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This article introduces a new approach to prediction by bringing together two different nonparametric ideas: distribution-free inference and nonparametric smoothing. Specifically, we consider the problem of constructing nonparametric tolerance/prediction sets. We start from the general conformal prediction approach, and we use a kernel density estimator as a measure of agreement between a sample point and the underlying distribution. The resulting prediction set is shown to be closely related to plug-in density level sets with carefully chosen cutoff values. Under standard smoothness conditions, we get an asymptotic efficiency result that is near optimal for a wide range of function classes. But the coverage is guaranteed whether or not the smoothness conditions hold and regardless of the sample size. The performance of our method is investigated through simulation studies and illustrated in a real data example.

In the context of density level set estimation, we study the convergence of general plug-in methods under two main assumptions on the density for a given level λ. More precisely, it is assumed that the density (i) is smooth in a neighborhood of λ and (ii) has γ-exponent at level λ. Condition (i) ensures that the density can be estimated at a standard nonparametric rate and condition (ii) is similar to Tsybakov's margin assumption which is stated for the classification framework. Under these assumptions, we derive optimal rates of convergence for plug-in estimators. Explicit convergence rates are given for plug-in estimators based on kernel density estimators when the underlying measure is the Lebesgue measure. Lower bounds proving optimality of the rates in a minimax sense when the density is Hölder smooth are also provided.

Consider the problem of estimating the γ-level set $G_{\\gamma}^{*} = {x : f(x) \\geq \\gamma}$ of an unknown d-dimensional density function f based on n independent observations $X_{1},..., X_{n}$ from the density. This problem has been addressed under global error criteria related to the symmetric set difference. However, in certain applications a spatially uniform mode of convergence is desirable to ensure that the estimated set is close to the target set everywhere. The Hausdorff error criterion provides this degree of uniformity and, hence, is more appropriate in such situations. It is known that the minimax optimal rate of error convergence for the Hausdorff metric is $(n/logn)^{-1/(d+2\\alpha)$ for level sets with boundaries that have a Lipschitz functional form, where the parameter α characterizes the regularity of the density around the level of interest. However, the estimators proposed in previous work are nonadaptive to the density regularity and require knowledge of the parameter α. Furthermore, previously developed estimators achieve the minimax optimal rate for rather restricted classes of sets (e.g., the boundary fragment and star-shaped sets) that effectively reduce the set estimation problem to a function estimation problem. This characterization precludes level sets with multiple connected components, which are fundamental to many applications. This paper presents a fully data-driven procedure that is adaptive to unknown regularity conditions and achieves near minimax optimal Hausdorff error control for a class of density level sets with very general shapes and multiple connected components.

We study kernel estimation of highest-density regions (HDR). Our main contributions are two-fold. First, we derive a uniform-in-bandwidth asymptotic approximation to a risk that is appropriate for HDR estimation. This approximation is then used to derive a bandwidth selection rule for HDR estimation possessing attractive asymptotic properties. We also present the results of numerical studies that illustrate the benefits of our theory and methodology.

A minimum volume set of a probability density is a region of minimum size among the regions covering a given probability mass of the density. Effective methods for finding the minimum volume sets are very useful for detecting failures or anomalies in commercial and security applications-a problem known as novelty detection . One theoretical approach of estimating the minimum volume set is to use a density level set where a kernel density estimator is plugged into the optimization problem that yields the appropriate level. Such a plug-in estimator is not of practical use because solving the corresponding minimization problem is usually intractable. A modified plug-in estimator was proposed by Hyndman in 1996 to overcome the computation difficulty of the theoretical approach but is not well studied in the literature. In this paper, we provide theoretical support to this estimator by showing its asymptotic consistency. We also show that this estimator is very competitive to other existing novelty detection methods through an extensive empirical study.

Stiffened structures have great potential for improving mechanical performance, and the study of their stability is of great interest. In this paper, the optimization of the critical buckling load factor for curved grid stiffeners is solved by using the level set based density method, where the shape and cross section (including thickness and width) of the stiffeners can be optimized simultaneously. The grid stiffeners are a combination of many single stiffeners which are projected by the corresponding level set functions. The thickness and width of each stiffener are designed to be independent variables in the projection applied to each level set function. Besides, the path of each single stiffener is described by the zero iso-contour of the level set function. All the single stiffeners are combined together by using the p-norm method to obtain the stiffener grid. The proposed method is validated by several numerical examples to optimize the critical buckling load factor.

Let X1, ..., Xnbe independent identically distributed observations from an unknown probability density f(·). Consider the problem of estimating the level set G = Gf(λ) = {x ∈ R2: f(x) ≥ λ} from the sample X1, ..., Xn, under the assumption that the boundary of G has a certain smoothness. We propose piecewise-polynomial estimators of G based on the maximization of local empirical excess masses. We show that the estimators have optimal rates of convergence in the asymptotically minimax sense within the studied classes of densities. We find also the optimal convergence rates for estimation of convex level sets. A generalization to the N-dimensional case, where$N > 2$, is given.

The level set and density methods for topology optimization are often perceived as two very different approaches. This has to some extent led to two competing research directions working in parallel with only little overlap and knowledge exchange. In this paper, we conjecture that this is a misconception and that the overlap and similarities are far greater than the differences. To verify this claim, we employ, without significant modifications, many of the base ingredients from the density method to construct a crisp interface level set optimization approach using a simple cut element method. That is, we use the same design field representation, the same projection filters, the same optimizer, and the same so-called robust approach as used in density-based optimization for length scale control. The only noticeable difference lies in the finite element and sensitivity analysis, here based on a cut element method, which provides an accurate tool to model arbitrary, crisp interfaces on a structured mesh based on the thresholding of a level set—or density—field. The presented work includes a heuristic hole generation scheme and we demonstrate the design approach on several numerical examples covering compliance minimization and a compliant force inverter. Finally, we provide our MATLAB code, downloadable from www.topopt.dtu.dk, to facilitate further extension of the proposed method to, e.g., multiphysics problems.

Topology optimization has undergone a tremendous development since its introduction in the seminal paper by Bendsøe and Kikuchi in 1988. By now, the concept is developing in many different directions, including “density”, “level set”, “topological derivative”, “phase field”, “evolutionary” and several others. The paper gives an overview, comparison and critical review of the different approaches, their strengths, weaknesses, similarities and dissimilarities and suggests guidelines for future research.

Topology optimization is the process of determining the optimal layout of material and connectivity inside a design domain. This paper surveys topology optimization of continuum structures from the year 2000 to 2012. It focuses on new developments, improvements, and applications of finite element-based topology optimization, which include a maturation of classical methods, a broadening in the scope of the field, and the introduction of new methods for multiphysics problems. Four different types of topology optimization are reviewed: (1) density-based methods, which include the popular Solid Isotropic Material with Penalization (SIMP) technique, (2) hard-kill methods, including Evolutionary Structural Optimization (ESO), (3) boundary variation methods (level set and phase field), and (4) a new biologically inspired method based on cellular division rules. We hope that this survey will provide an update of the recent advances and novel applications of popular methods, provide exposure to lesser known, yet promising, techniques, and serve as a resource for those new to the field. The presentation of each method’s focuses on new developments and novel applications.