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Materials Science vs. Data Science
Published in Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef, Statistical Methods for Materials Science, 2019
Jeffrey P. Simmons, Lawrence F. Drummy, Charles A. Bouman, Marc De Graef
The word “sparsity” is familiar to many computational modelers in the sense of “sparse matrices,” which have many zero entries. This is the same concept as in sparse data, except there are not so many zero entries. Rather, many measurements may be formed from combinations of others. In the proper coordinate system, they would show zeroes as entries, but the computational cost of rotating coordinates this way is prohibitive. For example, when diagonalizing a matrix, i.e., expressing the measurements in terms of linear combinations of basis functions, on the order of N3 operations must be performed, where N is the number of dimensions of the matrix. In imaging, N is of the order of millions of pixel intensities, so sparse methods generally do not seek a diagonal representation, but estimate both the basis functions and their coefficients from the data.
Computation in Practice
Published in Taylor Arnold, Michael Kane, Bryan W. Lewis, A Computational Approach to Statistical Learning, 2019
Taylor Arnold, Michael Kane, Bryan W. Lewis
A matrix is called sparse if most of its entries are zero. Matrices that are not sparse are called dense. The sparsity of a matrix is a number indicating the proportion of elements that are equal to zero. There is no formal threshold for the minimal sparsity value before a matrix can be called sparse. Typically a ‘sparse’ matrix has at least 90% of its terms equal to zero, with most applications having a sparsity of 99% or more. Sparse matrices arise in many statistical learning applications. When constructing a numeric data matrix as indicator variables from a categorical input, as seen in Section 2.1, only one variable in each row of X is allowed to be non‐zero. Datasets representing counts are often sampled from zero‐inflated distributions; term frequency matrices, for instance, often have less than 0.01% of their entries equal to a non‐zero value. The kernel and similarity matrices we saw in Chapter 9 can become sparse in two ways. Some metrics such as symmetric nearest neighbors lead to naturally sparse similarity scores. Others, such as the radial kernel, lead to many very small entries, which can be hard thresholded to zero with minimal loss of performance.
Modeling of Flow Problems
Published in Krishnan Murugesan, Modeling and Simulation in Thermal and Fluids Engineering, 2023
Conjugate gradient methods are highly attractive schemes to solve a large sparse system of equations because the solution scheme depends only on vectors obtained as products of coefficient matrices and vectors [7]. When these products are performed at the element level, a significant saving in computational time and effort can be achieved. Sheu et al. [8] implemented the BICGSTAB iterative solver in an element-by-element format to achieve computational efficiency in a parallel computation of three-dimensional Navier-Stokes equations using the finite element method. Thiagarajan and Aravamuthan [9] proposed a pre-conditioner for the conjugate gradient method along with an element-by-element solution scheme for parallel computation. Phoon [10] developed a generalized Jacobi (diagonal) preconditioning approach to implement the conjugate gradient iterative solver using an element-by-element strategy. The implementation of the element-by-element iterative solution procedure to solve large-scale problems on a personal computer is not straightforward though the scheme has been efficiently exploited in parallel computations [8–10]. The main reason for this restriction is the requirement of huge computer memory to store the large size global matrices. Even a compact vector storage scheme requires a memory space of 1,336,694 to store only the non-zero entries for a 3D flow problem with a mesh of size 313. Hence, the necessity of storing such huge size vectors restricts the use of the conjugate gradient iterative solvers such as the BICG iterative solvers [2] on personal computers.
An adaptive approach for compression format based on bagging algorithm
Published in International Journal of Parallel, Emergent and Distributed Systems, 2023
Cui Huanyu, Han Qilong, Wang Nianbin, Wang Ye
Sparse matrix is a basic matrix type commonly used to solve linear equations in numerical simulation and engineering applications, such as chemical processes [1], heat conduction [2], circuit simulation [3] and Computational Fluid Dynamics (CFD) [4]. Among them, sparse matrix has a large number of zero elements, which will lead to calculation redundancy and storage redundancy in the process of matrix operation, which will reduce the iterative efficiency of numerical simulation and increase the memory pressure. The sparse matrix compression format can effectively reduce the filling of zero elements, to improve the continuity of data distribution, and improve the parallel computing efficiency of SPMV and other related matrix operations on the GPU platform. In addition, the sparse matrix produced in numerical simulation and other fields has a variety of sparse characteristics, so it will also lead to the diversity and complexity of matrix types. How to effectively improve the efficiency of SPMV parallel computing is one of the hot and difficult problems in the field of high performance computing.
Online directed-structural change-point detection: A segment-wise time-varying dynamic Bayesian network approach
Published in IISE Transactions, 2023
In reality, the network structure is unknown and should be estimated from data. A good structure should guarantee a good description of data, hence, a small value of the fitting error over all the time points and nodes should be ensured. Denote with and In general, we consider that the parent set for each node would be small, particularly for large-scale networks. Then the nonzero components of would be very few in number, which indicates that would be a sparse matrix. Therefore, a regularization on the -norm of is imposed as a criterion for a good structure, which can also avoid overfitting.
Patient-specific fluid–structure interaction model of bile flow: comparison between 1-way and 2-way algorithms
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2021
Alex G. Kuchumov, Vasily Vedeneev, Vladimir Samartsev, Aleksandr Khairulin, Oleg Ivanov
For the FSI, the two solvers, ANSYS Mechanical and ANSYS CFX, were coupled and solved iteratively. In the one-way simulations, the full Newton method with 2400 time iterations using a time step of 1 second was used within the fluid solver to capture the gallbladder emptying time. A sparse matrix solver based on Gaussian elimination was used for solving the system. The fluid field is solved until the convergence criteria were reached. The calculated forces at the structure boundaries were then transferred to the structure side. The structure side was calculated until the convergence criterion was reached: a relative change of 10−4 in the norm of all field variables. The solution was finished when the maximum number of time steps was reached.