1,837 research outputs found
Accurate Solutions of Polynomial Eigenvalue Problems
Quadratic eigenvalue problems (QEP) and more generally polynomial eigenvalue
problems (PEP) are among the most common types of nonlinear eigenvalue
problems. Both problems, especially the QEP, have extensive applications. A
typical approach to solve QEP and PEP is to use a linearization method to
reformulate the problem as a higher dimensional linear eigenvalue problem. In
this article, we use homotopy continuation to solve these nonlinear eigenvalue
problems without passing to higher dimensions. Our main contribution is to show
that our method produces substantially more accurate results, and finds all
eigenvalues with a certificate of correctness via Smale's -theory. To
explain the superior accuracy, we show that the nonlinear eigenvalue problem we
solve is better conditioned than its reformulated linear eigenvalue problem,
and our homotopy continuation algorithm is more stable than QZ algorithm -
theoretical findings that are borne out by our numerical experiments. Our
studies provide yet another illustration of the dictum in numerical analysis
that, for reasons of conditioning and stability, it is sometimes better to
solve a nonlinear problem directly even when it could be transformed into a
linear problem with the same solution mathematically
A Blackbox Polynomial System Solver on Parallel Shared Memory Computers
A numerical irreducible decomposition for a polynomial system provides
representations for the irreducible factors of all positive dimensional
solution sets of the system, separated from its isolated solutions. Homotopy
continuation methods are applied to compute a numerical irreducible
decomposition. Load balancing and pipelining are techniques in a parallel
implementation on a computer with multicore processors. The application of the
parallel algorithms is illustrated on solving the cyclic -roots problems, in
particular for , and~12.Comment: Accepted for publication in the proceedings of CASC 201
Paramotopy: Parameter homotopies in parallel
Numerical algebraic geometry provides a number of efficient tools for
approximating the solutions of polynomial systems. One such tool is the
parameter homotopy, which can be an extremely efficient method to solve
numerous polynomial systems that differ only in coefficients, not monomials.
This technique is frequently used for solving a parameterized family of
polynomial systems at multiple parameter values. Parameter homotopies have
recently been useful in several areas of application and have been implemented
in at least two software packages. This article describes Paramotopy, a new,
parallel, optimized implementation of this technique, making use of the Bertini
software package. The novel features of this implementation, not available
elsewhere, include allowing for the simultaneous solutions of arbitrary
polynomial systems in a parameterized family on an automatically generated (or
manually provided) mesh in the parameter space of coefficients, front ends and
back ends that are easily specialized to particular classes of problems, and
adaptive techniques for solving polynomial systems near singular points in the
parameter space. This last feature automates and simplifies a task that is
important but often misunderstood by non-experts.Comment: Long version of ICMS extended abstrac
Accelerating Polynomial Homotopy Continuation on a Graphics Processing Unit with Double Double and Quad Double Arithmetic
Numerical continuation methods track a solution path defined by a homotopy.
The systems we consider are defined by polynomials in several variables with
complex coefficients. For larger dimensions and degrees, the numerical
conditioning worsens and hardware double precision becomes often insufficient
to reach the end of the solution path. With double double and quad double
arithmetic, we can solve larger problems that we could not solve with hardware
double arithmetic, but at a higher computational cost. This cost overhead can
be compensated by acceleration on a Graphics Processing Unit (GPU). We describe
our implementation and report on computational results on benchmark polynomial
systems.Comment: Accepted for publication in the Proceedings of the 7th International
Workshop on Parallel Symbolic Computation (PASCO 2015
Unmixing the mixed volume computation
Computing mixed volume of convex polytopes is an important problem in
computational algebraic geometry. This paper establishes sufficient conditions
under which the mixed volume of several convex polytopes exactly equals the
normalized volume of the convex hull of their union. Under these conditions the
problem of computing mixed volume of several polytopes can be transformed into
a volume computation problem for a single polytope in the same dimension. We
demonstrate through problems from real world applications that substantial
reduction in computational costs can be achieved via this transformation in
situations where the convex hull of the union of the polytopes has less complex
geometry than the original polytopes. We also discuss the important
implications of this result in the polyhedral homotopy method for solving
polynomial systems
Polynomial Homotopies for Dense, Sparse and Determinantal Systems
Numerical homotopy continuation methods for three classes of polynomial
systems are presented. For a generic instance of the class, every path leads to
a solution and the homotopy is optimal. The counting of the roots mirrors the
resolution of a generic system that is used to start up the deformations.
Software and applications are discussed
Locating Power Flow Solution Space Boundaries: A Numerical Polynomial Homotopy Approach
The solution space of any set of power flow equations may contain different
number of real-valued solutions. The boundaries that separate these regions are
referred to as power flow solution space boundaries. Knowledge of these
boundaries is important as they provide a measure for voltage stability.
Traditionally, continuation based methods have been employed to compute these
boundaries on the basis of initial guesses for the solution. However, with
rapid growth of renewable energy sources these boundaries will be increasingly
affected by variable parameters such as penetration levels, locations of the
renewable sources, and voltage set-points, making it difficult to generate an
initial guess that can guarantee all feasible solutions for the power flow
problem. In this paper we solve this problem by applying a numerical polynomial
homotopy based continuation method. The proposed method guarantees to find all
solution boundaries within a given parameter space up to a chosen level of
discretization, independent of any initial guess. Power system operators can
use this computational tool conveniently to plan the penetration levels of
renewable sources at different buses. We illustrate the proposed method through
simulations on 3-bus and 10-bus power system examples with renewable
generation.Comment: 9 pages, 5 figure
Open quantum systems are harder to track than open classical systems
For a Markovian open quantum system it is possible, by continuously
monitoring the environment, to know the stochastically evolving pure state of
the system without altering the master equation. In general, even for a system
with a finite Hilbert space dimension , the pure state trajectory will
explore an infinite number of points in Hilbert space, meaning that the
dimension of the classical memory required for the tracking is infinite.
However, Karasik and Wiseman [Phys. Rev. Lett., 106(2):020406, 2011] showed
that tracking of a qubit () is always possible with a bit (), and
gave a heuristic argument implying that a finite should be sufficient for
any , although beyond it would be necessary to have . Our paper
is concerned with rigorously investigating the relationship between and
, the smallest feasible . We confirm the long-standing
conjecture of Karasik and Wiseman that, for generic systems with , , by a computational proof (via Hilbert Nullstellensatz certificates of
infeasibility). That is, beyond , -dimensional open quantum systems are
provably harder to track than -dimensional open classical systems. Moreover,
we develop, and better justify, a new heuristic to guide our expectation of
as a function of , taking into account the number of
Lindblad operators as well as symmetries in the problem. The use of invariant
subspace and Wigner symmetries makes it tractable to conduct a numerical
search, using the method of polynomial homotopy continuation, to find finite
physically realizable ensembles (as they are known) in . The results of
this search support our heuristic. We thus have confidence in the most
interesting feature of our heuristic: in the absence of symmetries, , implying a quadratic gap between the classical and quantum
tracking problems.Comment: 35 pages, 3 figures, Accepted in Quantum Journal, minor change
Numerical Polynomial Homotopy Continuation Method to Locate All The Power Flow Solutions
The manuscript addresses the problem of finding all solutions of power flow
equations or other similar nonlinear system of algebraic equations. This
problem arises naturally in a number of power systems contexts, most
importantly in the context of direct methods for transient stability analysis
and voltage stability assessment. We introduce a novel form of homotopy
continuation method called the numerical polynomial homotopy continuation
(NPHC) method that is mathematically guaranteed to find all the solutions
without ever encountering a bifurcation. The method is based on embedding the
real form of power flow equation in complex space, and tracking the generally
unphysical solutions with complex values of real and imaginary parts of the
voltage. The solutions converge to physical real form in the end of the
homotopy. The so-called -trick mathematically rigorously ensures that
all the paths are well-behaved along the paths, so unlike other continuation
approaches, no special handling of bifurcations is necessary. The method is
\textit{embarrassingly parallelizable} and can be applied to reasonably large
sized systems. We demonstrate the technique by analysis of several standard
test cases up to the 14-bus system size. Finally, we discuss possible
strategies for scaling the method to large size systems, and propose several
applications for transient stability analysis and voltage stability assessment.Comment: 9 pages, 3 figures, submitted to Transactions on Power Systems, 201
Solving Polynomial Systems in the Cloud with Polynomial Homotopy Continuation
Polynomial systems occur in many fields of science and engineering.
Polynomial homotopy continuation methods apply symbolic-numeric algorithms to
solve polynomial systems. We describe the design and implementation of our web
interface and reflect on the application of polynomial homotopy continuation
methods to solve polynomial systems in the cloud. Via the graph isomorphism
problem we organize and classify the polynomial systems we solved. The
classification with the canonical form of a graph identifies newly submitted
systems with systems that have already been solved.Comment: Accepted for publication in the Proceedings of CASC 201
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