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Solving the Boltzmann-BGK equation with traditional numerical methods suffers from high computational and memory costs due to the curse of dimensionality. In this paper, we propose a novel accuracy-preserved tensor-train (APTT) method to efficiently solve the Boltzmann-BGK equation. A second-order finite difference scheme is applied to discretize the Boltzmann-BGK equation, resulting in a tensor algebraic system at each time step. Based on the low-rank TT representation, the tensor algebraic system is then approximated as a TT-based low-rank system, which is efficiently solved using the TT-modified alternating least-squares (TT-MALS) solver. Thanks to the low-rank TT representation, the APTT method can significantly reduce the computational and memory costs compared to traditional numerical methods. Theoretical analysis demonstrates that the APTT method maintains the same convergence rate as that of the finite difference scheme. The convergence rate and efficiency of the APTT method are validated by several benchmark test cases.

Best rank-one approximation is one of the most fundamental tasks in tensor computation. In order to fully exploit modern multi-core parallel computers, it is necessary to develop decoupling algorithms for computing the best rank-one approximation of higher-order tensors at large scales. In this paper, we first build a bridge between the rank-one approximation of tensors and the eigenvector-dependent nonlinear eigenvalue problem (NEPv), and then develop an efficient decoupling algorithm, namely the higher-order self-consistent field (HOSCF) algorithm, inspired by the famous self-consistent field (SCF) iteration frequently used in computational chemistry. The convergence theory of the HOSCF algorithm and an estimation of the convergence speed are further presented. In addition, we propose an improved HOSCF (iHOSCF) algorithm that incorporates the Rayleigh quotient iteration, which can significantly accelerate the convergence of HOSCF. Numerical experiments show that the proposed algorithms can efficiently converge to the best rank-one approximation of both synthetic and real-world tensors and can scale with high parallel scalability on a modern parallel computer.

Low-rank approximation in data streams is a fundamental and significant task in computing science, machine learning and statistics. Multiple streaming algorithms have emerged over years and most of them are inspired by randomized algorithms, more specifically, sketching methods. However, many algorithms are not able to leverage information of data streams and consequently suffer from low accuracy. Existing data-driven methods improve accuracy but the training cost is expensive in practice. In this paper, from a subspace perspective, we propose a tensor-based sketching method for low-rank approximation of data streams. The proposed algorithm fully exploits the structure of data streams and obtains quasi-optimal sketching matrices by performing tensor decomposition on training data. A series of experiments are carried out and show that the proposed tensor-based method can be more accurate and much faster than the previous work., Comment: 12 pages, 2 figures

We propose a novel rank-adaptive higher-order orthogonal iteration (HOOI) algorithm to compute the truncated Tucker decomposition of higher-order tensors with a given error tolerance, and prove that the method is locally optimal and monotonically convergent. A series of numerical experiments related to both synthetic and real-world tensors are carried out to show that the proposed rank-adaptive HOOI algorithm is advantageous in terms of both accuracy and efficiency. Some further analysis on the HOOI algorithm and the classical alternating least squares method are presented to further understand why rank adaptivity can be introduced into the HOOI algorithm and how it works.

Tucker decomposition is one of the most popular models for analyzing and compressing large-scale tensorial data. Existing Tucker decomposition algorithms usually rely on a single solver to compute the factor matrices and core tensor, and are not flexible enough to adapt with the diversities of the input data and the hardware. Moreover, to exploit highly efficient GEMM kernels, most Tucker decomposition implementations make use of explicit matricizations, which could introduce extra costs in terms of data conversion and memory usage. In this paper, we present a-Tucker, a new framework for input-adaptive and matricization-free Tucker decomposition of dense tensors. A mode-wise flexible Tucker decomposition algorithm is proposed to enable the switch of different solvers for the factor matrices and core tensor, and a machine-learning adaptive solver selector is applied to automatically cope with the variations of both the input data and the hardware. To further improve the performance and enhance the memory efficiency, we implement a-Tucker in a fully matricization-free manner without any conversion between tensors and matrices. Experiments with a variety of synthetic and real-world tensors show that a-Tucker can substantially outperform existing works on both CPUs and GPUs., Comment: 10pages

The low multilinear rank approximation, also known as the truncated Tucker decomposition, has been extensively utilized in many applications that involve higher-order tensors. Popular methods for low multilinear rank approximation usually rely directly on matrix SVD, therefore often suffer from the notorious intermediate data explosion issue and are not easy to parallelize, especially when the input tensor is large. In this paper, we propose a new class of truncated HOSVD algorithms based on alternating least squares (ALS) for efficiently computing the low multilinear rank approximation of tensors. The proposed ALS-based approaches are able to eliminate the redundant computations of the singular vectors of intermediate matrices and are therefore free of data explosion. Also, the new methods are more flexible with adjustable convergence tolerance and are intrinsically parallelizable on high-performance computers. Theoretical analysis reveals that the ALS iteration in the proposed algorithms is q-linear convergent with a relatively wide convergence region. Numerical experiments with large-scale tensors from both synthetic and real-world applications demonstrate that ALS-based methods can substantially reduce the total cost of the original ones and are highly scalable for parallel computing.