12,972 research outputs found

    Kernel Methods are Competitive for Operator Learning

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    We present a general kernel-based framework for learning operators between Banach spaces along with a priori error analysis and comprehensive numerical comparisons with popular neural net (NN) approaches such as Deep Operator Net (DeepONet) [Lu et al.] and Fourier Neural Operator (FNO) [Li et al.]. We consider the setting where the input/output spaces of target operator G† : U→V\mathcal{G}^\dagger\,:\, \mathcal{U}\to \mathcal{V} are reproducing kernel Hilbert spaces (RKHS), the data comes in the form of partial observations ϕ(ui),φ(vi)\phi(u_i), \varphi(v_i) of input/output functions vi=G†(ui)v_i=\mathcal{G}^\dagger(u_i) (i=1,…,Ni=1,\ldots,N), and the measurement operators ϕ : U→Rn\phi\,:\, \mathcal{U}\to \mathbb{R}^n and φ : V→Rm\varphi\,:\, \mathcal{V} \to \mathbb{R}^m are linear. Writing ψ : Rn→U\psi\,:\, \mathbb{R}^n \to \mathcal{U} and χ : Rm→V\chi\,:\, \mathbb{R}^m \to \mathcal{V} for the optimal recovery maps associated with ϕ\phi and φ\varphi, we approximate G†\mathcal{G}^\dagger with Gˉ=χ∘fˉ∘ϕ\bar{\mathcal{G}}=\chi \circ \bar{f} \circ \phi where fˉ\bar{f} is an optimal recovery approximation of f†:=φ∘G†∘ψ : Rn→Rmf^\dagger:=\varphi \circ \mathcal{G}^\dagger \circ \psi\,:\,\mathbb{R}^n \to \mathbb{R}^m. We show that, even when using vanilla kernels (e.g., linear or Mat\'{e}rn), our approach is competitive in terms of cost-accuracy trade-off and either matches or beats the performance of NN methods on a majority of benchmarks. Additionally, our framework offers several advantages inherited from kernel methods: simplicity, interpretability, convergence guarantees, a priori error estimates, and Bayesian uncertainty quantification. As such, it can serve as a natural benchmark for operator learning.Comment: 35 pages, 10 figure

    Explicit formula for the solution of the Szeg\"o equation on the real line and applications

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    We consider the cubic Szeg\"o equation i u_t=Pi(|u|^2u) in the Hardy space on the upper half-plane, where Pi is the Szeg\"o projector on positive frequencies. It is a model for totally non-dispersive evolution equations and is completely integrable in the sense that it admits a Lax pair. We find an explicit formula for solutions of the Szeg\"o equation. As an application, we prove soliton resolution in H^s for all s>0, for generic data. As for non-generic data, we construct an example for which soliton resolution holds only in H^s, 0<s<1/2, while the high Sobolev norms grow to infinity over time, i.e. \lim_{t\to\pm\infty}|u(t)|_{H^s}=\infty if s>1/2. As a second application, we construct explicit generalized action-angle coordinates by solving the inverse problem for the Hankel operator H_u appearing in the Lax pair. In particular, we show that the trajectories of the Szeg\"o equation with generic data are spirals around Lagrangian toroidal cylinders T^N \times R^N.Comment: Small modifications in the proof of Proposition 1.3, changed the order in the proof of Theorem 1.9, replaced the proof of \chi proper mapping in Theorem 1.

    The impact of California table grapes on intestinal inflammation and microbiota in mice fed an american type diet

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    Obesity is a growing health concern in the United States and worldwide. The chronic, low grade inflammation associated with increased white adipose tissue mass has been linked to chronic metabolic disorders such as insulin resistance, hypertension, and hyperlipidemia. Consumption of high fat diets leads to intestinal disorders such as microbial dysbiosis and gut barrier dysfunction that can adversely impact systemic metabolism. One potential dietary strategy to alleviate the high fat-induced chronic inflammation is increased consumption of fruits and vegetables rich in polyphenols due to their antioxidant and anti-inflammatory properties. Notably, several studies have demonstrated that supplementation with grape products or anthocyanins found in grapes reduced inflammation systemically and increased the abundance of beneficial bacteria such as Lactobacillus, and Bifidobacterium in culture. However, the beneficial properties of whole grape powder and one or more of its fractions on intestinal inflammation, microbial populations, and barrier function in mice fed an American type diet rich in saturated fat are unclear. Therefore, the specific aims of this research were to; (i) determine the impact of consuming California table grapes on intestinal health in mice fed an American type diet rich in one type of saturated fat (Aim 1), and (ii) identify a key fraction (i.e., extractable polyphenol (EP) or non-extractable polyphenol (NEP) fractions) of California table grapes that improves markers of intestinal inflammation in mice fed an American type diet rich in four types of saturated fats (Aim 2). In Aim 1, consumption of at least one of the powder grape diets; (i) reduced body fat percentage, the total weight of all four fat depots, and inguinal fat depot weight, (ii) increased localization of a tight junction protein linked to improved barrier function, (iii) reduced the abundance of a deleterious sulfidogenic bacteria, and (iv) increased the abundance of beneficial bacteria (e.g., Akkermansia muciniphila, Bifidobacterium, and Lactobacillus), compared to high fat controls. However, the high fat diet did not significantly increase the abundance of inflammatory markers in the intestine nor did the powdered grapes significantly decrease their abundance. Taken together, these data acquired in Aim 1 demonstrate that whole powdered California table grapes improved a marker of gut barrier function and a metabolic profile that was positively correlated with changes in microbiota in mice fed a butter-rich diet. In Aim 2, the polyphenol-rich EP fraction alone or in combination with NEP (EP+NEP), but not powdered grapes; (i) decreased body fat percentage, body fat depot weights, and liver triglyceride levels, (ii) improved insulin sensitivity and glucose disposal, (iii) decreased the mRNA levels of several inflammatory genes in WAT, (iv) decreased the expression of the proinflammatory gene Cluster of differentiation 68 in the colon, and (v) increased expression of G-protein receptor 43 in the ileum compared to high fat controls. Taken together, these data demonstrate that the; (i) potential health benefits of consuming grape powder are dependent on the type and amount of fat in the diet, and (ii) extraction of polyphenols from powdered grapes results in improved metabolic profile and decreased systemic inflammation in conjunction with consuming a rich in saturated fats. Collectively, these findings are expected to contribute insight for the development of dietary strategies using table grapes for decreasing obesity and some of its metabolic complications, possibly by altering populations of gut microbes. However, clinical trials are needed to determine the extent to which these findings are applicable to humans

    Examining the effect of health behaviors on wages and healthcare utilization in models with endogeneity

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    This dissertation contains three essays on applied health economics. Although each essay is independent of the others, all three address the issue of estimating models where the relationship of interest is confounded by factors that are unobservable to the researcher. The first essay is an econometric simulation study while essays 2 and 3 address behavioral health topics. Essay 1 compares the accuracy and efficiency of parametric count data specifications paired with the Extended Olsen Model (EOM; Terza, 1998, 2009). The EOM is a nonlinear instrumental variables approach that allows for consistent estimation of model parameters when the data suffer from binary endogenous switching (e.g., endogenous sample selection or endogenous treatment). Count data models are ubiquitous in the health literature for estimating non-negative, discrete outcomes such as physician visits, hospital admissions, cigarettes smoked, etc. Essay 1 provides insight into the model selection process by informing practitioners which specification is likely to provide the most accurate parameter estimates under a variety of data configurations. Essay 1 also demonstrates the applicability of the Conway-Maxwell Poisson (CMP), a flexible count model developed in the field of industrial engineering that has yet to be utilized in the economic literature. In Essay 2 I apply a count version of the Extended Olsen Model to estimate the relationship between marijuana use disorder (MUD) and ER visits among US Medicaid recipients. This essay is the first in the literature to estimate the relationship between marijuana consumption and the demand for ER visits in isolation from other illicit drugs, thus providing an important addition to the ongoing policy regarding the potential relaxation of marijuana regulation. This study is also the first in the illicit substance literature to use an instrumental variables count data model to estimate the full distribution of ER visits, thus accounting for unobserved factors that may be jointly correlated between individual propensity for MUD and demand for ER visits. I fail to find a positive relationship between MUD and ER visits, instead uncovering suggestive, but inconclusive, evidence that MUD and ER visits may rather be negatively correlated. Essay 3 considers the relationship between wages and obesity. Although prior literature has firmly established a negative relationship between wages and obesity, it is equivocal with regard to the underlying pathway(s) through which obesity results in lower wages. Using firm-level data that gives me unique access to proxies for productivity and discrimination against obese individuals, I find that inputs to productivity, particularly health, are important confounders of the wage-obesity relationship. I fail to find any evidence of discrimination against obese employees, but I do find that among females the negative relationship between wages and obesity exists only among mothers

    Impact of California table grapes on systemic inflammation, insulin resistance, and hepatic steatosis in mice fed an American-type diet

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    The incidence of obesity is increasing worldwide, currently affecting over 400 million people. With obesity, expansion of white adipose tissue (WAT) contributes to a chronic, low-grade inflammatory response that is associated with the formation of complications such as type 2 diabetes, cardiovascular disease, and hypertension. One potential method of reducing chronic inflammation associated with obesity is through consumption of table grapes, which contain fiber and are rich in phytochemicals with potential health promoting properties. Several clinical trials and animal studies have demonstrated that consumption of grape products (i.e. grape seed extract, grape juice, or California powdered table grapes) can reduce oxidative stress, insulin resistance, and inflammation, as well as improve heart health. However, there is still much to be elucidated concerning the mechanisms by which grapes or grape products exert beneficial effects and which compounds within grapes are active in the mediation of these effects. Additionally, the influence of grape polyphenols and fiber on gut microbiota and the link of this with improvements in adiposity, systemic inflammation and insulin resistance are unclear. Therefore, the specific aims of this research were to (i) determine the extent to which California table grapes attenuate body fat accumulation, systemic inflammation, and insulin resistance, and impact gut microbiota in mice fed and American type diet rich in butter (Aim 1), and (ii) identify the key bioactive fraction(s) responsible for reducing adiposity, inflammation, and insulin resistance, and modulating gut microbiota in mice fed an American-type diet rich in four types of saturated fat (Aim 2). In Aim 1, grape consumption at one or both levels (i.e. 3% or 5% w/w) attenuated accumulation of body and liver fat, but these lipid lowering effects were not associated in improvements in glucose tolerance or markers of inflammation in WAT. Alterations in microbial species (i.e. reductions in deleterious sulfidogenic bacteria and increases in beneficial bacteria) were observed in grape-fed mice. Taken together these data demonstrate that consuming grapes results in attenuations in adiposity and hepatic steatosis, and also alterations gut microbial populations in mice fed a butter-rich diet. In Aim 2, consumption of the extractable polyphenol fraction (EP) alone or with the non-extractable polyphenol fraction (EP+NEP), but not powdered grapes (GP), reduced adiposity, liver and plasma triglycerides, markers of inflammation within WAT, and improved insulin sensitivity in mice fed a diet rich in saturated fat from four sources. Taken together, these data demonstrate that (i) consumption of polyphenols extracted from powdered grapes is effective at preventing the complications of diet-induced obesity, and that (ii) the effects of powdered grapes differ based on the amount and source of dietary fat. Overall, the findings from Aim 1 and Aim 2 are anticipated to contribute to the development of novel dietary strategies using grapes or grape products to manage or treat diet-induced obesity and associated conditions. Further research studies, including clinical trials, are still warranted to determine the applicability of these findings to the human population

    Bayesian model criticism: prior sensitivity of the posterior predictive checks method

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    Use of noninformative priors with the Posterior Predictive Checks (PPC) method requires more attention. Previous research of the PPC has treated noninformative priors as always noninformative in relation to the likelihood, regardless of model-data fit. However, as model-data fit deteriorates, and the steepness of the likelihood's curvature diminishes, the prior can become more informative than initially intended. The objective of this dissertation was to investigate whether specification of the prior distribution has an effect on the conclusions drawn from the PPC method. Findings indicated that the choice of discrepancy measure is an important factor in the overall success of the method, and that different discrepancy measures are affected more than others by prior specification

    Effects of edge roughness on optical scattering from periodic microstructures

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    Planar photonic crystals and other microstructured surfaces have important applications in a number of emerging technologies. However, these structures can be difficult to fabricate in a consistent manner. Rapid, precise measurements of critical parameters are needed to control the fabrication process, but current measurement techniques tend to be slow and often require that the sample be modified in order to make the measurement. Optical scattering can provide a rapid, non-destructive, and precise method for measuring these structures, and optical scatterometry is a good candidate technique for measuring micro-structured surfaces for process control. However, variations in the profile, such as those caused by edge roughness, can make significant contributions to the uncertainty in scatterometry measurements. Because of the multi- dimensional nature of the problem, modeling these variations can be computationally expensive. This dissertation examines the effects of edge roughness on optical scatterometry signals. Rigorous numerical simulations show that the effects of edge roughness are sensitive to the correlation length and the frequency content of the roughness as well as its amplitude. However, these rigorous calculations are computationally expensive. A less computationally expensive model based on a generalized Bruggeman effective medium approximation is developed and shown to be effective for modeling the effects of short correlation length edge roughness on optical scatterometry signals

    Adaptation of a moral judgment interview for a pen and paper test

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    The purpose of the present study was to design and validate a written instrument which would approximate the Kohlberg (1958) Moral Judgment Interview in yielded scores. Validity of the instrument was determined by whether it yielded age differences in moral stage scores. Kohlberg found age differences and he later postulated a relationship between moral stages and cognitive stages which predicts age differences. The subjects were a non-random sampling of 57 seventh-grade students, 61 twelfth-grade students, and 50 college students. The data were collected, using two comparable written instruments, one of which was an open-end questionnaire requiring self-structured responses; the other was an objective multiple-choice instrument. Analyses of score differences within groups and between groups were done by the statistical procedures of analysis of variance and t&gt;test for significance of the differences. The open-end Instrument discriminated the scores of seventh-graders and twelfth-graders, whereas the objective instrument did not discriminate among the three different groups at all

    Using the area between two item response functions to index differential item functioning : a generalized approach

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    The area between two item response functions was used to construct two indicators of differential item functioning. The first indicator was designed from the integral of the difference of the two response functions across six standard deviations of ability. The second index was designed from the integral of the square of the difference of the two response functions across the same range of ability. Both indices were developed using the three-parameter item response theory model such that they subsume the two-and one-parameter models as special cases; and both were designed to be sensitive to uniform and non-uniform differential item functioning. The standard errors of the item response theory parameters were used to estimate the standard error of each integral. The ratio of the integral to the estimate of its standard error was shown to be normally distributed using Monte-Carlo data. Hence, both indices provide a multivariate assessment, a z test, of differences in the sets of item response theory parameters
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