Abstract. Often, parameter estimation problems of parameter-dependent PDEs involve multiple right-hand sides. The computational cost and memory requirements of such problems increase linearly with the number of right-hand sides. For many applications this is the main bottleneck of the computation. In this paper we show that problems with multiple right-hand sides can be reformulated as stochastic programming problems by combining the right-hand sides into a few "simultaneous" sources. This effectively reduces the cost of the forward problem and results in problems that are much cheaper to solve. We discuss two solution methodologies: namely sample average approximation and stochastic approximation. To illustrate the effectiveness of our approach we present two model problems, direct current resistivity and seismic tomography.

Gut microbiota and the immune system interact to maintain tissue homeostasis, but whether this interaction is involved in the pathogenesis of systemic lupus erythematosus (SLE) is unclear. Here we report that oral antibiotics given during active disease removed harmful bacteria from the gut microbiota and attenuated SLE-like disease in lupus-prone mice. Using MRL/lpr mice, we showed that antibiotics given after disease onset ameliorated systemic autoimmunity and kidney histopathology. They decreased IL-17-producing cells and increased the level of circulating IL-10. In addition, antibiotics removed Lachnospiraceae and increased the relative abundance of Lactobacillus spp., two groups of bacteria previously shown to be associated with deteriorated or improved symptoms in MRL/lpr mice, respectively. Moreover, we showed that the attenuated disease phenotype could be recapitulated with a single antibiotic vancomycin, which reshaped the gut microbiota and changed microbial functional pathways in a time-dependent manner. Furthermore, vancomycin treatment increased the barrier function of the intestinal epithelium, thus preventing the translocation of lipopolysaccharide, a cell wall component of Gram-negative Proteobacteria and known inducer of lupus in mice, into the circulation. These results suggest that mixed antibiotics or a single antibiotic vancomycin ameliorate SLE-like disease in MRL/lpr mice by changing the composition of gut microbiota.

Spectral filtering suppresses the amplification of errors when computing solutions to ill-posed inverse problems; however, selecting good regularization parameters is often expensive. In many applications, data are available from calibration experiments. In this paper, we describe how to use such data to precompute optimal spectral filters. We formulate the problem in an empirical Bayes risk minimization framework and use efficient methods from stochastic and numerical optimization to compute optimal filters. Our formulation of the optimal filter problem is general enough to use a variety of assessments of goodness of the solution estimate, not just the mean square error. The relationship with the Wiener filter is discussed, and numerical examples from signal and image deconvolution illustrate that our proposed filters perform consistently better than well-established filtering methods. Furthermore, we show how our approach leads to easily computed uncertainty estimates for the pixel values.

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The analysis of survival endpoints subject to right-censoring is an important research area in statistics, particularly among econometricians and biostatisticians. The two most popular semiparametric models are the proportional hazards model and the accelerated failure time (AFT) model. Rank-based estimation in the AFT model is computationally challenging due to optimization of a non-smooth loss function. Previous work has shown that rank-based estimators may be written as solutions to linear programming (LP) problems. However, the size of the LP problem is O(n2 + p) subject to n2 linear constraints, where n denotes sample size and p denotes the dimension of parameters. As n and/or p increases, the feasibility of such solution in practice becomes questionable. Among data mining and statistical learning enthusiasts, there is interest in extending ordinary regression coefficient estimators for low-dimensions into high-dimensional data mining tools through regularization. Applying this recipe to rank-based coefficient estimators leads to formidable optimization problems which may be avoided through smooth approximations to non-smooth functions. We review smooth approximations and quasi-Newton methods for rank-based estimation in AFT models. The computational cost of our method is substantially smaller than the corresponding LP problem and can be applied to small- or large-scale problems similarly. The algorithm described here allows one to couple rank-based estimation for censored data with virtually any regularization and is exemplified through four case studies.