augmented models display lower out-of-family error rates. GPT-3.5 identifies ICD-10 codes by their prompted descriptions but underperforms on real data. Evaluators highlight the correctness of generated concepts while suffering in variety, supporting information, and narrative. While GPT-3.5 alone given our prompt setting is unsuitable for ICD-10 coding, it supports data augmentation for training neural models. Augmentation positively affects generation code families but mainly benefits codes with existing examples. Augmentation reduces out-of-family errors. Documents generated by GPT-3.5 state prompted concepts correctly but lack variety, and authenticity in narratives.

Configuration design methodof hybrid electric agricultural machinery system based on layered two-dimensional matrix

The working environment and objects of agricultural machinery are different from those of automobiles, andagricultural machinery is greatly affected by the working environment and working conditions, and the power output systemismore complex. Agricultural machinery not only has drive output, but also PTO output and hydraulic output, whichtogetherconstitute the output system of agricultural machinery. Agricultural machinery conditions can be dividedintoroadtransportation working conditions and field operation working conditions. The working conditions of agricultural machinerycan be divided into different load conditions according to the different traction tools and whether the hydraulic andPTOwork,such as ploughing, rotary tillage, fertilization, and transportation. Therefore, developing hybrid electric agricultural machinerysystems that are suitable for various complex working conditions holds great theoretical significance and practical value. Giventhe complex working conditions of agricultural machinery systems in agricultural work and the intricate challengesindesigning hybrid agricultural machinery systems. In this paper, the two-dimensional matrix is used to represent thephysicalstructure and dynamics of the multi-channel power output agricultural mechanism. A hierarchical two-dimensional matrixmethod for the generation and screening of hybrid electric agricultural machinery systems with multi-power output powerisproposed. The components of agricultural machinery are divided into an input layer and an output layer, and these componentsare coded and defined, and then transformed into a matrix. The hierarchical two-dimensional matrix method is usedtogenerateand screen the hybrid electric agricultural mechanism type. Through the stratification of the matrix, the complexityoftheconfiguration generation is reduced, and the constraints are applied to the basic screening of the generated configurations. Therationality of the configurations obtained after generation and screening is verified by Simulink simulation. The resultsshowthat the configuration screened by this method can meet the performance requirements of agricultural machinery.

A Hierarchical Matrix Factorization-Based Method for Intelligent Industrial Fault Diagnosis.

Data-driven fault diagnosis, identifying abnormality causes using collected industrial data, is one of the challenging tasks for intelligent industry safety management. It is worth noting that practical industrial data are usually related to a mixture of several physical attributes, such as the operating environment, product quality and working conditions. However, the traditional models may not be sufficient to leverage the coherent information for diagnostic performance enhancement, due to their shallow architecture. This paper presents a hierarchical matrix factorization (HMF) that relies on a succession of matrix factoring to find an efficient representation of industrial data for fault diagnosis. Specifically, HMF consecutively decomposes data into several hierarchies. The intermediate hierarchies play the role of analysis operators which automatically learn implicit characteristics of industrial data; the final hierarchy outputs high-level and discriminative features. Furthermore, HMF is also extended in a nonlinear manner by introducing activation functions, referred as NHMF, to deal with nonlinearities in practical industrial processes. The applications of HMF and NHMF to fault diagnosis are evaluated by the multiple-phase flow process. The experimental results show that our models achieve competitive performance against the considered shallow and deep models, consuming less computing time than deep models.

SmashGP: Large-Scale Spatial Modeling via Matrix-Free Gaussian Processes

Gaussian processes are essential for spatial data analysis. Not only do they allow the prediction of unknown values, but they also allow for uncertainty quantification. However, in the era of big data, directly using Gaussian processes has become computationally infeasible as cubic run times are required for dense matrix decomposition and inversion. Various alternatives have been proposed to reduce the computational burden of directly fitting Gaussian processes. These alternatives rely on assumptions on the underlying structure of the covariance or precision matrices, such as sparsity or low-rank. In contrast, this article uses hierarchical matrices and matrix-free methods to enable the computation of Gaussian processes for large spatial datasets by exploiting the underlying kernel properties. The proposed framework, smashGP, represents the covariance matrix as an H 2 matrix in O ( n ) time and is able to estimate the unknown parameters of the model and predict the values of spatial observations at unobserved locations in O ( n log n ) time thanks to fast matrix-vector products. Additionally, it can be parallelized to take full advantage of shared-memory computing environments. With simulations and case studies, we illustrate the advantage of smashGP to model large-scale spatial datasets. Supplementary materials for this article are available online.

Link prediction method for social networks based on a hierarchical and progressive user interaction matrix

Link prediction within social networks represents a notable research area. Leveraging semantic text and multidimensional user interaction features, this paper introduces a hierarchical and progressive user interaction matrix-based link prediction method in social networks. First, considering the dynamic nature of a user’s semantic text features, the word2vec algorithm was employed to extract text features. Given the long short-term memory algorithm’s proficiency in learning long-term sequence dependencies, we proposed a method to represent time-series text called “TT2vec” in order to capture the evolution pattern of a user’s semantic text features. Second, to address the multidimensionality of user interactions, we constructed a hierarchical and progressive interaction network adjacency matrix, thereby deriving advanced user interaction features using an enhanced graph convolutional network. Then, using TT2vec to further capture the interaction network’s evolution, we proposed HPIN2vec, which is a method for representing hierarchical and progressive user interaction networks. Finally, for the users’ semantic text features and multidimensional interaction features, we proposed a link prediction method based on graph attention networks (GATs) and users’ dynamic features, considering the advantage of GAT aggregating neighbor node features. Experimental findings indicated that the proposed method not only identifies immediate dynamics in user features but also enhances link prediction accuracy through complex hierarchical and progressive interaction patterns.

Hierarchical Design of Tissue-Mimetic Fibrillar Hydrogel Scaffolds.

Most tissues of the human body present hierarchical fibrillar extracellular matrices (ECMs) that have a strong influence over their physicochemical properties and biological behavior. Of great interest is the introduction of this fibrillar structure to hydrogels, particularly due to the water-rich composition, cytocompatibility, and tunable properties of this class of biomaterials. Here, the main bottom-up fabrication strategies for the design and production of hierarchical biomimetic fibrillar hydrogels and their most representative applications in the fields of tissue engineering and regenerative medicine are reviewed. For example, the controlled assembly/arrangement of peptides, polymeric micelles, cellulose nanoparticles (NPs), and magnetically responsive nanostructures, among others, into fibrillar hydrogels is discussed, as well as their potential use as fibrillar-like hydrogels (e.g., those from cellulose NPs) with key biofunctionalities such as electrical conductivity or remote stimulation. Finally, the major remaining barriers to the clinical translation of fibrillar hydrogels and potential future directions of research in this field are discussed.

BEM-based fast frequency sweep for acoustic scattering by periodic slab

This paper presents a boundary element method (BEM) for computing the energy transmittance of a singly-periodic grating in 2D for a wide frequency band, which is of engineering interest in various fields with possible applications to acoustic metamaterial design. The proposed method is based on the Padé approximants of the response. The high-order frequency derivatives of the sound pressure necessary to evaluate the approximants are evaluated by a novel fast BEM accelerated by the fast-multipole and hierarchical matrix methods combined with the automatic differentiation. The target frequency band is divided adaptively, and the Padé approximation is used in each subband so as to accurately estimate the transmittance for a wide frequency range. Through some numerical examples, we confirm that the proposed method can efficiently and accurately give the transmittance even when some anomalies and stopband exist in the target band.

Adaptive fast multiplication of ℋ²-matrices

Hierarchical matrices approximate a given matrix by a decomposition into low-rank submatrices that can be handled efficiently in factorized form. H 2 \mathcal {H}^2 -matrices refine this representation following the ideas of fast multipole methods in order to achieve linear, i.e., optimal complexity for a variety of important algorithms. The matrix multiplication, a key component of many more advanced numerical algorithms, has been a challenge: the only linear-time algorithms known so far either require the very special structure of HSS-matrices or need to know a suitable basis for all submatrices in advance. In this article, a new and fairly general algorithm for multiplying H 2 \mathcal {H}^2 -matrices in linear complexity with adaptively constructed bases is presented. The algorithm consists of two phases: first an intermediate representation with a generalized block structure is constructed, then this representation is re-compressed in order to match the structure prescribed by the application. The complexity and accuracy are analyzed and numerical experiments indicate that the new algorithm can indeed be significantly faster than previous attempts.

A Block Householder–Based Algorithm for the QR Decomposition of Hierarchical Matrices

A Block Householder–Based Algorithm for the QR Decomposition of Hierarchical Matrices

Morphological characterization of some local watermelon (Citrullus lanatus L.) genotypes of Turkey

The scope of this study, some of the local Turkish watermelon genotypes were evaluated morphologically. A total of 20 local genotypes and three commercial varieties were assessed regarding their cotyledon, leaf, fruit, and seed phenotypes. An extensive morphological analysis was performed on a diverse set of genotypes. Traits such as cotyledon shape and color intensity, leaf size and lobing, and fruit size, shape, and color were measured and compared. Hierarchical clustering analysis, correlation matrices, and principal component analysis were employed to interpret the data. The study revealed significant morphological variability among the watermelon genotypes. Cotyledons ranged from narrow to broad, with a predominance of large sizes and dark green coloration. Leaf characteristics varied widely, with a notable distribution across different sizes and degrees of lobing. Fruit analysis showed a broad spectrum of shapes, sizes, and colors, indicating a rich genetic diversity. Two main genotype clusters were identified, suggesting a clear distinction based on morphological traits.

Hypercorrelation Evolution for Video Class-Incremental Learning

Video class-incremental learning aims to recognize new actions while restricting the catastrophic forgetting of old ones, whose representative samples can only be saved in limited memory. Semantically variable subactions are susceptible to class confusion due to data imbalance. While existing methods address the problem by estimating and distilling the spatio-temporal knowledge, we further explores that the refinement of hierarchical correlations is crucial for the alignment of spatio-temporal features. To enhance the adaptability on evolved actions, we proposes a hierarchical aggregation strategy, in which hierarchical matching matrices are combined and jointly optimized to selectively store and retrieve relevant features from previous tasks. Meanwhile, a correlation refinement mechanism is presented to reinforce the bias on informative exemplars according to online hypercorrelation distribution. Experimental results demonstrate the effectiveness of the proposed method on three standard video class-incremental learning benchmarks, outperforming state-of-the-art methods. Code is available at: https://github.com/Lsen991031/HCE

Mitigating spectral bias for the multiscale operator learning

Neural operators have emerged as a powerful tool for learning the mapping between infinite-dimensional parameter and solution spaces of partial differential equations (PDEs). In this work, we focus on multiscale PDEs that have important applications such as reservoir modeling and turbulence prediction. We demonstrate that for such PDEs, the spectral bias towards low-frequency components presents a significant challenge for existing neural operators. To address this challenge, we propose a hierarchical attention neural operator (HANO) inspired by the hierarchical matrix approach. HANO features a scale-adaptive interaction range and self-attentions over a hierarchy of levels, enabling nested feature computation with controllable linear cost and encoding/decoding of multiscale solution space. We also incorporate an empirical H1 loss function to enhance the learning of high-frequency components. Our numerical experiments demonstrate that HANO outperforms state-of-the-art (SOTA) methods for representative multiscale problems.

Decoration on the inner surface of low-tortuosity microchannels derived from wood plate for highly stable lithium sulfur batteries

The low-tortuosity microchannels of wood-based carbon matrix in free-standing cathodes for lithium sulfur batteries (LSBs) were a double-edged sword, which brought in both a high energy density due to a high sulfur loading and severe shuttle effect for poor cycling stability. Herein, a layer of cellulose aerogel extracted from the wood wall was coated on the inner surface of the low-tortuosity microchannels of the wood plate, which was further transformed into a hierarchical carbon matrix using in free-standing cathode for LSBs. The decorated cathode exhibited a maximal specific capacity of 1377.2 mAh g−1 due to the enhanced utilizing ratio of active materials and exceptional cycling stability even under a high current density (1 C) for more than 500 cycles. Furthermore, the decorated cathode maintained good cycling stability with a higher sulfur areal loading (6.3 mg cm−2). The cellulose-based carbon aerogel coated on the inner surface of low-tortuosity microchannels provided a large specific surface area. This decoration strategy not only provided physical restriction on polysulfides but also increased conversion sites for polysulfides, improving the cycling performance of the wood-based free-standing cathode of LSBs. This work provided a potential structural design strategy for the advanced free-standing cathode of LSBs.