Guaranteed affine lower bound functions for multivariate polynomials

Rigorous Affine Lower Bound Functions for Multivariate Polynomials and Their Use in Global Optimisation

This paper addresses the problem of finding tight affine lower bound functions for multivariate polynomials, which may be employed when global optimisation problems involving polynomials are solved with a branch and bound method. These bound functions are constructed by using the expansion of the given polynomial into Bernstein polynomials. The coefficients of this expansion over a given box yi...

Lower Bound Functions for Polynomials

Lower bounds for multivariate approximation by affine-invariant dictionaries

The problem of approximating locally smooth multivariate functions by linear combinations of elements from an affine-invariant redundant dictionary is considered. Augmenting recent upper bound results for approximation, we establish lower bounds on the performance of such schemes. The lower bounds are tight to within a logarithmic factor in the number of elements used in the approximation. Usin...

A lower bound for the root separation of polynomials

An experimental study by Collins (JSC; 2001) suggested the conjecture that the minimum separation of real zeros of irreducible integer polynomials is about the square root of Mahler’s bound for general integer polynomials. We prove that a power of about two thirds of the Mahler bound is already a lower bound for the minimum root separation of all integer polynomials.

Guaranteed lower bounds for eigenvalues

This paper introduces fully computable two-sided bounds on the eigenvalues of the Laplace operator on arbitrarily coarse meshes based on some approximation of the corresponding eigenfunction in the nonconforming Crouzeix-Raviart finite element space plus some postprocessing. The efficiency of the guaranteed error bounds involves the global mesh-size and is proven for the large class of graded m...