Spectral methods for multiscale stochastic differential equations

This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented

Spectral Methods for Multiscale Stochastic Differential Equations

This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented

Optimal importance sampling for overdamped Langevin dynamics

Calculating averages with respect to multimodal probability distributions is often necessary in applications. Markov chain Monte Carlo (MCMC) methods to this end, which are based on time averages along a realization of a Markov process ergodic with respect to the target probability distribution, are usually plagued by a large variance due to the metastability of the process. In this work, we mathematically analyze an importance sampling approach for MCMC methods that rely on the overdamped Langevin dynamics. Specifically, we study an estimator based on an ergodic average along a realization of an overdamped Langevin process for a modified potential. The estimator we consider incorporates a reweighting term in order to rectify the bias that would otherwise be introduced by this modification of the potential. We obtain an explicit expression in dimension 1 for the biasing potential that minimizes the asymptotic variance of the estimator for a given observable, and propose a general numerical approach for approximating the optimal potential in the multi-dimensional setting. We also investigate an alternative approach where, instead of the asymptotic variance for a given observable, a weighted average of the asymptotic variances corresponding to a class of observables is minimized. Finally, we demonstrate the capabilities of the proposed method by means of numerical experiments

The Mean Field Ensemble Kalman Filter: Near-Gaussian Setting

The ensemble Kalman filter is widely used in applications because, for high dimensional filtering problems, it has a robustness that is not shared for example by the particle filter; in particular it does not suffer from weight collapse. However, there is no theory which quantifies its accuracy as an approximation of the true filtering distribution, except in the Gaussian setting. To address this issue we provide the first analysis of the accuracy of the ensemble Kalman filter beyond the Gaussian setting. Our analysis is developed for the mean field ensemble Kalman filter. We rewrite the update equations for this filter, and for the true filtering distribution, in terms of maps on probability measures. We introduce a weighted total variation metric to estimate the distance between the two filters and we prove various stability estimates for the maps defining the evolution of the two filters, in this metric. Using these stability estimates we demonstrate that if the true filtering distribution is close to Gaussian in the joint space of state and data, in the weighted total variation metric, then the true-filter is well approximated by the ensemble Kalman filter, in the same metric. Finally, we provide a generalization of these results to the Gaussian projected filter, which can be viewed as a mean field description of the unscented Kalman filter

A Characterization of right coideals of quotient type and its application to classification of Poisson boundaries

Let $G$ be a co-amenable compact quantum group. We show that a right coideal of $G$ is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on $SU_q(N)$ for co-amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q-deformed classical compact Lie group.Comment: 28 pages, Remark 4.9 adde

Folate catabolites in spot urine as non-invasive biomarkers of folate status during habitual intake and folic acid supplementation.

Folate status, as reflected by red blood cell (RCF) and plasma folates (PF), is related to health and disease risk. Folate degradation products para-aminobenzoylglutamate (pABG) and para-acetamidobenzoylglutamate (apABG) in 24 hour urine have recently been shown to correlate with blood folate. Since blood sampling and collection of 24 hour urine are cumbersome, we investigated whether the determination of urinary folate catabolites in fasted spot urine is a suitable non-invasive biomarker for folate status in subjects before and during folic acid supplementation. Immediate effects of oral folic acid bolus intake on urinary folate catabolites were assessed in a short-term pre-study. In the main study we included 53 healthy men. Of these, 29 were selected for a 12 week folic acid supplementation (400 µg). Blood, 24 hour and spot urine were collected at baseline and after 6 and 12 weeks and PF, RCF, urinary apABG and pABG were determined. Intake of a 400 µg folic acid bolus resulted in immediate increase of urinary catabolites. In the main study pABG and apABG concentrations in spot urine correlated well with their excretion in 24 hour urine. In healthy men consuming habitual diet, pABG showed closer correlation with PF (rs = 0.676) and RCF (rs = 0.649) than apABG (rs = 0.264, ns and 0.543). Supplementation led to significantly increased folate in plasma and red cells as well as elevated urinary folate catabolites, while only pABG correlated significantly with PF (rs = 0.574) after 12 weeks. Quantification of folate catabolites in fasted spot urine seems suitable as a non-invasive alternative to blood or 24 hour urine analysis for evaluation of folate status in populations consuming habitual diet. In non-steady-state conditions (folic acid supplementation) correlations between folate marker (RCF, PF, urinary catabolites) decrease due to differing kinetics

Association of Thyroid dysfunction with cognitive function an individual participant data analysis

IMPORTANCE In clinical guidelines, overt and subclinical thyroid dysfunction are mentioned as causal and treatable factors for cognitive decline. However, the scientific literature on these associations shows inconsistent findings.OBJECTIVE To assess cross-sectional and longitudinal associations of baseline thyroid dysfunction with cognitive function and dementia.DESIGN, SETTING, AND PARTICIPANTS This multicohort individual participant data analysis assessed 114 267 person-years (median, 1.7-11.3 years) of follow-up for cognitive function and 525 222 person-years (median, 3.8-15.3 years) for dementia between 1989 and 2017. Analyses on cognitive function included 21 cohorts comprising 38 144 participants. Analyses on dementia included eight cohorts with a total of 2033 cases with dementia and 44 573 controls. Data analysis was performed from December 2016 to January 2021.EXPOSURES Thyroid function was classified as overt hyperthyroidism, subclinical hyperthyroidism, euthyroidism, subclinical hypothyroidism, and overt hypothyroidism based on uniform thyrotropin cutoff values and study-specific free thyroxine values.MAIN OUTCOMES AND MEASURES The primary outcome was global cognitive function, mostly measured using the Mini-Mental State Examination. Executive function, memory, and dementia were secondary outcomes. Analyses were first performed at study level using multivariable linear regression and multivariable Cox regression, respectively. The studies were combined with restricted maximum likelihood meta-analysis. To overcome the use of different scales, results were transformed to standardized mean differences. For incident dementia, hazard ratios were calculated.RESULTS Among 74 565 total participants, 66 567 (89.3%) participants had normal thyroid function, 577 (0.8%) had overt hyperthyroidism, 2557 (3.4%) had subclinical hyperthyroidism, 4167 (5.6%) had subclinical hypothyroidism, and 697 (0.9%) had overt hypothyroidism. The study-specific median age at baseline varied from 57 to 93 years; 42 847 (57.5%) participants were women. Thyroid dysfunction was not associated with global cognitive function; the largest differences were observed between overt hypothyroidism and euthyroidism-cross-sectionally (-0.06 standardized mean difference in score; 95% CI, -0.20 to 0.08; P = .40) and longitudinally (0.11 standardized mean difference higher decline per year; 95% CI, -0.01 to 0.23; P = .09). No consistent associations were observed between thyroid dysfunction and executive function, memory, or risk of dementia.CONCLUSIONS AND RELEVANCE In this individual participant data analysis of more than 74 000 adults, subclinical hypothyroidism and hyperthyroidism were not associated with cognitive function, cognitive decline, or incident dementia. No rigorous conclusions can be drawn regarding the role of overt thyroid dysfunction in risk of dementia. These findings do not support the practice of screening for subclinical thyroid dysfunction in the context of cognitive decline in older adults as recommended in current guidelines.Molecular Epidemiolog