We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown function. The observational noise is assumed to be Gaussian; as a consequence the prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We assume that the forward operator and the prior and noise covariance operators commute with one another. We show how, for given smoothness assumptions on the truth, the scale parameter of the prior, which is a constant multiplier of the prior covariance operator, can be adjusted to optimize the rate of posterior contraction to the truth, and we explicitly compute the logarithmic rate.

Helicid (4-formylphenyl-O-β-D-allopyranoside), an active component found in seeds from the Chinese herb Helicia nilagirica, has been reported to exert sedative, analgesic, hypnotic and antidepressant effects. The present study was designed to evaluate the antidepressant, learning and cognitive improvement effects of helicid in a chronic unpredictable mild stress (CUMS) model of depression in rats and to explore cAMP/protein kinase A (PKA)/cAMP response element-binding (CREB) signaling pathway. Sprague-Dawley rats were randomly assigned to six groups (n 10): control; CUMS; CUMS fluoxetine (5 mg/kg) and CUMS helicid at 8, 16 and 32 mg/kg. All rats were subjected to 12 weeks of CUMS protocols and drug administration during the last 6 weeks of CUMS. Our results showed that helicid, at a dose of 32 mg/kg, significantly reversed decreases in body weight and sucrose consumption, increased the distance and number of crossings in the open-field test (OFT), reduced immobility times in the forced swimming test (FST) and improved spatial memory in the Morris water maze (MWM); all of these effects had been induced by CUMS paradigm. Immunohistochemistry showed that administration of helicid could promoted the proliferation of neurons in the hippocampal CA1 and dentate gyrus (DG) regions. CUMS rats treated with helicid had dramatically decreased protein levels of serotonin transporters (SERTs). In addition, CUMS resulted in a significant reduction in the expression of cAMP, PKA C-α and p-CREB, each of which were partially attenuated by helicid administration. These results indicated that helicid could improve depressive behaviors, learning and cognitive deficits and increase hippocampal neurogenesis, which may be mediated by the regulation of SERTs, activation of the cAMP/PKA/CREB signaling pathway and upregulation of p-CREB levels in hippocampal.

The major challenges in the Bayesian inverse problems arise from the need for repeated evaluations of the forward model, as required by Markov chain Monte Carlo (MCMC) methods for posterior sampling. Many attempts at accelerating Bayesian inference have relied on surrogates for the forward model, typically constructed through repeated forward simulations that are performed in an offline phase. Although such approaches can be quite effective at reducing computation cost, there has been little analysis of the approximation on posterior inference. In this work, we prove error bounds on the Kullback-Leibler (KL) distance between the true posterior distribution and the approximation based on surrogate models. Our rigorous error analysis show that if the forward model approximation converges at certain rate in the prior-weighted L 2 norm, then the posterior distribution generated by the approximation converges to the true posterior at least two times faster in the KL sense. The error bound on the Hellinger distance is also provided. To provide concrete examples focusing on the use of the surrogate model based methods, we present an efficient technique for constructing stochastic surrogate models to accelerate the Bayesian inference approach. The Christoffel least squares algorithms, based on generalized polynomial chaos, are used to construct a polynomial approximation of the forward solution over the support of the prior distribution. The numerical strategy and the predicted convergence rates are then demonstrated on the nonlinear inverse problems, involving the inference of parameters appearing in partial differential equations.

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A medium-Mn steel (Fe–0.3C–4Mn–1.82Al–0.6Si wt-%) was produced by a novel processing route involving twin-roll strip casting, hot rolling and intercritical annealing (IA). Macrosegregation was absent in the as-cast strip. The microstructure of the as-cast strip consisted of martensite and austenite (∼10 vol.-%), and the solidification structure was characterised by dendritic structure. With an increase in IA temperature from 680 to 725 and to 755°C, austenite fraction in intercritically annealed steels was increased from 22 to 45% and then decreased to 27%. The 710°C intercritically annealed steel yielded excellent mechanical properties with a tensile strength of ∼1007 MPa and total elongation of ∼48%, achieved by a high volume fraction of austenite (∼42%) with appropriate mechanical stability.