Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem

We consider the statistical non-linear inverse problem of recovering the absorption term f>0 in the heat equation with given sufficiently smooth functions describing boundary and initial values respectively. The data consists of N discrete noisy point evaluations of the solution u_f. We study the statistical performance of Bayesian nonparametric procedures based on a large class of Gaussian process priors. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate for the reconstruction error of the associated posterior means. We also consider the optimality of the contraction rates and prove a lower bound for the minimax convergence rate for inferring f from the data, and show that optimal rates can be achieved with truncated Gaussian priors.

Evaluation of Bi-Layer Silk Fibroin Grafts for Penile Tunica Albuginea Repair in a Rabbit Corporoplasty Model

The use of autologous tissue grafts for tunica albuginea repair in Peyronie’s disease and congenital chordee is often restricted by limited tissue availability and donor site morbidity, therefore new biomaterial options are needed. In this study, bi-layer silk fibroin (BLSF) scaffolds were investigated to support functional tissue regeneration of tunica albuginea in a rabbit corporoplasty model. Eighteen adult male, New Zealand white rabbits were randomized to nonsurgical controls (NSC, N = 3), or subjected to corporoplasty with BLSF grafts (N = 5); decellularized small intestinal submucosa (SIS) matrices (N = 5); or autologous tunica vaginalis (TV) flaps (N = 5). End-point evaluations were cavernosography, cavernosometry, histological, immunohistochemical, and histomorphometric assessments. Maximum intracorporal pressures (ICP) following papaverine-induced erection were similar between all groups. Eighty percent of rabbits repaired with BLSF scaffolds or TV flaps achieved full rigid erections, compared to 40% of SIS reconstructed animals. Five-minute peak erections were maintained in 60% of BLSF rabbits, compared to 20% of SIS and TV flap reconstructed rabbits. Graft perforation occurred in 60% of TV group at maximum ICP compared to 20% of BLSF cohort. Neotissues supported by SIS and BLSF scaffolds were composed of collagen type I and elastin fibers similar to NSC. SIS and TV flaps showed significantly elevated levels of corporal fibrosis relative to NSC with a corresponding decrease in corporal smooth muscle cells expressing contractile proteins. BLSF biomaterials represent emerging platforms for corporoplasty and produce superior functional and histological outcomes in comparison to TV flaps and SIS matrices for tunica albuginea repair.

The space $D$ in several variables: random variables and higher moments

We study the Banach space $D([0,1]^m)$ of functions of several variables that are (in a certain sense) right-continuous with left limits, and extend several results previously known for the standard case $m=1$. We give, for example, a description of the dual space, and we show that a bounded multilinear form always is measurable with respect to the $\sigma$-field generated by the point evaluations. These results are used to study random functions in the space. (I.e., random elements of the space.) In particular, we give results on existence of moments (in different senses) of such random functions, and we give an application to the Zolotarev distance between two such random functions.

Research on Tire Marking Point Completeness Evaluation Based on K-Means Clustering Image Segmentation

The tire marking points of dynamic balance and uniformity play a crucial guiding role in tire installation. Incomplete marking points block the recognition of tire marking points, and then affect the installation of tires. It is usually necessary to evaluate the marking point completeness during the quality inspection of finished tires. In order to meet the high-precision requirements of the evaluation of tire marking point completeness in the smart factories, the K-means clustering algorithm is introduced to segment the image of marking points in this paper. The pixels within the contour of the marking point are weighted to calculate the marking point completeness on the basis of the image segmentation. The completeness is rated and evaluated by completeness calculation. The experimental results show that the accuracy of the marking point completeness ratings is 95%, and the accuracy of the marking point evaluations is 99%. The proposed method has an important guiding significance of practice to evaluate the tire marking point completeness during the tire quality inspection based on machine vision.

A Characterization of Polynomial Density on Curves via Matrix Algebra

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

A Generic Approach for Accelerating Stochastic Zeroth-Order Convex Optimization

In this paper, we propose a generic approach for accelerating the convergence of existing algorithms to solve the problem of stochastic zeroth-order convex optimization (SZCO). Standard techniques for accelerating the convergence of stochastic zeroth-order algorithms are by exploring multiple functional evaluations (e.g., two-point evaluations), or by exploiting global conditions of the problem (e.g., smoothness and strong convexity). Nevertheless, these classic acceleration techniques are necessarily restricting the applicability of newly developed algorithms. The key of our proposed generic approach is to explore a local growth condition (or called local error bound condition) of the objective function in SZCO. The benefits of the proposed acceleration technique are: (i) it is applicable to both settings with one-point evaluation and two-point evaluations; (ii) it does not necessarily require strong convexity or smoothness condition of the objective function; (iii) it yields an improvement on convergence for a broad family of problems. Empirical studies in various settings demonstrate the effectiveness of the proposed acceleration approach.

The algebras of bounded and essentially bounded Lebesgue measurable functions

Let X be a set in ℝn with positive Lebesgue measure. It is well known that the spectrum of the algebra L∞(X) of (equivalence classes) of essentially bounded, complex-valued, measurable functions on X is an extremely disconnected compact Hausdorff space.We show, by elementary methods, that the spectrum M of the algebra ℒb(X, ℂ) of all bounded measurable functions on X is not extremely disconnected, though totally disconnected. Let ∆ = { δx : x ∈ X} be the set of point evaluations and let g be the Gelfand topology on M. Then (∆, g) is homeomorphic to (X, Τdis),where Tdis is the discrete topology. Moreover, ∆ is a dense subset of the spectrum M of ℒb(X, ℂ). Finally, the hull h(I), (which is homeomorphic to M(L∞(X))), of the ideal of all functions in ℒb(X, ℂ) vanishing almost everywhere on X is a nowhere dense and extremely disconnected subset of the Corona M \ ∆ of ℒb(X, ℂ).