Informal Numerical Analysis Seminar

University of Bath

Friday 11 April 2008

- BICS talk by Jon Trevelyan last Monday (7 April)
- About method for solving the Helmholtz equation
- very inspiring
- got me thinking

**Helmholtz equation****boundary integral**equation formulation- standard finite elements (
**FE**) for function on boundary - boundary element method
**BEM** **enrich**with plane wave expansion**decompose**solution as- in general write solution as linear combination of
**basis**functions **collocation**of the boundary integral equation- gives rise to
**system**for unknowns on the boundary - curve (1D) for 2D, surface (2D) for 3D
- Dirichlet/Neumann
- building
**matrix**, involves solving**integrals** - his approach subdivide integration domain, Gauss along wave crests,
- rule for oscillatory integrals (steepest descent) in direction of wave
- 1D integral for 2D, 2D integral for 3D

- idea: can we choose
**beter basis**functions? - sidestep
- standard finite element (
**FE**) method (e.g. for elliptic equations) - Dirichlet problem
- choose basis functions that satisfy boundary conditions (
**BC**) (but don't solve PDE) - e.g. triangulation and
**hat**functions for nodes in interior - then do Galerkin or collocation inside domain
- homogeneous
**Helmholtz**equation - basis functions that
**solve PDE**? (but don't satisfy boundary conditions) - for homogeneous Helmholtz (Laplace) this is possible
- 2D Hankel function \(H_0^{(1)}(k r)\)
**spherical**wave 3D \(\exp(i k r)/r\)- point source in 3D
**cylindrical**wave, line source in 3D- essentially point source in 2D
**fundamental**solutions**approximate**solution solves**PDE**in domain- need to find
**coefficients**such that boundary conditions (**BC**) satisfied **collocation**- very similar in flavour to boundary element method (
**BEM**) - but
**no integration**necessary

- of course I wasn't the first to come up with this
- some searching
- memory of talk at
**Dundee** - method of
**particular solutions**for PDE eigenvalue problems - paper by
**Trefethen** - work by
**Moler**,**Matlab logo** **Method of Fundamental Solutions**- paper by
**Barnett and Betcke** - authors used
**Matlab** - they make the
**link**with the boundary integral formulation (**BEM**) **analysis**for interior Helmholtz for disc- convergence, conditioning, stability
- conditioning of system, size of coefficients
- solution as analytic function
- e.g. point source outside of domain
- best to choose points "inbetween" singularity and domain
- choice of curves

**interior**or**exterior**problems- automatically satisfies boundary conditions at
**infinity**(**Sommerfeld**radiation conditions) - very easy to
**implement** - works in
**2D**,**3D**, ... **meshless**

**conditioning**- but B&B article shows that this can be alleviated by
- choosing
**source locations**well - also problem for other methods

- How hard is this to implement?
- walk through
**python**code - show that we can very easily get very
**similar**results to**Matlab** - many
**libraries**available - here we use
**numpy**,**scipy**and**matplotlib** - module/file doc string
- importing modules
- numpy, scipy, pylab (matlab-like plotting interface to matplotlib)
- importing specific functions, submodules (, variables, classes)
- defining new functions
- scipy documentation
- tutorials, some reference documentation, api docs, wiki, source
- here api docs were useful
- scipy.special Hankel, derivative of Hankel, Bessel
- unit_circle, circle (only points and gradients needed, meshless)
- fundamental solutions
- source using Bessel function of the second kind (\(Y_0\))
- helmholtz class
- example: unit disc, bc based on Bessel point source outside
- Dirichlet/Neumann
- numerical values for parameters
- plots using matplotlib, very similar to Matlab

- Figures 4a, 4b
- Figures 3a, 3b, 3c
- other examples: discs with Dirichlet and Neumann conditions
- Animations

- talk at Gene Golub memorial day by Godela Scherer
- about work with Gene Golub and Victor Pereyra on optimisation
- Gene Golub QR and SVD for LS
- generalised to separable nonlinear optimisation
- VARPRO/PORT
- paper illustrates that we want to avoid growth of coefficients
- minimise not just ||r||^2 but ||r||^2 + \nu ||c||^2
- standard regularisation
- Golub showed how to this for QR
- more stable for ill conditioned problems
- normal equations kappa^2, QR kappa
- same equations as before, but in addition n equations of the form \(\sqrt{\nu} c_j = 0\)