Approximate Approximations and their Applications
- Schmidt, Gunther
2010 Mathematics Subject Classification
- 65-02 41A30 41A63 45K05 65D32
- Quasi--interpolation, cubature, wavelets, numerical methods, integral equation method
This paper gives a survey of an approximation method which was proposed by V. Maz'ya as underlying procedure for numerical algorithms to solve initial and boundary value problems of mathematical physics. Due to a greater flexibility in the choice of approximating functions it allows efficient approximations of multi-dimensional integral operators often occuring in applied problems. Its application especially in connection with integral equation methods is very promising, which has been proved already for different classes of evolution equations. The survey describes some basic results concerning error estimates for quasi-interpolation and cubature of integral operators with singular kernels as well as a multiscale and wavelet approach to approximate those operators over bounded domains. Finally a general numerical method for solving nonlocal nonlinear evolution equations is presented.
- J. Rossmann, P. Takac, G. Wildenhain (eds): The Mazya Anniversary Collection Vol.1, Operator Theory: Advances and Applications, Vol. 109, Birhaeuser Basel, 1999, pp.75-92