TUHH Open Research (TORE)https://tore.tuhh.deTORE captures, stores, indexes, preserves, and distributes digital research material.Thu, 29 Sep 2022 08:49:02 GMT2022-09-29T08:49:02Z5021- Minimizing and maximizing the Euclidean norm of the product of two polynomialshttp://hdl.handle.net/11420/7919Title: Minimizing and maximizing the Euclidean norm of the product of two polynomials
Authors: Bünger, Florian
Abstract: We consider the problem of minimizing or maximizing the quotient, where p = p0 + p1x + ... + pmxm, q = q0 + q1x + ... + qnxn ∈ K[x], K ∈ R, C, are non-zero real or complex polynomials of maximum degree m, n ∈ ℕ respectively and double pipepdouble pipe := (pipep0pipe2 + ... + pipepmpipe2)1/2 is simply the Euclidean norm of the polynomial coefficients. Clearly fm,n is bounded and assumes its maximum and minimum values min fm,n = fm,n(pmin, qmin) and max fm,n = f(pmax, qmax). We prove that minimizers pmin, qmin for K = ¢ and maximizers pmax, qmax for arbitrary K fulfill deg(pmin) = m = deg(pmax), deg(qmin) = n = deg(qmax) and all roots of pmin, qmin, pmax, qmax have modulus one and are simple. For K = ℝ we can only prove the existence of minimizers pmin, qmin of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min fm,n for real polynomials which are slightly better than the known ones and inclusions for max fm,n. © 2010 Springer Science+Business Media, LLC.
Wed, 09 Jun 2010 00:00:00 GMThttp://hdl.handle.net/11420/79192010-06-09T00:00:00Z
- Reconstruction of low-rank aggregation kernels in univariate population balance equationshttp://hdl.handle.net/11420/9572Title: Reconstruction of low-rank aggregation kernels in univariate population balance equations
Authors: Ahrens, Robin; Le Borne, Sabine
Abstract: The dynamics of particle processes can be described by population balance equations which are governed by phenomena including growth, nucleation, breakage and aggregation. Estimating the kinetics of the aggregation phenomena from measured density data constitutes an ill-conditioned inverse problem. In this work, we focus on the aggregation problem and present an approach to estimate the aggregation kernel in discrete, low rank form from given (measured or simulated) data. The low-rank assumption for the kernel allows the application of fast techniques for the evaluation of the aggregation integral (O(nlogn) instead of O(n ) where n denotes the number of unknowns in the discretization) and reduces the dimension of the optimization problem, allowing for efficient and accurate kernel reconstructions. We provide and compare two approaches which we will illustrate in numerical tests. 2
Sun, 02 May 2021 00:00:00 GMThttp://hdl.handle.net/11420/95722021-05-02T00:00:00Z