2,284 research outputs found
Rigorous numerics for NLS: bound states, spectra, and controllability
In this paper it is demonstrated how rigorous numerics may be applied to the
one-dimensional nonlinear Schr\"odinger equation (NLS); specifically, to
determining bound--state solutions and establishing certain spectral properties
of the linearization. Since the results are rigorous, they can be used to
complete a recent analytical proof [6] of the local exact controllability of
NLS.Comment: 30 pages, 2 figure
Formal Proofs for Nonlinear Optimization
We present a formally verified global optimization framework. Given a
semialgebraic or transcendental function and a compact semialgebraic domain
, we use the nonlinear maxplus template approximation algorithm to provide a
certified lower bound of over . This method allows to bound in a modular
way some of the constituents of by suprema of quadratic forms with a well
chosen curvature. Thus, we reduce the initial goal to a hierarchy of
semialgebraic optimization problems, solved by sums of squares relaxations. Our
implementation tool interleaves semialgebraic approximations with sums of
squares witnesses to form certificates. It is interfaced with Coq and thus
benefits from the trusted arithmetic available inside the proof assistant. This
feature is used to produce, from the certificates, both valid underestimators
and lower bounds for each approximated constituent. The application range for
such a tool is widespread; for instance Hales' proof of Kepler's conjecture
yields thousands of multivariate transcendental inequalities. We illustrate the
performance of our formal framework on some of these inequalities as well as on
examples from the global optimization literature.Comment: 24 pages, 2 figures, 3 table
Field Theory on Noncommutative Spacetimes: Quasiplanar Wick Products
We give a definition of admissible counterterms appropriate for massive
quantum field theories on the noncommutative Minkowski space, based on a
suitable notion of locality. We then define products of fields of arbitrary
order, the so-called quasiplanar Wick products, by subtracting only such
admissible counterterms. We derive the analogue of Wick's theorem and comment
on the consequences of using quasiplanar Wick products in the perturbative
expansion.Comment: 22 pages, 2 figures, v2: minor changes, v3: minor changes, reference
adde
Mathematical computer programs: A compilation
Computer programs, routines, and subroutines for aiding engineers, scientists, and mathematicians in direct problem solving are presented. Also included is a group of items that affords the same users greater flexibility in the use of software
Kodiak: An Implementation Framework for Branch and Bound Algorithms
Recursive branch and bound algorithms are often used to refine and isolate solutions to several classes of global optimization problems. A rigorous computation framework for the solution of systems of equations and inequalities involving nonlinear real arithmetic over hyper-rectangular variable and parameter domains is presented. It is derived from a generic branch and bound algorithm that has been formally verified, and utilizes self-validating enclosure methods, namely interval arithmetic and, for polynomials and rational functions, Bernstein expansion. Since bounds computed by these enclosure methods are sound, this approach may be used reliably in software verification tools. Advantage is taken of the partial derivatives of the constraint functions involved in the system, firstly to reduce the branching factor by the use of bisection heuristics and secondly to permit the computation of bifurcation sets for systems of ordinary differential equations. The associated software development, Kodiak, is presented, along with examples of three different branch and bound problem types it implements
Computation of maximal local (un)stable manifold patches by the parameterization method
In this work we develop some automatic procedures for computing high order
polynomial expansions of local (un)stable manifolds for equilibria of
differential equations. Our method incorporates validated truncation error
bounds, and maximizes the size of the image of the polynomial approximation
relative to some specified constraints. More precisely we use that the manifold
computations depend heavily on the scalings of the eigenvectors: indeed we
study the precise effects of these scalings on the estimates which determine
the validated error bounds. This relationship between the eigenvector scalings
and the error estimates plays a central role in our automatic procedures. In
order to illustrate the utility of these methods we present several
applications, including visualization of invariant manifolds in the Lorenz and
FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable
manifolds in a suspension bridge problem. In the present work we treat
explicitly the case where the eigenvalues satisfy a certain non-resonance
condition.Comment: Revised version, typos corrected, references adde
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