F Ur Mathematik in Den Naturwissenschaften Leipzig Stable Focal Inverse Source Localization Using Combinatorial Optimization Techniques Combined with Regularization Methods Stable Focal Inverse Source Localization Using Combinatorial Optimization Techniques Combined with Regularization Methods Running Title: Stable Focal Inverse Source Localization

The inverse problem arising from EEG and MEG is largely underdetermined. One strategy to alleviate this problem is the restriction to a limited number of point-like sources, the focal source model. Although the singular value decomposition of the spatio-temporal data gives an estimate of the minimal number of dipoles contributing to the measurement, the exact number is unknown in advance and noise complicates the reconstruction. Classical nonlinear dipole t algorithms do not give an estimate for the correct number because they are not stable with regard to an overestimation of this parameter. Too many sources may only describe noise, but can still attain a large magnitude during the inverse procedure and may be indiscernible from the true sources. This paper describes a nonlinear dipole t reconstruction algorithm with a new regularization approach for the embedded linear problem, automatically controlled by the noise in the data and the condition of the occuring least square problems. The algorithm is stable with regard to source components which \nearly" lie in the kernel of the projection or lead eld operator and it thus gives an estimate of the unknown number parameter. Simulation studies in a simulated sulcus structure are carried out for an instantaneous dipole model and spatial resolution in the sulcus and stability of the new method are compared with a classical reconstruction algorithm without regularization.