19 research outputs found
A convex polynomial that is not sos-convex
A multivariate polynomial is sos-convex if its Hessian
can be factored as with a possibly nonsquare
polynomial matrix . It is easy to see that sos-convexity is a sufficient
condition for convexity of . Moreover, the problem of deciding
sos-convexity of a polynomial can be cast as the feasibility of a semidefinite
program, which can be solved efficiently. Motivated by this computational
tractability, it has been recently speculated whether sos-convexity is also a
necessary condition for convexity of polynomials. In this paper, we give a
negative answer to this question by presenting an explicit example of a
trivariate homogeneous polynomial of degree eight that is convex but not
sos-convex. Interestingly, our example is found with software using sum of
squares programming techniques and the duality theory of semidefinite
optimization. As a byproduct of our numerical procedure, we obtain a simple
method for searching over a restricted family of nonnegative polynomials that
are not sums of squares.Comment: 15 page
NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems
We show that unless P=NP, there exists no polynomial time (or even
pseudo-polynomial time) algorithm that can decide whether a multivariate
polynomial of degree four (or higher even degree) is globally convex. This
solves a problem that has been open since 1992 when N. Z. Shor asked for the
complexity of deciding convexity for quartic polynomials. We also prove that
deciding strict convexity, strong convexity, quasiconvexity, and
pseudoconvexity of polynomials of even degree four or higher is strongly
NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd
degree polynomials can be decided in polynomial time.Comment: 20 page
On solving quasilinear elliptic equations by variational methods, using piecewise polynomial trial functions
Imperial Users onl
Recovery methods for evolution and nonlinear problems
Functions in finite dimensional spaces are, in general, not smooth enough to be differentiable in the classical sense and “recovered” versions of their first and second derivatives must be sought for certain applications. In this work we make use of recovered derivatives for applications in finite element schemes for two different purposes. We thus split this Thesis into two distinct parts.
In the first part we derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error for fully discrete schemes of the linear heat equation. To our knowledge this is the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique introduced as an aposteriori analog to the elliptic (Ritz) projection.
Our theoretical results are backed up with extensive numerical experimentation aimed at (1) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (2) deriving an adaptive method based on our estimators.
An extra novelty is an implementation of a coarsening error “preindicator”, with a complete implementation guide in ALBERTA (versions 1.0–2.0).
In the second part of this Thesis we propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galërkin type using conforming finite elements and applied directly to the nonvariational(or nondivergence) form of a second order linear elliptic problem. The key tools are an
appropriate concept of the “finite element Hessian” based on a Hessian recovery and a Schur complement approach to solving the resulting linear algebra problem. The method
is illustrated with computational experiments on linear PDEs in nonvariational form.
We then use the nonvariational finite element method to build a numerical method for fully nonlinear elliptic equations. We linearise the problem via Newton’s method resulting in a sequence of nonvariational elliptic problems which are then approximated with the nonvariational finite element method. This method is applicable to general fully nonlinear PDEs who admit a unique solution without constraint.
We also study fully nonlinear PDEs when they are only uniformly elliptic on a certain class of functions. We construct a numerical method for the Monge–Ampère equation
based on using “finite element convexity” as a constraint for the aforementioned nonvariational finite element method. This method is backed up with numerical experimentation
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum