ADAPTIVE METHODS FOR SOLVING OPERATOR EQUATIONS BY USING FRAMES OF SUBSPACES

Year 2017, Volume: 7 Issue: 1, 142 - 153, 01.06.2017

H. Jamali K.h. Shokri Ternoniz

Abstract

In this paper, using a frame of subspaces we transform an operator equation to an equivalent l2-problem. Then, we propose an adaptive algorithm to solve the problem and investigate the optimality and complexity properties of the algorithm.

Keywords

Hilbert space, dual space, frame of subspaces, best N-term approximation, adaptive algorithm.

References

• Casazza,P.G., (2000), The art of frame theory, Taiwaness J. Math., 4, pp.129-201.
• Casazza,P.G. and Kutyniok, G., (2004), Frames of subspaces, Wavelets, Frames and Operator Theory, Contemp. Math. , Amer. Math. Soc, 345, pp.87-113.
• Christensen,O., (2003), An Introduction to Frames and Riesz Bases, Birkhauser, Boston.
• Cohen,A., Dahmen,W., and DeVore,R., (2001), Adaptive wavelet methods for elliptic operator equa- tions: convergence rates, Math. of comp., 70, pp.27-75.
• Cohen,A., Dahmen,W., and DeVore,R., (2002), Adaptive wavelets methods II-beyond the elliptic case, Found. of Comp. Math., 2, pp.203-245.
• Dahlke,S., Dahmen,W., and Urban,K., (2002), Adaptive wavelet methods for saddle point problems- optimal convergence rates, SIAM J. Numer. Anal., 40, pp.1230–1262.
• Dahlke,S., Fornasier,M., and Raasch,T., (2007), Adaptive frame methods for elliptic operator equa- tions, Advances in comp. Math., 27, pp.27-63.
• Dahlke,S., Raasch,T., and Werner,M., (2007), Adaptive frame methods for elliptic operator equations: the steepest descent approach, IMA J. Numer. Anal., 27, pp.717-740.
• DeVore,R., (1998), Nonlinear approximation, Acta Numer., 7, pp. 51-150.