Asymptotic Conditional Distribution of Exceedance Counts: Fragility Index with Different Margins

Let $\bm X=(X_1,...,X_d)$ be a random vector, whose components are not necessarily independent nor are they required to have identical distribution functions $F_1,...,F_d$. Denote by $N_s$ the number of exceedances among $X_1,...,X_d$ above a high threshold $s$. The fragility index, defined by $FI=\lim_{s\nearrow}E(N_s\mid N_s>0)$ if this limit exists, measures the asymptotic stability of the stochastic system $\bm X$ as the threshold increases. The system is called stable if $FI=1$ and fragile otherwise. In this paper we show that the asymptotic conditional distribution of exceedance counts (ACDEC) $p_k=\lim_{s\nearrow}P(N_s=k\mid N_s>0)$, $1\le k\le d$, exists, if the copula of $\bm X$ is in the domain of attraction of a multivariate extreme value distribution, and if $\lim_{s\nearrow}(1-F_i(s))/(1-F_\kappa(s))=\gamma_i\in[0,\infty)$ exists for $1\le i\le d$ and some $\kappa\in{1,...,d}$. This enables the computation of the FI corresponding to $\bm X$ and of the extended FI as well as of the asymptotic distribution of the exceedance cluster length also in that case, where the components of $\bm X$ are not identically distributed

Developmental design, fabrication, and test of acoustic suppressors for fans of high bypass turbofan engines

An analysis procedure was developed for design of acoustically treated nacelles for high bypass turbofan engines. The plan was applied to the conceptual design of a nacelle for the quiet engine typical of a 707/DC-8 airplane installation. The resultant design was modified to a test nacelle design for the NASA Lewis quiet fan. The acoustic design goal was a 10 db reduction in effective perceived fan noise levels during takoff and approach. Detailed nacelle designs were subsequently developed for both the quiet engine and the quiet fan. The acoustic design goal for each nacelle was 15 db reductions in perceived fan noise levels from the inlet and fan duct. Acoustically treated nacelles were fabricated for the quiet engine and quiet fan for testing. Performance of selected inlet and fan duct lining configurations was experimentally evaluated in a flow duct. Results of the tests show that the linings perform as designed

The Random Feature Model for Input-Output Maps between Banach Spaces

Well known to the machine learning community, the random feature model, originally introduced by Rahimi and Recht in 2008, is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to the real line. In this paper, we instead propose a methodology for use of the random feature model as a data-driven surrogate for operators that map an input Banach space to an output Banach space. Although the methodology is quite general, we consider operators defined by partial differential equations (PDEs); here, the inputs and outputs are themselves functions, with the input parameters being functions required to specify the problem, such as initial data or coefficients, and the outputs being solutions of the problem. Upon discretization, the model inherits several desirable attributes from this infinite-dimensional, function space viewpoint, including mesh-invariant approximation error with respect to the true PDE solution map and the capability to be trained at one mesh resolution and then deployed at different mesh resolutions. We view the random feature model as a non-intrusive data-driven emulator, provide a mathematical framework for its interpretation, and demonstrate its ability to efficiently and accurately approximate the nonlinear parameter-to-solution maps of two prototypical PDEs arising in physical science and engineering applications: viscous Burgers' equation and a variable coefficient elliptic equation

The Random Feature Model for Input-Output Maps between Banach Spaces

Well known to the machine learning community, the random feature model is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to the real line. In this paper, we instead propose a methodology for use of the random feature model as a data-driven surrogate for operators that map an input Banach space to an output Banach space. Although the methodology is quite general, we consider operators defined by partial differential equations (PDEs); here, the inputs and outputs are themselves functions, with the input parameters being functions required to specify the problem, such as initial data or coefficients, and the outputs being solutions of the problem. Upon discretization, the model inherits several desirable attributes from this infinite-dimensional viewpoint, including mesh-invariant approximation error with respect to the true PDE solution map and the capability to be trained at one mesh resolution and then deployed at different mesh resolutions. We view the random feature model as a non-intrusive data-driven emulator, provide a mathematical framework for its interpretation, and demonstrate its ability to efficiently and accurately approximate the nonlinear parameter-to-solution maps of two prototypical PDEs arising in physical science and engineering applications: viscous Burgers' equation and a variable coefficient elliptic equation.Comment: To appear in SIAM Journal on Scientific Computing; 32 pages, 9 figure

Meniscal tissue explants response depends on level of dynamic compressive strain

SummaryObjectiveFollowing partial meniscectomy, the remaining meniscus is exposed to an altered loading environment. In vitro 20% dynamic compressive strains on meniscal tissue explants has been shown to lead to an increase in release of glycosaminoglycans from the tissue and increased expression of interleukin-1α (IL-1α). The goal of this study was to determine if compressive loading which induces endogenously expressed IL-1 results in downstream changes in gene expression of anabolic and catabolic molecules in meniscal tissue, such as MMP expression.MethodRelative changes in gene expression of MMP-1, MMP-3, MMP-9, MMP-13, A Disintegrin and Metalloproteinase with ThromboSpondin 4 (ADAMTS4), ADAMTS5, TNFα, TGFβ, COX-2, Type I collagen (COL-1) and aggrecan and subsequent changes in the concentration of prostaglandin E2 released by meniscal tissue in response to varying levels of dynamic compression (0%, 10%, and 20%) were measured. Porcine meniscal explants were dynamically compressed for 2h at 1Hz.Results20% dynamic compressive strains upregulated MMP-1, MMP-3, MMP-13 and ADAMTS4 compared to no dynamic loading. Aggrecan, COX-2, and ADAMTS5 gene expression were upregulated under 10% strain compared to no dynamic loading while COL-1, TIMP-1, and TGFβ gene expression were not dependent on the magnitude of loading.ConclusionThis data suggests that changes in mechanical loading of the knee joint meniscus from 10% to 20% dynamic strain can increase the catabolic activity of the meniscus