80,938 research outputs found
Designing Competent Mutation Operators via Probabilistic Model Building of Neighborhoods
This paper presents a competent selectomutative genetic algorithm (GA), that
adapts linkage and solves hard problems quickly, reliably, and accurately. A
probabilistic model building process is used to automatically identify key
building blocks (BBs) of the search problem. The mutation operator uses the
probabilistic model of linkage groups to find the best among competing building
blocks. The competent selectomutative GA successfully solves additively
separable problems of bounded difficulty, requiring only subquadratic number of
function evaluations. The results show that for additively separable problems
the probabilistic model building BB-wise mutation scales as O(2^km^{1.5}), and
requires O(k^{0.5}logm) less function evaluations than its selectorecombinative
counterpart, confirming theoretical results reported elsewhere (Sastry &
Goldberg, 2004).Comment: Genetic and Evolutionary Computation Conference (GECCO-2004
Effective linkage learning using low-order statistics and clustering
The adoption of probabilistic models for the best individuals found so far is
a powerful approach for evolutionary computation. Increasingly more complex
models have been used by estimation of distribution algorithms (EDAs), which
often result better effectiveness on finding the global optima for hard
optimization problems. Supervised and unsupervised learning of Bayesian
networks are very effective options, since those models are able to capture
interactions of high order among the variables of a problem. Diversity
preservation, through niching techniques, has also shown to be very important
to allow the identification of the problem structure as much as for keeping
several global optima. Recently, clustering was evaluated as an effective
niching technique for EDAs, but the performance of simpler low-order EDAs was
not shown to be much improved by clustering, except for some simple multimodal
problems. This work proposes and evaluates a combination operator guided by a
measure from information theory which allows a clustered low-order EDA to
effectively solve a comprehensive range of benchmark optimization problems.Comment: Submitted to IEEE Transactions on Evolutionary Computatio
Let's Get Ready to Rumble: Crossover Versus Mutation Head to Head
This paper analyzes the relative advantages between crossover and mutation on
a class of deterministic and stochastic additively separable problems. This
study assumes that the recombination and mutation operators have the knowledge
of the building blocks (BBs) and effectively exchange or search among competing
BBs. Facetwise models of convergence time and population sizing have been used
to determine the scalability of each algorithm. The analysis shows that for
additively separable deterministic problems, the BB-wise mutation is more
efficient than crossover, while the crossover outperforms the mutation on
additively separable problems perturbed with additive Gaussian noise. The
results show that the speed-up of using BB-wise mutation on deterministic
problems is O(k^{0.5}logm), where k is the BB size, and m is the number of BBs.
Likewise, the speed-up of using crossover on stochastic problems with fixed
noise variance is O(mk^{0.5}log m).Comment: Genetic and Evolutionary Computation Conference (GECCO-2004
Efficiency Enhancement of Probabilistic Model Building Genetic Algorithms
This paper presents two different efficiency-enhancement techniques for
probabilistic model building genetic algorithms. The first technique proposes
the use of a mutation operator which performs local search in the sub-solution
neighborhood identified through the probabilistic model. The second technique
proposes building and using an internal probabilistic model of the fitness
along with the probabilistic model of variable interactions. The fitness values
of some offspring are estimated using the probabilistic model, thereby avoiding
computationally expensive function evaluations. The scalability of the
aforementioned techniques are analyzed using facetwise models for convergence
time and population sizing. The speed-up obtained by each of the methods is
predicted and verified with empirical results. The results show that for
additively separable problems the competent mutation operator requires O(k 0.5
logm)--where k is the building-block size, and m is the number of building
blocks--less function evaluations than its selectorecombinative counterpart.
The results also show that the use of an internal probabilistic fitness model
reduces the required number of function evaluations to as low as 1-10% and
yields a speed-up of 2--50.Comment: Optimization by Building and Using Probabilistic Models. Workshop at
the 2004 Genetic and Evolutionary Computation Conferenc
A Fast Propagation Method for the Helmholtz equation
A fast method is proposed for solving the high frequency Helmholtz equation.
The building block of the new fast method is an overlapping source transfer
domain decomposition method for layered medium, which is an extension of the
source transfer domain decomposition method proposed by Chen and Xiang
\cite{Chen2013a,Chen2013b}. The new fast method contains a setup phase and a
solving phase. In the setup phase, the computation domain is decomposed
hierarchically into many subdomains of different levels, and the mapping from
incident traces to field traces on all the subdomains are set up bottom-up. In
the solving phase, first on the bottom level, the local problem on the
subdomains with restricted source is solved, then the wave propagates on the
boundaries of all the subdomains bottom-up, at last the local solutions on all
the subdomains are summed up top-down. The total computation cost of the new
fast method is for 2D problem. Numerical
experiments shows that with the new fast method, Helmholtz equations with half
billion unknowns could be solved efficiently on massively parallel machines.Comment: 20 pages, 11 figure
On the honeycomb conjecture and the Kepler problem
This paper views the honeycomb conjecture and the Kepler problem essentially
as extreme value problems and solves them by partitioning 2-space and 3-space
into building blocks and determining those blocks that have the universal
extreme values that one needs. More precisely, we proved two results. First, we
proved that the regular hexagons are the only 2-dim blocks that have unit area
and the least perimeter (or contain a unit circle and have the least area) that
tile the plane. Secondly, we proved that the rhombic dodecahedron and the
rhombus-isosceles trapezoidal dodecahedron are the only two 3-dim blocks that
contain a unit sphere and have the least volume that can fill 3-space without
either overlapping or leaving gaps. Finally, the Kepler conjecture can also be
proved to be true by introducing the concept of the minimum 2-dim and 3-dim
Kepler building blocks.Comment: 20 pages,14 figures Note: the title has been change
A Random Sample Partition Data Model for Big Data Analysis
Big data sets must be carefully partitioned into statistically similar data
subsets that can be used as representative samples for big data analysis tasks.
In this paper, we propose the random sample partition (RSP) data model to
represent a big data set as a set of non-overlapping data subsets, called RSP
data blocks, where each RSP data block has a probability distribution similar
to the whole big data set. Under this data model, efficient block level
sampling is used to randomly select RSP data blocks, replacing expensive record
level sampling to select sample data from a big distributed data set on a
computing cluster. We show how RSP data blocks can be employed to estimate
statistics of a big data set and build models which are equivalent to those
built from the whole big data set. In this approach, analysis of a big data set
becomes analysis of few RSP data blocks which have been generated in advance on
the computing cluster. Therefore, the new method for data analysis based on RSP
data blocks is scalable to big data.Comment: 9 pages, 7 figure
Randomized Block Coordinate Descent for Online and Stochastic Optimization
Two types of low cost-per-iteration gradient descent methods have been
extensively studied in parallel. One is online or stochastic gradient descent
(OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this
paper, we combine the two types of methods together and propose online
randomized block coordinate descent (ORBCD). At each iteration, ORBCD only
computes the partial gradient of one block coordinate of one mini-batch
samples. ORBCD is well suited for the composite minimization problem where one
function is the average of the losses of a large number of samples and the
other is a simple regularizer defined on high dimensional variables. We show
that the iteration complexity of ORBCD has the same order as OGD or SGD. For
strongly convex functions, by reducing the variance of stochastic gradients, we
show that ORBCD can converge at a geometric rate in expectation, matching the
convergence rate of SGD with variance reduction and RBCD.Comment: The errors in the proof of ORBCD with variance reduction have been
correcte
Purely algebraic domain decomposition methods for the incompressible Navier-Stokes equations
In the context of non overlapping domain decomposition methods, several
algebraic approximations of the Dirichlet-to-Neumann (DtN) map are proposed in
[F. X. Roux, et. al. Algebraic approximation of Dirichlet- to-Neumann maps for
the equations of linear elasticity, Comput. Methods Appl. Mech. Engrg., 195,
2006, 3742-3759]. For the case of non overlapping domains, approximation to the
DtN are analogous to the approximation of the Schur complements in the
incomplete multilevel block factorization. In this work, several original and
purely algebraic (based on graph of the matrix) domain decomposition techniques
are investigated for steady state incompressible Navier-Stokes equation defined
on uniform and stretched grid for low viscosity. Moreover, the methods proposed
are highly parallel during both setup and application phase. Spectral and
numerical analysis of the methods are also presented.Comment: Introduction rewritten, Comparison with state-of-art methods added,
figure on overlapping case added, Complete algorithms added to build and
solve with the preconditioners, Tests with Reynold number 3000 added, some
observations with block jacobi method in analysis sectio
A Massively Parallel Algebraic Multigrid Preconditioner based on Aggregation for Elliptic Problems with Heterogeneous Coefficients
This paper describes a massively parallel algebraic multigrid method based on
non-smoothed aggregation. It is especially suited for solving heterogeneous
elliptic problems as it uses a greedy heuristic algorithm for the aggregation
that detects changes in the coefficients and prevents aggregation across them.
Using decoupled aggregation on each process with data agglomeration onto fewer
processes on the coarse level, it weakly scales well in terms of both total
time to solution and time per iteration to nearly 300,000 cores. Because of
simple piecewise constant interpolation between the levels, its memory
consumption is low and allows solving problems with more than 100,000,000,000
degrees of freedom.Comment: 22 pages, 1 figur
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