4,869 research outputs found
DOLFIN: Automated Finite Element Computing
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness with efficient computation. Finite element variational forms may be expressed in near mathematical notation, from which low-level code is automatically generated, compiled and seamlessly integrated with efficient implementations of
computational meshes and high-performance linear algebra. Easy-to-use object-oriented interfaces to the library are provided in the form of a C++ library and a Python module. This paper discusses the mathematical abstractions and methods used in the design of the library and its implementation. A number of examples are presented to demonstrate the use of the library in application code
Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem
Alternating current optimal power flow (AC OPF) is one of the most
fundamental optimization problems in electrical power systems. It can be
formulated as a semidefinite program (SDP) with rank constraints. Solving AC
OPF, that is, obtaining near optimal primal solutions as well as high quality
dual bounds for this non-convex program, presents a major computational
challenge to today's power industry for the real-time operation of large-scale
power grids. In this paper, we propose a new technique for reformulation of the
rank constraints using both principal and non-principal 2-by-2 minors of the
involved Hermitian matrix variable and characterize all such minors into three
types. We show the equivalence of these minor constraints to the physical
constraints of voltage angle differences summing to zero over three- and
four-cycles in the power network. We study second-order conic programming
(SOCP) relaxations of this minor reformulation and propose strong cutting
planes, convex envelopes, and bound tightening techniques to strengthen the
resulting SOCP relaxations. We then propose an SOCP-based spatial
branch-and-cut method to obtain the global optimum of AC OPF. Extensive
computational experiments show that the proposed algorithm significantly
outperforms the state-of-the-art SDP-based OPF solver and on a simple personal
computer is able to obtain on average a 0.71% optimality gap in no more than
720 seconds for the most challenging power system instances in the literature
Solving rank-constrained semidefinite programs in exact arithmetic
We consider the problem of minimizing a linear function over an affine
section of the cone of positive semidefinite matrices, with the additional
constraint that the feasible matrix has prescribed rank. When the rank
constraint is active, this is a non-convex optimization problem, otherwise it
is a semidefinite program. Both find numerous applications especially in
systems control theory and combinatorial optimization, but even in more general
contexts such as polynomial optimization or real algebra. While numerical
algorithms exist for solving this problem, such as interior-point or
Newton-like algorithms, in this paper we propose an approach based on symbolic
computation. We design an exact algorithm for solving rank-constrained
semidefinite programs, whose complexity is essentially quadratic on natural
degree bounds associated to the given optimization problem: for subfamilies of
the problem where the size of the feasible matrix is fixed, the complexity is
polynomial in the number of variables. The algorithm works under assumptions on
the input data: we prove that these assumptions are generically satisfied. We
also implement it in Maple and discuss practical experiments.Comment: Published at ISSAC 2016. Extended version submitted to the Journal of
Symbolic Computatio
Block stochastic gradient iteration for convex and nonconvex optimization
The stochastic gradient (SG) method can minimize an objective function
composed of a large number of differentiable functions, or solve a stochastic
optimization problem, to a moderate accuracy. The block coordinate
descent/update (BCD) method, on the other hand, handles problems with multiple
blocks of variables by updating them one at a time; when the blocks of
variables are easier to update individually than together, BCD has a lower
per-iteration cost. This paper introduces a method that combines the features
of SG and BCD for problems with many components in the objective and with
multiple (blocks of) variables.
Specifically, a block stochastic gradient (BSG) method is proposed for
solving both convex and nonconvex programs. At each iteration, BSG approximates
the gradient of the differentiable part of the objective by randomly sampling a
small set of data or sampling a few functions from the sum term in the
objective, and then, using those samples, it updates all the blocks of
variables in either a deterministic or a randomly shuffled order. Its
convergence for both convex and nonconvex cases are established in different
senses. In the convex case, the proposed method has the same order of
convergence rate as the SG method. In the nonconvex case, its convergence is
established in terms of the expected violation of a first-order optimality
condition. The proposed method was numerically tested on problems including
stochastic least squares and logistic regression, which are convex, as well as
low-rank tensor recovery and bilinear logistic regression, which are nonconvex
Self-Calibration and Biconvex Compressive Sensing
The design of high-precision sensing devises becomes ever more difficult and
expensive. At the same time, the need for precise calibration of these devices
(ranging from tiny sensors to space telescopes) manifests itself as a major
roadblock in many scientific and technological endeavors. To achieve optimal
performance of advanced high-performance sensors one must carefully calibrate
them, which is often difficult or even impossible to do in practice. In this
work we bring together three seemingly unrelated concepts, namely
Self-Calibration, Compressive Sensing, and Biconvex Optimization. The idea
behind self-calibration is to equip a hardware device with a smart algorithm
that can compensate automatically for the lack of calibration. We show how
several self-calibration problems can be treated efficiently within the
framework of biconvex compressive sensing via a new method called SparseLift.
More specifically, we consider a linear system of equations y = DAx, where both
x and the diagonal matrix D (which models the calibration error) are unknown.
By "lifting" this biconvex inverse problem we arrive at a convex optimization
problem. By exploiting sparsity in the signal model, we derive explicit
theoretical guarantees under which both x and D can be recovered exactly,
robustly, and numerically efficiently via linear programming. Applications in
array calibration and wireless communications are discussed and numerical
simulations are presented, confirming and complementing our theoretical
analysis
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