# Algorithms

## Finite Element-Wavelet Hybrid Algorithm (FEWHA)

The next generation ground-based telescopes rely heavily on adaptive optics (AO) for overcoming the limitations of atmospheric turbulence. Atmospheric tomography, or the reconstruction of the refractive index fluctuations in the atmosphere, is a major mathematical and computational challenge in AO. In this severely ill-posed problem, a stable and fast reconstruction algorithm that can take into account many real-life phenomena of telescope imaging is needed. For the future extremely large telescopes, such as the European Extremely Large Telescope (E-ELT), the algorithm must additionally process a large amount of data in the millisecond time frame, a substantial hurdle for standard reconstruction techniques even with heavy parallelization.

To this end, a new algorithm for AO, called the Finite Element-Wavelet Hybrid Algorithm (FEWHA), with an emphasis on atmospheric tomography was developed by the team. The goal of this algorithm is to be as effective as current techniques while, at the same time, greatly reducing the computing load on the computer that performs the calculations.

The FEWHA is a conjugate gradient based approach in which the Bayesian MAP estimate of the turbulence layers of the atmosphere is discretized using a finite element and a wavelet basis simultaneously. This dual-domain strategy induces a very efficient "matrix-free'' representation of the underlying operators. The method utilizes the locality properties of compactly supported orthonormal wavelets, both in spatial and frequency domains.

The convergence of this iterative scheme is accelerated by an efficient preconditioner and by utilizing multi-scale techniques. Altogether, the computational complexity of the FEWHA scales linearly with the dimension of the problem; the method converges in a few iterations, is highly parallelizable and has a small memory footprint. Moreover, the algorithm is versatile as it can be applied to different AO systems: GLAO, LTAO, MOAO, MCAO and even SCAO.

The performance of this method was verified in terms of quality and speed for several AO systems in the E-ELT configuration using numerical simulations. In terms of quality, the method is comparable to the state of the art algorithms, the MVM and the FrIM, and in some configurations the FEWHA outperforms these benchmark approaches. The speed capabilities of the method were demonstrated on off-the-shelf computing systems. On these hardware platforms the FEWHA was from 5 to 50 times faster than the MVM, depending on the AO system.

In an MCAO system for the E-ELT that uses a mix of LGS and NGS and three DMs, the FEWHA (wavelet method) outperforms other benchmark approaches.

### References

[1] T. Helin and M. Yudytskiy. Wavelet methods in multi-conjugate adaptive optics. Inverse Problems, 29(8):085003, 2013.

[2] M. Yudytskiy, T. Helin, R. Ramlau. A frequency dependent preconditioned wavelet method for atmospheric tomography, in Third AO4ELT Conference - Adaptive Optics for Extremely Large Telescopes, 2013.

[3] M. Yudytskiy, T. Helin, R. Ramlau. Finite element-wavelet hybrid algorithm for atmospheric tomography, J. Opt. Soc. Am. A, 31(3):550-560, Mar 2014.

[4] R. Ramlau, D. Saxenhuber, M. Yudytskiy. Iterative reconstruction methods in atmospheric tomography: FEWHA, Kaczmarz and Gradient-based algorithm, in SPIE 9148, Adaptive Optics Systems IV, 91480Q, 2014.

[5] M. Yudytskiy. Wavelet methods in adaptive optics. PhD thesis, Johannes Kepler University Linz, 2014.