Bayesian modeling and analysis of the magnetoencephalography and electroencephalography modalities provide a flexible framework for introducing prior information complementary to the measured data. This prior information is often qualitative in nature, making the translation of the available information into a computational model a challenging task. We propose a generalized gamma family of hyperpriors which allows the impressed currents to be focal and we advocate a fast and efficient iterative algorithm, the iterative alternating sequential algorithm for computing maximum a posteriori (MAP) estimates. Furthermore, we show that for particular choices of the scalar parameters specifying the hyperprior, the algorithm effectively approximates popular regularization strategies such as the minimum current estimate and the minimum support estimate. The connection between priorconditioning and adaptive regularization methods is also pointed out. The posterior densities are explored by means of a Markov chain Monte Carlo strategy suitable for this family of hypermodels. The computed experiments suggest that the known preference of regularization methods for superficial sources over deep sources is a property of the MAP estimators only, and that estimation of the posterior mean in the hierarchical model is better adapted for localizing deep sources.

No abstract

No abstract

Moduli of rings and quadrilaterals are frequently applied in geometric function theory, see e.g. the Handbook by Kühnau. Yet their exact values are known only in a few special cases. Previously, the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation. We present here a new hp-FEM algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the Schwarz-Christoffel Toolbox of Driscoll and Trefethen. We also demonstrate that the hp-FEM algorithm applies to the case of non-polygonal boundary and report results with concrete error bounds.for which numerical results were reported, too. Again, it is also possible to use the Schwarz-Christoffel method for doubly connected domains [25].We next consider the case of simply connected plane domains. For such a domain D and for a quadruple {z 1 , z 2 , z 3 , z 4 } of its boundary points we call (D; z 1 , z 2 , z 3 , z 4 ) a quadrilateral if z 1 , z 2 , z 3 , z 4 occur in this order when the boundary curve is traversed in the positive direction. The points z k , k = 1, .., 4 , are called the vertices and the part of the oriented boundary between two successive vertices such as z 1 and z 2 is called a boundary arc (z 1 , z 2 ) . The modulus M(D; z 1 , z 2 , z 3 , z 4 ) of the quadrilateral (D; z 1 , z 2 , z 3 , z 4 ) is defined to be the unique h > 0 for which there exists a conformal mapping of D onto the rectangle with vertices 1 + ih, ih, 0, 1 such that the points z 1 , z 2 , z 3 , z 4 correspond to the vertices in this order. This conformal mapping is called the canonical conformal mapping associated with the quadrilateral. As in the case of doubly connected domains discussed above, it is well-known that the computation of the modulus h of the quadrilateral may be reduced to solving the Dirichlét-Neumann boundary value problem in the original domain D with the Dirichlét boundary values 1 on the boundary arc (z 1 , z 2 ) and 0 on the arc (z 3 , z 4 ) and Neumann boundary values 0 on the arcs (z 3 , z 4 ) and (z 4 , z 1 ) .Conformal moduli of rings and quadrilaterals have independent theoretical interest because of their crucial role in the theory of quasiconformal mappings [30]. These quantities are closely related to certain physical constants, e.g. they play an important role in applications involving the measurement of resistance values of integrated circuit networks. But the problem of computing the moduli is also interesting in the wider engineering context. The reciprocal identities (4.1) and (6.1) can be used to generate test cases for curvilinear Lipschitz domains and thus should be standard tools in the FEM-software development community. Unfortunately these identities are missing from the introductory FEM textbooks. Although the experimental results in this paper show that the reciprocal identities provide error estimates similar to the true error (in cases where the exact analytic result is known) more in...

No abstract