Some uniform stability and convergence results for integral equations on the real line and projection methods for their solution

The paper considers second kind equations of the form (abbreviated x=y + K2x) in which and the factor z is bounded but otherwise arbitrary so that equations of Wiener-Hopf type are included as a special case. Conditions on a set are obtained such that a generalized Fredholm alternative is valid: if W satisfies these conditions and I − Kz, is injective for each z ε W then I − Kz is invertible for each z ε W and the operators (I − Kz)−1 are uniformly bounded. As a special case some classical results relating to Wiener-Hopf operators are reproduced. A finite section version of the above equation (with the range of integration reduced to [−a, a]) is considered, as are projection and iterated projection methods for its solution. The operators (where denotes the finite section version of Kz) are shown uniformly bounded (in z and a) for all a sufficiently large. Uniform stability and convergence results, for the projection and iterated projection methods, are obtained. The argument generalizes an idea in collectively compact operator theory. Some new results in this theory are obtained and applied to the analysis of projection methods for the above equation when z is compactly supported and k(s − t) replaced by the general kernel k(s,t). A boundary integral equation of the above type, which models outdoor sound propagation over inhomogeneous level terrain, illustrates the application of the theoretical results developed.

Microbial−mammalian cometabolites dominate the age-associated urinary metabolic phenotype in Taiwanese and American populations

Understanding the metabolic processes associated with aging is key to developing effective management and treatment strategies for age-related diseases. We investigated the metabolic profiles associated with age in a Taiwanese and an American population. 1H NMR spectral profiles were generated for urine specimens collected from the Taiwanese Social Environment and Biomarkers of Aging Study (SEBAS; n = 857; age 54–91 years) and the Mid-Life in the USA study (MIDUS II; n = 1148; age 35–86 years). Multivariate and univariate linear projection methods revealed some common age-related characteristics in urinary metabolite profiles in the American and Taiwanese populations, as well as some distinctive features. In both cases, two metabolites—4-cresyl sulfate (4CS) and phenylacetylglutamine (PAG)—were positively associated with age. In addition, creatine and β-hydroxy-β-methylbutyrate (HMB) were negatively correlated with age in both populations (p < 4 × 10–6). These age-associated gradients in creatine and HMB reflect decreasing muscle mass with age. The systematic increase in PAG and 4CS was confirmed using ultraperformance liquid chromatography–mass spectrometry (UPLC–MS). Both are products of concerted microbial–mammalian host cometabolism and indicate an age-related association with the balance of host–microbiome metabolism.

Optimal convergence estimates for the trace of the polynomial L2-projection operator on a simplex

In this paper we study convergence of the L2-projection onto the space of polynomials up to degree p on a simplex in Rd, d >= 2. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in Rd from [P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163].

A comparative review of dimension reduction methods in approximate Bayesian computation

Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC requires computations based on vectors of summary statistics, rather than full data sets, a central question is how to derive low-dimensional summary statistics from the observed data with minimal loss of information. In this article we provide a comprehensive review and comparison of the performance of the principal methods of dimension reduction proposed in the ABC literature. The methods are split into three nonmutually exclusive classes consisting of best subset selection methods, projection techniques and regularization. In addition, we introduce two new methods of dimension reduction. The first is a best subset selection method based on Akaike and Bayesian information criteria, and the second uses ridge regression as a regularization procedure. We illustrate the performance of these dimension reduction techniques through the analysis of three challenging models and data sets.