101,682 research outputs found
Numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation
In this article, we are concerned with the analysis on the numerical
reconstruction of the spatial component in the source term of a time-fractional
diffusion equation. This ill-posed problem is solved through a stabilized
nonlinear minimization system by an appropriately selected Tikhonov
regularization. The existence and the stability of the optimization system are
demonstrated. The nonlinear optimization problem is approximated by a fully
discrete scheme, whose convergence is established under a novel result verified
in this study that the -norm of the solution to the discrete forward
system is uniformly bounded. The iterative thresholding algorithm is proposed
to solve the discrete minimization, and several numerical experiments are
presented to show the efficiency and the accuracy of the algorithm.Comment: 17 pages, 2 figures, 2 table
Operator norm convergence of spectral clustering on level sets
Following Hartigan, a cluster is defined as a connected component of the
t-level set of the underlying density, i.e., the set of points for which the
density is greater than t. A clustering algorithm which combines a density
estimate with spectral clustering techniques is proposed. Our algorithm is
composed of two steps. First, a nonparametric density estimate is used to
extract the data points for which the estimated density takes a value greater
than t. Next, the extracted points are clustered based on the eigenvectors of a
graph Laplacian matrix. Under mild assumptions, we prove the almost sure
convergence in operator norm of the empirical graph Laplacian operator
associated with the algorithm. Furthermore, we give the typical behavior of the
representation of the dataset into the feature space, which establishes the
strong consistency of our proposed algorithm
Theoretical analysis of a Stochastic Approximation approach for computing Quasi-Stationary distributions
This paper studies a method, which has been proposed in the Physics
literature by [8, 7, 10], for estimating the quasi-stationary distribution. In
contrast to existing methods in eigenvector estimation, the method eliminates
the need for explicit transition matrix manipulation to extract the principal
eigenvector. Our paper analyzes the algorithm by casting it as a stochastic
approximation algorithm (Robbins-Monro) [23, 16]. In doing so, we prove its
convergence and obtain its rate of convergence. Based on this insight, we also
give an example where the rate of convergence is very slow. This problem can be
alleviated by using an improved version of the algorithm that is given in this
paper. Numerical experiments are described that demonstrate the effectiveness
of this improved method
Diffusion Approximations for Online Principal Component Estimation and Global Convergence
In this paper, we propose to adopt the diffusion approximation tools to study
the dynamics of Oja's iteration which is an online stochastic gradient descent
method for the principal component analysis. Oja's iteration maintains a
running estimate of the true principal component from streaming data and enjoys
less temporal and spatial complexities. We show that the Oja's iteration for
the top eigenvector generates a continuous-state discrete-time Markov chain
over the unit sphere. We characterize the Oja's iteration in three phases using
diffusion approximation and weak convergence tools. Our three-phase analysis
further provides a finite-sample error bound for the running estimate, which
matches the minimax information lower bound for principal component analysis
under the additional assumption of bounded samples.Comment: Appeared in NIPS 201
Robust distributed linear programming
This paper presents a robust, distributed algorithm to solve general linear
programs. The algorithm design builds on the characterization of the solutions
of the linear program as saddle points of a modified Lagrangian function. We
show that the resulting continuous-time saddle-point algorithm is provably
correct but, in general, not distributed because of a global parameter
associated with the nonsmooth exact penalty function employed to encode the
inequality constraints of the linear program. This motivates the design of a
discontinuous saddle-point dynamics that, while enjoying the same convergence
guarantees, is fully distributed and scalable with the dimension of the
solution vector. We also characterize the robustness against disturbances and
link failures of the proposed dynamics. Specifically, we show that it is
integral-input-to-state stable but not input-to-state stable. The latter fact
is a consequence of a more general result, that we also establish, which states
that no algorithmic solution for linear programming is input-to-state stable
when uncertainty in the problem data affects the dynamics as a disturbance. Our
results allow us to establish the resilience of the proposed distributed
dynamics to disturbances of finite variation and recurrently disconnected
communication among the agents. Simulations in an optimal control application
illustrate the results
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