Matti Leinonen

Y. Sarcheshmehpour, M. Leinonen, A. Jung

This paper formulates and studies a novel algorithm for federated learning from large collections of local datasets. This algorithm capitalizes on an intrinsic network structure that relates the local datasets via an undirected "empirical" graph. We model such big data over networks using a networked linear regression model. Each local dataset has individual regression weights. The weights of close-knit sub-collections of local datasets are enforced to deviate only little. This lends naturally to a network Lasso problem which we solve using a primal-dual method. We obtain a distributed federated learning algorithm via a message passing implementation of this primal-dual method. We provide a detailed analysis of the statistical and computational properties of the resulting federated learning algorithm.

Harri Hakula, Matti Leinonen

We consider the construction of the stochastic moment matrices that appear in the typical elliptic diffusion problem considered in the setting of stochastic Galerkin finite element method (sGFEM). Algorithms for the efficient construction of the stochastic moment matrices are presented for certain combinations of affine/non-affine diffusion coefficients and multivariate polynomial spaces. We report the performance of various standard polynomial spaces for three different non-affine diffusion coefficients in a one-dimensional spatial setting and compare observed Legendre coefficient convergence rates to theoretical results.

Nuutti Hyvönen, Matti Leinonen

The objective of electrical impedance tomography is to deduce information about the conductivity inside a physical body from electrode measurements of current and voltage at the object boundary. In this work, the unknown conductivity is modeled as a random field parametrized by its values at a set of pixels. The uncertainty in the pixel values is propagated to the electrode measurements by numerically solving the forward problem of impedance tomography by a stochastic Galerkin finite element method in the framework of the complete electrode model. For a given set of electrode measurements, the stochastic forward solution is employed in approximately parametrizing the posterior probability density of the conductivity and contact resistances. Subsequently, the conductivity is reconstructed by computing the maximum a posteriori and conditional mean estimates as well as the posterior covariance. The functionality of this approach is demonstrated with experimental water tank data.