18,759 research outputs found
Stationary probability density of stochastic search processes in global optimization
A method for the construction of approximate analytical expressions for the
stationary marginal densities of general stochastic search processes is
proposed. By the marginal densities, regions of the search space that with high
probability contain the global optima can be readily defined. The density
estimation procedure involves a controlled number of linear operations, with a
computational cost per iteration that grows linearly with problem size
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods
Many problems in computational science and engineering are simultaneously
characterized by the following challenging issues: uncertainty, nonlinearity,
nonstationarity and high dimensionality. Existing numerical techniques for such
models would typically require considerable computational and storage
resources. This is the case, for instance, for an optimization problem governed
by time-dependent Navier-Stokes equations with uncertain inputs. In particular,
the stochastic Galerkin finite element method often leads to a prohibitively
high dimensional saddle-point system with tensor product structure. In this
paper, we approximate the solution by the low-rank Tensor Train decomposition,
and present a numerically efficient algorithm to solve the optimality equations
directly in the low-rank representation. We show that the solution of the
vorticity minimization problem with a distributed control admits a
representation with ranks that depend modestly on model and discretization
parameters even for high Reynolds numbers. For lower Reynolds numbers this is
also the case for a boundary control. This opens the way for a reduced-order
modeling of the stochastic optimal flow control with a moderate cost at all
stages.Comment: 29 page
Domain Decomposition for Stochastic Optimal Control
This work proposes a method for solving linear stochastic optimal control
(SOC) problems using sum of squares and semidefinite programming. Previous work
had used polynomial optimization to approximate the value function, requiring a
high polynomial degree to capture local phenomena. To improve the scalability
of the method to problems of interest, a domain decomposition scheme is
presented. By using local approximations, lower degree polynomials become
sufficient, and both local and global properties of the value function are
captured. The domain of the problem is split into a non-overlapping partition,
with added constraints ensuring continuity. The Alternating Direction
Method of Multipliers (ADMM) is used to optimize over each domain in parallel
and ensure convergence on the boundaries of the partitions. This results in
improved conditioning of the problem and allows for much larger and more
complex problems to be addressed with improved performance.Comment: 8 pages. Accepted to CDC 201
Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method
This paper addresses optimization problems constrained by partial
differential equations with uncertain coefficients. In particular, the robust
control problem and the average control problem are considered for a tracking
type cost functional with an additional penalty on the variance of the state.
The expressions for the gradient and Hessian corresponding to either problem
contain expected value operators. Due to the large number of uncertainties
considered in our model, we suggest to evaluate these expectations using a
multilevel Monte Carlo (MLMC) method. Under mild assumptions, it is shown that
this results in the gradient and Hessian corresponding to the MLMC estimator of
the original cost functional. Furthermore, we show that the use of certain
correlated samples yields a reduction in the total number of samples required.
Two optimization methods are investigated: the nonlinear conjugate gradient
method and the Newton method. For both, a specific algorithm is provided that
dynamically decides which and how many samples should be taken in each
iteration. The cost of the optimization up to some specified tolerance
is shown to be proportional to the cost of a gradient evaluation with requested
root mean square error . The algorithms are tested on a model elliptic
diffusion problem with lognormal diffusion coefficient. An additional nonlinear
term is also considered.Comment: This work was presented at the IMG 2016 conference (Dec 5 - Dec 9,
2016), at the Copper Mountain conference (Mar 26 - Mar 30, 2017), and at the
FrontUQ conference (Sept 5 - Sept 8, 2017
Optimization techniques in respiratory control system models
One of the most complex physiological systems whose modeling is still an open study is the respiratory control system where different models have been proposed based on the criterion of minimizing the work of breathing (WOB). The aim of this study is twofold: to compare two known models of the respiratory control system which set the breathing pattern based on quantifying the respiratory work; and to assess the influence of using direct-search or evolutionary optimization algorithms on adjustment of model parameters. This study was carried out using experimental data from a group of healthy volunteers under CO2 incremental inhalation, which were used to adjust the model parameters and to evaluate how much the equations of WOB follow a real breathing pattern. This breathing pattern was characterized by the following variables: tidal volume, inspiratory and expiratory time duration and total minute ventilation. Different optimization algorithms were considered to determine the most appropriate model from physiological viewpoint. Algorithms were used for a double optimization: firstly, to minimize the WOB and secondly to adjust model parameters. The performance of optimization algorithms was also evaluated in terms of convergence rate, solution accuracy and precision. Results showed strong differences in the performance of optimization algorithms according to constraints and topological features of the function to be optimized. In breathing pattern optimization, the sequential quadratic programming technique (SQP) showed the best performance and convergence speed when respiratory work was low. In addition, SQP allowed to implement multiple non-linear constraints through mathematical expressions in the easiest way. Regarding parameter adjustment of the model to experimental data, the evolutionary strategy with covariance matrix and adaptation (CMA-ES) provided the best quality solutions with fast convergence and the best accuracy and precision in both models. CMAES reached the best adjustment because of its good performance on noise and multi-peaked fitness functions. Although one of the studied models has been much more commonly used to simulate respiratory response to CO2 inhalation, results showed that an alternative model has a more appropriate cost function to minimize WOB from a physiological viewpoint according to experimental data.Postprint (author's final draft
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