Aproximace maticemi malé hodnosti a jejich aplikace

Outrata, Michal

Consider the problem of solving a large system of linear algebraic equations, using the Krylov subspace methods. In order to find the solution efficiently, the system often needs to be preconditioned, i.e., transformed prior to the iterative scheme. A feature of the system that often enables fast solution with efficient preconditioners is the structural sparsity of the corresponding matrix. A recent development brought another and a slightly different phe- nomenon called the data sparsity. In contrast to the classical (structural) sparsity, the data sparsity refers to an uneven distribution of extractable information inside the matrix. In practice, the data sparsity of a matrix ty- pically means that its blocks can be successfully approximated by matrices of low rank. Naturally, this may significantly change the character of the numerical computations involving the matrix. The thesis focuses on finding ways to construct Cholesky-based preconditioners for the conjugate gradi- ent method to solve systems with symmetric and positive definite matrices, exploiting a combination of the data and structural sparsity. Methods to exploit the data sparsity are evolving very fast, influencing not only iterative solvers but direct solvers as well. Hierarchical schemes based on the data sparsity concepts can be derived...

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