8,148 research outputs found
Proximal Gradient Method for Nonsmooth Optimization over the Stiefel Manifold
We consider optimization problems over the Stiefel manifold whose objective
function is the summation of a smooth function and a nonsmooth function.
Existing methods for solving this kind of problems can be classified into three
classes. Algorithms in the first class rely on information of the subgradients
of the objective function and thus tend to converge slowly in practice.
Algorithms in the second class are proximal point algorithms, which involve
subproblems that can be as difficult as the original problem. Algorithms in the
third class are based on operator-splitting techniques, but they usually lack
rigorous convergence guarantees. In this paper, we propose a retraction-based
proximal gradient method for solving this class of problems. We prove that the
proposed method globally converges to a stationary point. Iteration complexity
for obtaining an -stationary solution is also analyzed. Numerical
results on solving sparse PCA and compressed modes problems are reported to
demonstrate the advantages of the proposed method
Non-overlapping domain decomposition methods in structural mechanics
The modern design of industrial structures leads to very complex simulations
characterized by nonlinearities, high heterogeneities, tortuous geometries...
Whatever the modelization may be, such an analysis leads to the solution to a
family of large ill-conditioned linear systems. In this paper we study
strategies to efficiently solve to linear system based on non-overlapping
domain decomposition methods. We present a review of most employed approaches
and their strong connections. We outline their mechanical interpretations as
well as the practical issues when willing to implement and use them. Numerical
properties are illustrated by various assessments from academic to industrial
problems. An hybrid approach, mainly designed for multifield problems, is also
introduced as it provides a general framework of such approaches
Parallelizable Algorithms for Optimization Problems with Orthogonality Constraints
To construct a parallel approach for solving optimization problems with
orthogonality constraints is usually regarded as an extremely difficult
mission, due to the low scalability of the orthonormalization procedure.
However, such demand is particularly huge in some application areas such as
materials computation. In this paper, we propose a proximal linearized
augmented Lagrangian algorithm (PLAM) for solving optimization problems with
orthogonality constraints. Unlike the classical augmented Lagrangian methods,
in our algorithm, the prime variables are updated by minimizing a proximal
linearized approximation of the augmented Lagrangian function, meanwhile the
dual variables are updated by a closed-form expression which holds at any
first-order stationary point. The orthonormalization procedure is only invoked
once at the last step of the above mentioned algorithm if high-precision
feasibility is needed. Consequently, the main parts of the proposed algorithm
can be parallelized naturally. We establish global subsequence convergence,
worst-case complexity and local convergence rate for PLAM under some mild
assumptions. To reduce the sensitivity of the penalty parameter, we put forward
a modification of PLAM, which is called parallelizable column-wise block
minimization of PLAM (PCAL). Numerical experiments in serial illustrate that
the novel updating rule for the Lagrangian multipliers significantly
accelerates the convergence of PLAM and makes it comparable with the existent
feasible solvers for optimization problems with orthogonality constraints, and
the performance of PCAL does not highly rely on the choice of the penalty
parameter. Numerical experiments under parallel environment demonstrate that
PCAL attains good performance and high scalability in solving discretized
Kohn-Sham total energy minimization problems
A Projected Preconditioned Conjugate Gradient Algorithm for Computing Many Extreme Eigenpairs of a Hermitian Matrix
We present an iterative algorithm for computing an invariant subspace
associated with the algebraically smallest eigenvalues of a large sparse or
structured Hermitian matrix A. We are interested in the case in which the
dimension of the invariant subspace is large (e.g., over several hundreds or
thousands) even though it may still be small relative to the dimension of A.
These problems arise from, for example, density functional theory based
electronic structure calculations for complex materials. The key feature of our
algorithm is that it performs fewer Rayleigh--Ritz calculations compared to
existing algorithms such as the locally optimal precondition conjugate gradient
or the Davidson algorithm. It is a block algorithm, hence can take advantage of
efficient BLAS3 operations and be implemented with multiple levels of
concurrency. We discuss a number of practical issues that must be addressed in
order to implement the algorithm efficiently on a high performance computer
Compressed Modes for Variational Problems in Mathematics and Physics
This paper describes a general formalism for obtaining localized solutions to
a class of problems in mathematical physics, which can be recast as variational
optimization problems. This class includes the important cases of
Schr\"odinger's equation in quantum mechanics and electromagnetic equations for
light propagation in photonic crystals. These ideas can also be applied to
develop a spatially localized basis that spans the eigenspace of a differential
operator, for instance, the Laplace operator, generalizing the concept of plane
waves to an orthogonal real-space basis with multi-resolution capabilities.Comment: 18 page
SOFAR: large-scale association network learning
Many modern big data applications feature large scale in both numbers of
responses and predictors. Better statistical efficiency and scientific insights
can be enabled by understanding the large-scale response-predictor association
network structures via layers of sparse latent factors ranked by importance.
Yet sparsity and orthogonality have been two largely incompatible goals. To
accommodate both features, in this paper we suggest the method of sparse
orthogonal factor regression (SOFAR) via the sparse singular value
decomposition with orthogonality constrained optimization to learn the
underlying association networks, with broad applications to both unsupervised
and supervised learning tasks such as biclustering with sparse singular value
decomposition, sparse principal component analysis, sparse factor analysis, and
spare vector autoregression analysis. Exploiting the framework of
convexity-assisted nonconvex optimization, we derive nonasymptotic error bounds
for the suggested procedure characterizing the theoretical advantages. The
statistical guarantees are powered by an efficient SOFAR algorithm with
convergence property. Both computational and theoretical advantages of our
procedure are demonstrated with several simulation and real data examples
Primal-Dual Optimization Algorithms over Riemannian Manifolds: an Iteration Complexity Analysis
In this paper we study nonconvex and nonsmooth multi-block optimization over
Riemannian manifolds with coupled linear constraints. Such optimization
problems naturally arise from machine learning, statistical learning,
compressive sensing, image processing, and tensor PCA, among others. We develop
an ADMM-like primal-dual approach based on decoupled solvable subroutines such
as linearized proximal mappings. First, we introduce the optimality conditions
for the afore-mentioned optimization models. Then, the notion of
-stationary solutions is introduced as a result. The main part of the
paper is to show that the proposed algorithms enjoy an iteration complexity of
to reach an -stationary solution. For prohibitively
large-size tensor or machine learning models, we present a sampling-based
stochastic algorithm with the same iteration complexity bound in expectation.
In case the subproblems are not analytically solvable, a feasible curvilinear
line-search variant of the algorithm based on retraction operators is proposed.
Finally, we show specifically how the algorithms can be implemented to solve a
variety of practical problems such as the NP-hard maximum bisection problem,
the regularized sparse tensor principal component analysis and the
community detection problem. Our preliminary numerical results show great
potentials of the proposed methods
Towards a Complexity-through-Realisability Theory
We explain how recent developments in the fields of realisability models for
linear logic -- or geometry of interaction -- and implicit computational
complexity can lead to a new approach of implicit computational complexity.
This semantic-based approach should apply uniformly to various computational
paradigms, and enable the use of new mathematical methods and tools to attack
problem in computational complexity. This paper provides the background,
motivations and perspectives of this complexity-through-realisability theory to
be developed, and illustrates it with recent results
Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation
We develop an efficient and reliable adaptive finite element method (AFEM)
for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the
regularization technique of Chen, Holst, and Xu; this technique made possible
the first a priori pointwise estimates and the first complete solution and
approximation theory for the Poisson-Boltzmann equation. It also made possible
the first provably convergent discretization of the PBE, and allowed for the
development of a provably convergent AFEM for the PBE. However, in practice the
regularization turns out to be numerically ill-conditioned. In this article, we
examine a second regularization, and establish a number of basic results to
ensure that the new approach produces the same mathematical advantages of the
original regularization, without the ill-conditioning property. We then design
an AFEM scheme based on the new regularized problem, and show that the
resulting AFEM scheme is accurate and reliable, by proving a contraction result
for the error. This result, which is one of the first results of this type for
nonlinear elliptic problems, is based on using continuous and discrete a priori
pointwise estimates to establish quasi-orthogonality. To provide a high-quality
geometric model as input to the AFEM algorithm, we also describe a class of
feature-preserving adaptive mesh generation algorithms designed specifically
for constructing meshes of biomolecular structures, based on the intrinsic
local structure tensor of the molecular surface. The stability advantages of
the new regularization are demonstrated using an FETK-based implementation,
through comparisons with the original regularization approach for a model
problem. The convergence and accuracy of the overall AFEM algorithm is also
illustrated by numerical approximation of electrostatic solvation energy for an
insulin protein.Comment: 28 pages, 7 figure
An optimization approach for dynamical Tucker tensor approximation
An optimization-based approach for the Tucker tensor approximation of
parameter-dependent data tensors and solutions of tensor differential equations
with low Tucker rank is presented. The problem of updating the tensor
decomposition is reformulated as fitting problem subject to the tangent space
without relying on an orthogonality gauge condition. A discrete Euler scheme is
established in an alternating least squares framework, where the quadratic
subproblems reduce to trace optimization problems, that are shown to be
explicitly solvable and accessible using SVD of small size. In the presence of
small singular values, instability for larger ranks is reduced, since the
method does not need the (pseudo) inverse of matricizations of the core tensor.
Regularization of Tikhonov type can be used to compensate for the lack of
uniqueness in the tangent space. The method is validated numerically and shown
to be stable also for larger ranks in the case of small singular values of the
core unfoldings. Higher order explicit integrators of Runge-Kutta type can be
composed.Comment: 18 pages, 10 figure
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