A quasinonlocal coupling method for nonlocal and local diffusion models

In this paper, we extend the idea of "geometric reconstruction" to couple a nonlocal diffusion model directly with the classical local diffusion in one dimensional space. This new coupling framework removes interfacial inconsistency, ensures the flux balance, and satisfies energy conservation as well as the maximum principle, whereas none of existing coupling methods for nonlocal-to-local coupling satisfies all of these properties. We establish the well-posedness and provide the stability analysis of the coupling method. We investigate the difference to the local limiting problem in terms of the nonlocal interaction range. Furthermore, we propose a first order finite difference numerical discretization and perform several numerical tests to confirm the theoretical findings. In particular, we show that the resulting numerical result is free of artifacts near the boundary of the domain where a classical local boundary condition is used, together with a coupled fully nonlocal model in the interior of the domain

Ratio-consistent estimation for long range dependent Toeplitz covariance with application to matrix data whitening

We consider a data matrix $X:=R_M^{1/2}ZC_N^{1/2}$ where $R_M$ is a $M\times M$ Toeplitz matrix, $Z$ is a $M\times N$ random matrix of uncorrelated standardized white noise, and $C_N$ a $N\times N$ positive semi-definite matrix. The model $X$ can be interpreted as a multivariate stationary time series with a separable covariance function. When this series is short range dependent, two estimators $\hat{R}_M$ and $\hat{R}_M^b$ of $R_M$, constructed by toeplitzifying the sample covariance matrix $S=N^{-1}XX^*$, are commonly used to whiten the correlation $R_M$ in $X$. Both are proved to be consistent in spectral norm in previous articles under mild conditions. In this paper, we establish that when the time series is long range dependent, the above spectral norm consistency does not always hold, but a weaker {\it ratio consistency} for the unbiased estimator $\hat{R}_M$ still holds. It is shown that this ratio consistency is sufficient for the whitening procedure. For the biased estimator $\hat{R}_M^b$, such ratio consistency does not hold either, but a weaker {\it ratio LSD consistency} does. Numeric simulations are also provided to illustrate these new phenomena and their impact on applications such as the whitening procedure. Finally we apply our results to signal detection and high-dimensional PCA. Let $X=[YR_M^{1/2}]^*$ with $Y=A\mathbf{M}+\sigma^2\mathbf{N}$ a complex Gaussian signal plus noise model. Using the whitened sample covariance matrix $\underline{S}_w=M^{-1}X^*\hat{R}_M^{-1}X$, we estimate the number of signals and their strengths contained in $A$. Then we proceed PCA on $X$ to obtain a compressed data matrix formed with its principal components

Increasing power for voxel-wise genome-wide association studies : the random field theory, least square kernel machines and fast permutation procedures

Imaging traits are thought to have more direct links to genetic variation than diagnostic measures based on cognitive or clinical assessments and provide a powerful substrate to examine the influence of genetics on human brains. Although imaging genetics has attracted growing attention and interest, most brain-wide genome-wide association studies focus on voxel-wise single-locus approaches, without taking advantage of the spatial information in images or combining the effect of multiple genetic variants. In this paper we present a fast implementation of voxel- and cluster-wise inferences based on the random field theory to fully use the spatial information in images. The approach is combined with a multi-locus model based on least square kernel machines to associate the joint effect of several single nucleotide polymorphisms (SNP) with imaging traits. A fast permutation procedure is also proposed which significantly reduces the number of permutations needed relative to the standard empirical method and provides accurate small p-value estimates based on parametric tail approximation. We explored the relation between 448,294 single nucleotide polymorphisms and 18,043 genes in 31,662 voxels of the entire brain across 740 elderly subjects from the Alzheimer's Disease Neuroimaging Initiative (ADNI). Structural MRI scans were analyzed using tensor-based morphometry (TBM) to compute 3D maps of regional brain volume differences compared to an average template image based on healthy elderly subjects. We find method to be more sensitive compared with voxel-wise single-locus approaches. A number of genes were identified as having significant associations with volumetric changes. The most associated gene was GRIN2B, which encodes the N-methyl-d-aspartate (NMDA) glutamate receptor NR2B subunit and affects both the parietal and temporal lobes in human brains. Its role in Alzheimer's disease has been widely acknowledged and studied, suggesting the validity of the approach. The various advantages over existing approaches indicate a great potential offered by this novel framework to detect genetic influences on human brains

A Secure Authentication Framework to Guarantee the Traceability of Avatars in Metaverse

Metaverse is a vast virtual environment parallel to the physical world in which users enjoy a variety of services acting as an avatar. To build a secure living habitat, it's vital to ensure the virtual-physical traceability that tracking a malicious player in the physical world via his avatars in virtual space. In this paper, we propose a two-factor authentication framework based on chameleon signature and biometric-based authentication. First, aiming at disguise in virtual space, we propose a chameleon collision signature algorithm to achieve the verifiability of the avatar's virtual identity. Second, facing at impersonation in physical world, we construct an avatar's identity model based on the player's biometric template and the chameleon key to realize the verifiability of the avatar's physical identity. Finally, we design two decentralized authentication protocols based on the avatar's identity model to ensure the consistency of the avatar's virtual and physical identities. Security analysis indicates that the proposed authentication framework guarantees the consistency and traceability of avatar's identity. Simulation experiments show that the framework not only completes the decentralized authentication between avatars but also achieves the virtual-physical tracking.Comment: 12 pages, 9 figure