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.

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.