Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Sparse adaptive approximation of high dimensional parametric initial value problems

by M. Hansen and Ch. Schwab

(Report number 2011-64)

Abstract
We consider nonlinear systems of ordinary differential equations (ODEs) on a state space \(S\). We consider the general setting when \(S\) is a Banach space over \(IR\) or \(IC\). We assume the right hand side depends affinely linear on a vector \(y = (y_j)_{j\ge 1}\) of possibly countably many parameters, normalized such that \(|y_i | \le 1\). Under suitable analyticity assumptions on the ODEs, we prove that the parametric solution \({X (t; y) : 0 \le t \le T } \subset S\) of the corresponding IVP depends holomorphically on the parameter vector \(y\), as a mapping from the infinite-dimensional parameter domain \(U = (-1,1)^IN\) into a suitable function space on \([0, T] \times S\). Such affine parameter dependence of the ODE arises, among others, in mass action models in computational biology (see, e.g. [18]) and in stochiometry with uncertain reaction rate constants. Using our analytic regularity result, we prove summability theorems for coefficient sequences of generalized polynomial chaos (gpc) expansions of the parametric solutions \({X (\cdot; y)}_{y\in U}\) with respect to tensor product orthogonal polynomial bases of \(L^2(U)\). We give sufficient conditions on the ODEs for \(N\) -term truncations of these expansions to converge on the entire parameter space with efficiency (i.e. accuracy versus complexity) being independent of the number of parameters viz. the dimension of the parameter space \(U\).

Keywords: Ordinary differential equations, initial value problem, parametric dependence, analyticity in infinite dimensional spaces, Taylor series, N-term approximation.

@Techreport{HS11_114,
  author = {M. Hansen and Ch. Schwab},
  title = {Sparse adaptive approximation of high dimensional parametric initial value problems },
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-64},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-64.pdf },
  year = {2011}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).