5,112 research outputs found
Sparse approximate inverse preconditioners on high performance GPU platforms
Simulation with models based on partial differential equations often requires the solution of (sequences of) large and sparse algebraic linear systems. In multidimensional domains, preconditioned Krylov iterative solvers are often appropriate for these duties. Therefore, the search for efficient preconditioners for Krylov subspace methods is a crucial theme. Recent developments, especially in computing hardware, have renewed the interest in approximate inverse preconditioners in factorized form, because their application during the solution process can be more efficient. We present here some experiences focused on the approximate inverse preconditioners proposed by Benzi and Tůma from 1996 and the sparsification and inversion proposed by van Duin in 1999. Computational costs, reorderings and implementation issues are considered both on conventional and innovative computing architectures like Graphics Programming Units (GPUs)
Efficient Quantum Pseudorandomness
Randomness is both a useful way to model natural systems and a useful tool
for engineered systems, e.g. in computation, communication and control. Fully
random transformations require exponential time for either classical or quantum
systems, but in many case pseudorandom operations can emulate certain
properties of truly random ones. Indeed in the classical realm there is by now
a well-developed theory of such pseudorandom operations. However the
construction of such objects turns out to be much harder in the quantum case.
Here we show that random quantum circuits are a powerful source of quantum
pseudorandomness. This gives the for the first time a polynomialtime
construction of quantum unitary designs, which can replace fully random
operations in most applications, and shows that generic quantum dynamics cannot
be distinguished from truly random processes. We discuss applications of our
result to quantum information science, cryptography and to understanding
self-equilibration of closed quantum dynamics.Comment: 6 pages, 1 figure. Short version of http://arxiv.org/abs/1208.069
Robust Dropping Criteria for F-norm Minimization Based Sparse Approximate Inverse Preconditioning
Dropping tolerance criteria play a central role in Sparse Approximate Inverse
preconditioning. Such criteria have received, however, little attention and
have been treated heuristically in the following manner: If the size of an
entry is below some empirically small positive quantity, then it is set to
zero. The meaning of "small" is vague and has not been considered rigorously.
It has not been clear how dropping tolerances affect the quality and
effectiveness of a preconditioner . In this paper, we focus on the adaptive
Power Sparse Approximate Inverse algorithm and establish a mathematical theory
on robust selection criteria for dropping tolerances. Using the theory, we
derive an adaptive dropping criterion that is used to drop entries of small
magnitude dynamically during the setup process of . The proposed criterion
enables us to make both as sparse as possible as well as to be of
comparable quality to the potentially denser matrix which is obtained without
dropping. As a byproduct, the theory applies to static F-norm minimization
based preconditioning procedures, and a similar dropping criterion is given
that can be used to sparsify a matrix after it has been computed by a static
sparse approximate inverse procedure. In contrast to the adaptive procedure,
dropping in the static procedure does not reduce the setup time of the matrix
but makes the application of the sparser for Krylov iterations cheaper.
Numerical experiments reported confirm the theory and illustrate the robustness
and effectiveness of the dropping criteria.Comment: 27 pages, 2 figure
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