## Holger Waubke

### Objective:

In the past a FWF project dealing with the basics of Stochastic Transformation Methods was executed at the ARI. Explicitly the Karhunen Loeve Expansion and the Transformation of a polynomial Chaos were applied in the wave number domain. The procedure is based on the assumption of Gaussian distributed variables. This assumption shall be generalized to arbitrary random variables.

### Method:

The assumption of a wave number domain limits the model to a horizontally layered half space. This limitation shall be overcome by Wavelets kernels in the transformation instead of Fourier kernels. The aim is the possibility to calculated one sided statistical distributions for the physical parameters and arbitrary boundaries with the new method.

### Objective:

One of the biggest problems encountered when building numerical models for layers is the lack of exact deterministic material parameters. Therefore, stochastic models should be use. However, these models have the general drawback of overusing computer resources. This project developed a stochastic model with the ability to use a shear modulus in conjunction with a special iteration scheme allowing efficient implementation.

### Method:

With the Karhunen Loeve Expansion (KLE), it is possible to split the stochastic shear modulus, and therefore the whole system, into a deterministic and a stochastic part. These parts can then be transformed into a linear system of equations using finite elements and Chaos Polynomial Decomposition. Combining the KLE and the Fourier Transformation in combination with Plancherel's theorem enables decoupling of the deterministic part into smaller subsystems. An iteration scheme was developed which narrows the application of "costly" routines to only these smaller deterministic subsystems, instead of the whole higher dimensional (up to a dimension of 10,000) system matrix.

### Application:

As concerns about vibrations produced by machinery and traffic have increased in past years, models that can predict vibrations in soil became more important. However, since material parameters for soil layers cannot be measured exactly in practice, it is reasonable to use stochastic models.

### Objective:

Beside the Fast Multipole Method wavelet based approaches are of increasing interest for the fast calculation of large matrices.

### Method:

The first part of the project is the implementation of the wavelet method for the compression of the data. Next step is the investigation whether wavelet and FMM approaches can be used together and whether an additional speed up is possible.

### Application:

The aim of the project is the development of an algorithm that allows for a fast calculation of large matrices. The final aim is the possibilities to handle large acoustic problems numerically.