Application of hierarchical matrices for computing the Karhunen-Loève expansion

Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Lo`eve expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented

Soy isoflavones and their relationship with microflora: beneficial effects on human health in equol producers

The bioavailability of soy isoflavones depends on the composition of the microflora for each subject. Bacteria act on different isoflavones with increased or reduced absorption and cause biotransformation of these compounds into metabolites with higher biological activity. S-equol is the most important metabolite and only 25–65 % of the population have the microflora that produces this compound. The presence of equol-producing bacteria in soy product consumers means that the consumption of such products for prolonged periods leads to lower cardiovascular risk, reduced incidence of prostate and breast cancer, and greater relief from symptoms related to the menopause such as hot flushes and osteoporosis