Wavelets in Physics

1. Wavelets in Physics Paul Nerenberg

2. Why Wavelets? Fourier analysis is inadequate � it loses time localization for a given frequency. Fourier analysis is uneconomical � still need infinite series even if function is almost flat. Fourier transform is unstable with respect to perturbation � small changes in original function can completely modify the Fourier spectrum. Because they�re fun, no doubt.

3. General Areas of Application Turbulence analysis � looking at fluid flows. Astrophysics � image analysis and processing. Systems of differential or polynomial equations (all physics) � wavelets can make solving them easier (it�s a deal, it�s a steal�). All types of things I probably can�t even conceive of, but if it involves a signal, it can probably be analyzed with wavelets.

4. Multi-dimensional Wavelets Let�s take a brief look at the 2-D CWT�because I�m a physicist, this isn�t mathematically rigorous, but it�s still cool. First, start out with s and s-hat for our 2-D space�

5. CWT in 2-D Only need to satisfy admissibility condition:

6. Choices of Wavelets Isotropic wavelets � good for pointwise analysis; pick a wavelet that is invariant under rotation; e.g. 2-D Mexican hat wavelet. Anisotropic wavelets � detect directional features in an image/signal; support is contained in a convex cone; e.g. 2-D Morlet wavelet and Cauchy wavelets.

7. CWT in 3-D Let�s have some fun in 3-D�

8. Wavelets in Space-Time (don�t ask) Necessary for analysis of time-dependent signals�translations and dilations in space and time independently. The interesting possibility here � relativistic wavelets, e.g. Poincar� wavelets, constrained within light cone, so useful for situations with EM fields.

9. Modeling Maxwell�s Equations and Magnetotellurics � A Specific Example Looking at solar wind-induced EM currents in the Earth�s magnetic core. Core problem � a set of differential equations with boundary conditions (a ?12...different media) that needs to be solved with an iterative process.

10. Beyond our basic equations and boundary conditions� Make some approximations (physicists love them!)�so now we get two simple equations�

11. Looking at secondary component of the electric field� So electric field has two components � we want the secondary one�

12. Yet another formulation� Now we have:

13. Formulating it with Lagrangians�and the final iteration� Almost there�define a space too:

14. The Good News This result does converge�it�s actually worth something.

15. So what does this all mean�and why do we care at all? We took a set of differential problems, performed a domain decomposition, and then using a hybrid finite element method, found an iterative algorithm for solving these problems. This reveals a core nature of wavelets � they can simplify extremely complex models � this is great for physics (or any area of applied science).