4,425 research outputs found
Low-rank updates and a divide-and-conquer method for linear matrix equations
Linear matrix equations, such as the Sylvester and Lyapunov equations, play
an important role in various applications, including the stability analysis and
dimensionality reduction of linear dynamical control systems and the solution
of partial differential equations. In this work, we present and analyze a new
algorithm, based on tensorized Krylov subspaces, for quickly updating the
solution of such a matrix equation when its coefficients undergo low-rank
changes. We demonstrate how our algorithm can be utilized to accelerate the
Newton method for solving continuous-time algebraic Riccati equations. Our
algorithm also forms the basis of a new divide-and-conquer approach for linear
matrix equations with coefficients that feature hierarchical low-rank
structure, such as HODLR, HSS, and banded matrices. Numerical experiments
demonstrate the advantages of divide-and-conquer over existing approaches, in
terms of computational time and memory consumption
A fast semi-direct least squares algorithm for hierarchically block separable matrices
We present a fast algorithm for linear least squares problems governed by
hierarchically block separable (HBS) matrices. Such matrices are generally
dense but data-sparse and can describe many important operators including those
derived from asymptotically smooth radial kernels that are not too oscillatory.
The algorithm is based on a recursive skeletonization procedure that exposes
this sparsity and solves the dense least squares problem as a larger,
equality-constrained, sparse one. It relies on a sparse QR factorization
coupled with iterative weighted least squares methods. In essence, our scheme
consists of a direct component, comprised of matrix compression and
factorization, followed by an iterative component to enforce certain equality
constraints. At most two iterations are typically required for problems that
are not too ill-conditioned. For an HBS matrix with
having bounded off-diagonal block rank, the algorithm has optimal complexity. If the rank increases with the spatial dimension as is
common for operators that are singular at the origin, then this becomes
in 1D, in 2D, and
in 3D. We illustrate the performance of the method on
both over- and underdetermined systems in a variety of settings, with an
emphasis on radial basis function approximation and efficient updating and
downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App
Self-consistent theory of large amplitude collective motion: Applications to approximate quantization of non-separable systems and to nuclear physics
The goal of the present account is to review our efforts to obtain and apply
a ``collective'' Hamiltonian for a few, approximately decoupled, adiabatic
degrees of freedom, starting from a Hamiltonian system with more or many more
degrees of freedom. The approach is based on an analysis of the classical limit
of quantum-mechanical problems. Initially, we study the classical problem
within the framework of Hamiltonian dynamics and derive a fully self-consistent
theory of large amplitude collective motion with small velocities. We derive a
measure for the quality of decoupling of the collective degree of freedom. We
show for several simple examples, where the classical limit is obvious, that
when decoupling is good, a quantization of the collective Hamiltonian leads to
accurate descriptions of the low energy properties of the systems studied. In
nuclear physics problems we construct the classical Hamiltonian by means of
time-dependent mean-field theory, and we transcribe our formalism to this case.
We report studies of a model for monopole vibrations, of Si with a
realistic interaction, several qualitative models of heavier nuclei, and
preliminary results for a more realistic approach to heavy nuclei. Other topics
included are a nuclear Born-Oppenheimer approximation for an {\em ab initio}
quantum theory and a theory of the transfer of energy between collective and
non-collective degrees of freedom when the decoupling is not exact. The
explicit account is based on the work of the authors, but a thorough survey of
other work is included.Comment: 203 pages, many figure
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