13,259 research outputs found
Understanding Search Trees via Statistical Physics
We study the random m-ary search tree model (where m stands for the number of
branches of a search tree), an important problem for data storage in computer
science, using a variety of statistical physics techniques that allow us to
obtain exact asymptotic results. In particular, we show that the probability
distributions of extreme observables associated with a random search tree such
as the height and the balanced height of a tree have a traveling front
structure. In addition, the variance of the number of nodes needed to store a
data string of a given size N is shown to undergo a striking phase transition
at a critical value of the branching ratio m_c=26. We identify the mechanism of
this phase transition, show that it is generic and occurs in various other
problems as well. New results are obtained when each element of the data string
is a D-dimensional vector. We show that this problem also has a phase
transition at a critical dimension, D_c= \pi/\sin^{-1}(1/\sqrt{8})=8.69363...Comment: 11 pages, 8 .eps figures included. Invited contribution to
STATPHYS-22 held at Bangalore (India) in July 2004. To appear in the
proceedings of STATPHYS-2
Parallel structurally-symmetric sparse matrix-vector products on multi-core processors
We consider the problem of developing an efficient multi-threaded
implementation of the matrix-vector multiplication algorithm for sparse
matrices with structural symmetry. Matrices are stored using the compressed
sparse row-column format (CSRC), designed for profiting from the symmetric
non-zero pattern observed in global finite element matrices. Unlike classical
compressed storage formats, performing the sparse matrix-vector product using
the CSRC requires thread-safe access to the destination vector. To avoid race
conditions, we have implemented two partitioning strategies. In the first one,
each thread allocates an array for storing its contributions, which are later
combined in an accumulation step. We analyze how to perform this accumulation
in four different ways. The second strategy employs a coloring algorithm for
grouping rows that can be concurrently processed by threads. Our results
indicate that, although incurring an increase in the working set size, the
former approach leads to the best performance improvements for most matrices.Comment: 17 pages, 17 figures, reviewed related work section, fixed typo
A Fast Algorithm for Parabolic PDE-based Inverse Problems Based on Laplace Transforms and Flexible Krylov Solvers
We consider the problem of estimating parameters in large-scale weakly
nonlinear inverse problems for which the underlying governing equations is a
linear, time-dependent, parabolic partial differential equation. A major
challenge in solving these inverse problems using Newton-type methods is the
computational cost associated with solving the forward problem and with
repeated construction of the Jacobian, which represents the sensitivity of the
measurements to the unknown parameters. Forming the Jacobian can be
prohibitively expensive because it requires repeated solutions of the forward
and adjoint time-dependent parabolic partial differential equations
corresponding to multiple sources and receivers. We propose an efficient method
based on a Laplace transform-based exponential time integrator combined with a
flexible Krylov subspace approach to solve the resulting shifted systems of
equations efficiently. Our proposed solver speeds up the computation of the
forward and adjoint problems, thus yielding significant speedup in total
inversion time. We consider an application from Transient Hydraulic Tomography
(THT), which is an imaging technique to estimate hydraulic parameters related
to the subsurface from pressure measurements obtained by a series of pumping
tests. The algorithms discussed are applied to a synthetic example taken from
THT to demonstrate the resulting computational gains of this proposed method
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