Accurate and Efficient Algorithms for Computing Exponentials of Large Matrices with Applications

Matrix exponential eA is an important linear algebra tool that has a wide range of applications. Its ecient computation is a classical numerical linear algebra problem that is of considerable importance to many elds. The main objective of this proposal is to systematically develop ecient preconditioning techniques for computing eAv for a given vector v and to develop accurate and ecient algorithms to compute some selected entries of eA for a large matrix A. We shall apply our methods to application problems arising in muscle models and large complex networks. At the conclusion of this project, robust softwares that implement our new methods will be made publicly available to serve scientic user community. Intellectual Merit of the Proposed Activity: The Krylov subspace methods are some of the most ecient methods to approximate the operation eAv. In spite of tremendous progresses made over the last two decades, some important numerical techniques such as preconditioning remain to be fully developed. The proposed research will advance theory and algorithms of the Krylov subspace methods for matrix exponentials to a level on par with those for other linear algebra problems. On the other hand, the proposed works on computing some entries of eA accurately for a large matrix A will open a new direction of research where numerical accuracy issues are studied in the setting of iterative methods for large scale problems. Broader Impacts Resulting from the Proposed Activity: The discoveries from this proposed research are expected to impact a wide range of application areas where matrix exponentials are used. A fully developed preconditioning technique would signicantly advance the state of the art in solving large scale initial value problems. Our proposed works on muscle models would lead to a signicantly more powerful simulation program with broadened applicability and increased scalabiltiy, which has the potential to accelerate the rate of biological discovery. Our proposed algorithms for accurate computations of selected entries of eA would remove the numerical accuracy issues that may present a signicant challenge in the large complex network application. The related research activities may also increase awareness on the potential problems of relative accuracy among researchers in network applications. In particular, the proposed software would be an important addition to the set of numerical tools available to the network research community.