Oblio: Design and Performance

Dobrian, Florin; Pothen, Alex

doi:10.1007/11558958_92

Cited by 5 publications

(5 citation statements)

References 9 publications

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“…The precomputed LDL factorizations of the stiffness matrices were computed using OBLIO, a sparse direct solver library [Dobrian and Pothen 2006]. Both the GMRES iterative solver used in solution…”

Section: Methodsmentioning

confidence: 99%

Interactively Cutting and Constraining Vertices in Meshes Using Augmented Matrices

2016

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We present a finite-element solution method that is well suited for interactive simulations of cutting meshes in the regime of linear elastic models. Our approach features fast updates to the solution of the stiffness system of equations to account for real-time changes in mesh connectivity and boundary conditions. Updates are accomplished by augmenting the stiffness matrix to keep it consistent with changes to the underlying model, without refactoring the matrix at each step of cutting. The initial stiffness matrix and its Cholesky factors are used to implicitly form and solve a Schur complement system using an iterative solver. As changes accumulate over many simulation timesteps, the augmented solution method slows down due to the size of the augmented matrix. However, by periodically refactoring the stiffness matrix in a concurrent background process, fresh Cholesky factors that incorporate recent model changes can replace the initial factors. This controls the size of the augmented matrices and provides a way to maintain a fast solution rate as the number of changes to a model grows. We exploit sparsity in the stiffness matrix, the right-hand-side vectors and the solution vectors to compute the solutions fast, and show that the time complexity of the update steps is bounded linearly by the size of the Cholesky factor of the initial matrix. Our complexity analysis and experimental results demonstrate that this approach scales well with problem size. Results for cutting and deformation of 3D linear elastic models are reported for meshes representing the brain, eye, and model problems with element counts up to 167,000; these show the potential of this method for real-time interactivity. An application to limbal incisions for surgical correction of astigmatism, for which linear elastic models and small deformations are sufficient, is included.

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Section: Methodsmentioning

confidence: 99%

Interactively Cutting and Constraining Vertices in Meshes Using Augmented Matrices

2016

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“…The precomputed LDL ⊤ factorizations of the stiffness matrices were computed using OBLIO, a sparse direct solver library 21 . All other basic linear algebra subroutines including matrix‐vector products, dense matrix factorization and solves, as well as the GMRES iterative solver used in the unsymmetric augmented matrix methods and the CG solver used for comparison purposes were from the Intel Math Kernel Library (MKL) 22 .…”

Section: Resultsmentioning

confidence: 99%

AMPS: Real‐time mesh cutting with augmented matrices for surgical simulations

Yeung

Pothen

Crouch

2020

Numerical Linear Algebra App

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SummaryWe present the augmented matrix for principal submatrix update (AMPS) algorithm, a finite element solution method that combines principal submatrix updates and Schur complement techniques, well‐suited for interactive simulations of deformation and cutting of finite element meshes. Our approach features real‐time solutions to the updated stiffness matrix systems to account for interactive changes in mesh connectivity and boundary conditions. Updates are accomplished by an augmented matrix formulation of the stiffness equations to maintain its consistency with changes to the underlying model without refactorization at each timestep. As changes accumulate over multiple simulation timesteps, the augmented solution algorithm enables tens or hundreds of updates per second. Acceleration schemes that exploit sparsity, memoization and parallelization lead to the updates being computed in real time. The complexity analysis and experimental results for this method demonstrate that it scales linearly with the number of nonzeros of the factors of the stiffness matrix. Results for cutting and deformation of three‐dimensional (3D) elastic models are reported for meshes with up to 50 000 nodes, and involve models of surgery for astigmatism and the brain.

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“…Removing a row and a column. To remove the jth row and column from Ax = b, we consider the system (5) A e j e ⊤ j 0…”

Section: 2mentioning

confidence: 99%

“…The precomputed LDL ⊤ factorizations of the admittance matrices were computed using Oblio, a direct solver library for solving sparse symmetric linear systems of equations with data structure support for dynamic pivoting using 1 × 1 and 2 × 2 pivots [5]. Both the GMRES iterative solver used in (19) and the PARDISO solver applied to (2) for comparison purposes were from the Intel Math Kernel Library (MKL) [8].…”

Section: 5mentioning

confidence: 99%

AMPS: An Augmented Matrix Formulation for Principal Submatrix Updates with Application to Power Grids

Yeung¹,

Pothen²,

Halappanavar³

et al. 2017

SIAM J. Sci. Comput.

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We present AMPS, an augmented matrix approach to update the solution to a linear system of equations when the matrix is modified by a few elements within a principal submatrix. This problem arises in the dynamic security analysis of a power grid, where operators need to perform N − k contingency analysis, i.e., determine the state of the system when exactly k links from N fail. Our algorithms augment the matrix to account for the changes in it, and then compute the solution to the augmented system without refactoring the modified matrix. We provide two algorithms, a direct method, and a hybrid direct-iterative method for solving the augmented system. We also exploit the sparsity of the matrices and vectors to accelerate the overall computation. We analyze the time complexity of both algorithms, and show that it is bounded by the number of nonzeros in a subset of the columns of the Cholesky factor that are selected by the nonzeros in the sparse right-handside vector. Our algorithms are compared on three power grids with PARDISO, a parallel direct solver, and CHOLMOD, a direct solver with the ability to modify the Cholesky factors of the matrix. We show that our augmented algorithms outperform PARDISO (by two orders of magnitude), and CHOLMOD (by a factor of up to 5). Further, our algorithms scale better than CHOLMOD as the number of elements updated increases. The solutions are computed with high accuracy. Our algorithms are capable of computing N −k contingency analysis on a 778 thousand bus grid, updating a solution with k = 20 elements in 16 milliseconds on an Intel Xeon processor.

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Oblio: Design and Performance

Cited by 5 publications

References 9 publications

Interactively Cutting and Constraining Vertices in Meshes Using Augmented Matrices

Interactively Cutting and Constraining Vertices in Meshes Using Augmented Matrices

AMPS: Real‐time mesh cutting with augmented matrices for surgical simulations

AMPS: An Augmented Matrix Formulation for Principal Submatrix Updates with Application to Power Grids

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