![]() By analyzing the time complexity of our solver, we show that the running time of AMPS is O(mρ+: L:), where m is the dimension of the principal submatrix, ρ is the number of nonzeros in a subset of the columns of the Cholesky factor L that are selected by the nonzeros in the sparse right-hand-side vector, and :L: is the number of nonzeros in L. To accelerate the computation, we utilize the precomputed factors of the initial matrix and the solutions to the original system of equations, exploit the sparsity of the matrices and vectors, and apply memoization and parallelization techniques. A number of approaches, including direct methods, which factor the Schur complement and iterative methods, which use Krylov space solvers, are then available to solve the system. ![]() The intactness of the initial matrix allows us to compute its factors only once, and then without refactoring the modified matrix, we solve the augmented system by means of the Schur complement. We characterize the situations when the matrix updates lead to nonsingular systems of equations when the initial matrix is nonsingular. Our approach keeps the initial matrix as a submatrix of the new system of equations and the subsequent updates are accounted for by augmenting the system in blocked form. AMPS, our augmented matrix solver, updates the solution by means of an augmented matrix formulation, in which all changes made to the coefficient matrix are represented by adding rows and columns to the initial matrix. In the power grid problem, the size of the matrix remains unchanged. In the surgical simulations, the dimension of the matrix changes while it is being updated, since the matrix arises from a finite element mesh that is being cut, and we have to remesh around the cut during the update. We observe that the changes in both applications result in the matrix's being modified by a low-rank update within a principal submatrix. ![]() The second is contingency analysis in the power grids, when their operators need to simulate a large number of scenarios to predict what could happen when elements of the grid fail. The first is surgical simulations, where a simulator used to train surgeons needs to provide haptic feedback by updating the system ten to hundred times per second. This problem arises in many computational science and engineering applications, and we consider two of them. In this dissertation, the problem of updating in real time the solution to a linear system of equations when a sequence of small changes is made to the data is considered. Augmented Matrix Solvers for Dynamic System of Equations
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