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991.
Colin Ponce Ruipeng Li Christina Mao Panayot Vassilevski 《Numerical Linear Algebra with Applications》2023,30(5):e2501
A common challenge in regression is that for many problems, the degrees of freedom required for a high-quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this paper, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. We draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. Training through V-cycles then encourage the neural networks to build a hierarchical understanding of the problem. We refer to this approach as multilevel-in-width to distinguish from prior multilevel works which hierarchically alter the depth of neural networks. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalize performance of the neural networks studied. 相似文献
992.
Alexander Litvinenko Youssef Marzouk Hermann G. Matthies Marco Scavino Alessio Spantini 《Numerical Linear Algebra with Applications》2023,30(3):e2467
Very often, in the course of uncertainty quantification tasks or data analysis, one has to deal with high-dimensional random variables. Here the interest is mainly to compute characterizations like the entropy, the Kullback–Leibler divergence, more general -divergences, or other such characteristics based on the probability density. The density is often not available directly, and it is a computational challenge to just represent it in a numerically feasible fashion in case the dimension is even moderately large. It is an even stronger numerical challenge to then actually compute said characteristics in the high-dimensional case. In this regard it is proposed to approximate the discretized density in a compressed form, in particular by a low-rank tensor. This can alternatively be obtained from the corresponding probability characteristic function, or more general representations of the underlying random variable. The mentioned characterizations need point-wise functions like the logarithm. This normally rather trivial task becomes computationally difficult when the density is approximated in a compressed resp. low-rank tensor format, as the point values are not directly accessible. The computations become possible by considering the compressed data as an element of an associative, commutative algebra with an inner product, and using matrix algorithms to accomplish the mentioned tasks. The representation as a low-rank element of a high order tensor space allows to reduce the computational complexity and storage cost from exponential in the dimension to almost linear. 相似文献
993.
Tobias Danczul Clemens Hofreither Joachim Schöberl 《Numerical Linear Algebra with Applications》2023,30(5):e2488
We present a unified framework to efficiently approximate solutions to fractional diffusion problems of stationary and parabolic type. After discretization, we can take the point of view that the solution is obtained by a matrix-vector product of the form , where is the discretization matrix of the spatial operator, a prescribed vector, and a parametric function, such as a fractional power or the Mittag-Leffler function. In the abstract framework of Stieltjes and complete Bernstein functions, to which the functions we are interested in belong to, we apply a rational Krylov method and prove uniform convergence when using poles based on Zolotarëv's minimal deviation problem. The latter are particularly suited for fractional diffusion as they allow for an efficient query of the map and do not degenerate as the fractional parameters approach zero. We also present a variety of both novel and existing pole selection strategies for which we develop a computable error certificate. Our numerical experiments comprise a detailed parameter study of space-time fractional diffusion problems and compare the performance of the poles with the ones predicted by our certificate. 相似文献
994.
I consider two cases where QCD string is described by an effective theory of long strings: the static potential and meson scattering amplitudes in the Regge regime. I show how they can be solved in the mean-field approximation, justified by the large number of space–time dimensions, and argue that it turns out to be exact. I compare contributions from QCD string and perturbative QCD and discuss experimental consequences for the scattering amplitudes. 相似文献
995.
996.
997.
The moving least-square approximation is discussed first. Sometimes the method can form an ill-conditioned equation system,
and thus the solution cannot be obtained correctly. A Hilbert space is presented on which an orthogonal function system mixed
a weight function is defined. Next the improved moving least-square approximation is discussed in detail. The improved method
has higher computational efficiency and precision than the old method, and cannot form an ill-conditioned equation system.
A boundary element-free method (BEFM) for elastodynamics problems is presented by combining the boundary integral equation
method for elastodynamics and the improved moving least-square approximation. The boundary element-free method is a meshless
method of boundary integral equation and is a direct numerical method compared with others, in which the basic unknowns are
the real solutions of the nodal variables and the boundary conditions can be applied easily. The boundary element-free method
has a higher computational efficiency and precision. In addition, the numerical procedure of the boundary element-free method
for elastodynamics problems is presented in this paper. Finally, some numerical examples are given. 相似文献
998.
999.
本文将移动粒子半隐式法(MPS)的基本算法由二维扩展至三维。将圆柱坐标系引入到初场粒子的布置中,避免了在笛卡儿坐标系下处理不规则形状(如斜边或曲边)问题时粒子初场布置困难和精确度较低的问题,改善了对计算边界条件表达的精确性。引入移动边界模型,对直叶片搅拌器的内部流动进行了三维数值模拟。还提出了一种新的初始粒子布置简易方法,明显简化粒子初始布置时的复杂程度,提高了对三维复杂几何形状问题的可操作性。 相似文献
1000.
从双目遥感凝视系统的视场重叠区进入系统的信息量大于通过非重叠区的信息量,根据这一特征建立了一种新的运动小目标双目并行快速实时自动检测算法。首先采用差分向量无穷范数算法对原始图像序列做预处理,去掉大量低频噪音和背景,然后采用光流场法对运动小目标进行分割,最后用所提出的空间时间并行快速判定算法对分割的可疑运动小目标进行判定。实验结果表明:由于识别判定算法的空间时间是并行处理的,所以识别判定的平均速度比单目视觉系统提高了50%;在图像信噪比不小于5dB的情况下,准确判定识别的概率为97%。 相似文献