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This is Part II of our paper in which we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. In Part I of our paper [ChenHou2023a], we establish an analytic framework to prove stability of an approximate self-similar blowup profile by a combination of a weighted $L^\infty$ norm and a weighted $C^{1/2}$ norm. Under the assumption that the stability constants, which depend on the approximate steady state, satisfy certain inequalities stated in our stability lemma, we prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. In Part II of our paper, we provide sharp stability estimates of the linearized operator by constructing space-time solutions with rigorous error control. We also obtain sharp estimates of the velocity in the regular case using computer assistance. These results enable us to verify that the stability constants obtained in Part I [ChenHou2023a] indeed satisfy the inequalities in our stability lemma. This completes the analysis of the finite time singularity of the axisymmetric Euler equations with smooth initial data and boundary.

Feature screening is an important tool in analyzing ultrahigh-dimensional data, particularly in the field of Omics and oncology studies. However, most attention has been focused on identifying features that have a linear or monotonic impact on the response variable. Detecting a sparse set of variables that have a nonlinear or non-monotonic relationship with the response variable is still a challenging task. To fill the gap, this paper proposed a robust model-free screening approach for right-censored survival data by providing a new perspective of quantifying the covariate effect on the restricted mean survival time, rather than the routinely used hazard function. The proposed measure, based on the difference between the restricted mean survival time of covariate-stratified and overall data, is able to identify comprehensive types of associations including linear, nonlinear, non-monotone, and even local dependencies like change points. This approach is highly interpretable and flexible without any distribution assumption. The sure screening property is established and an iterative screening procedure is developed to address multicollinearity between high-dimensional covariates. Simulation studies are carried out to demonstrate the superiority of the proposed method in selecting important features with a complex association with the response variable. The potential of applying the proposed method to handle interval-censored failure time data has also been explored in simulations, and the results have been promising. The method is applied to a breast cancer dataset to identify potential prognostic factors, which reveals potential associations between breast cancer and lymphoma.

Recent advances in self-supervised learning and neural network scaling have enabled the creation of large models -- known as foundation models -- which can be easily adapted to a wide range of downstream tasks. The current paradigm for comparing foundation models involves benchmarking them with aggregate metrics on various curated datasets. Unfortunately, this method of model comparison is heavily dependent on the choice of metric, which makes it unsuitable for situations where the ideal metric is either not obvious or unavailable. In this work, we present a metric-free methodology for comparing foundation models via their embedding space geometry. Our methodology is grounded in random graph theory, and facilitates both pointwise and multi-model comparison. Further, we demonstrate how our framework can be used to induce a manifold of models equipped with a distance function that correlates strongly with several downstream metrics.

Over the past decade, 3D graphics have become highly detailed to mimic the real world, exploding their size and complexity. Certain applications and device constraints necessitate their simplification and/or lossy compression, which can degrade their visual quality. Thus, to ensure the best Quality of Experience (QoE), it is important to evaluate the visual quality to accurately drive the compression and find the right compromise between visual quality and data size. In this work, we focus on subjective and objective quality assessment of textured 3D meshes. We first establish a large-scale dataset, which includes 55 source models quantitatively characterized in terms of geometric, color, and semantic complexity, and corrupted by combinations of 5 types of compression-based distortions applied on the geometry, texture mapping and texture image of the meshes. This dataset contains over 343k distorted stimuli. We propose an approach to select a challenging subset of 3000 stimuli for which we collected 148929 quality judgments from over 4500 participants in a large-scale crowdsourced subjective experiment. Leveraging our subject-rated dataset, a learning-based quality metric for 3D graphics was proposed. Our metric demonstrates state-of-the-art results on our dataset of textured meshes and on a dataset of distorted meshes with vertex colors. Finally, we present an application of our metric and dataset to explore the influence of distortion interactions and content characteristics on the perceived quality of compressed textured meshes.

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to full discrete conservation of mass, squared density, momentum, angular momentum and kinetic energy without the divergence-free constraint being strongly enforced. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.

Cholesky factorization is a widely used method for solving linear systems involving symmetric, positive-definite matrices, and can be an attractive choice in applications where a high degree of numerical stability is needed. One such application is numerical optimization, where direct methods for solving linear systems are widely used and often a significant performance bottleneck. An example where this is the case, and the specific type of optimization problem motivating this work, is radiation therapy treatment planning, where numerical optimization is used to create individual treatment plans for patients. To address this bottleneck, we propose a task-based multi-threaded method for Cholesky factorization of banded matrices with medium-sized bands. We implement our algorithm using OpenMP tasks and compare our performance with state-of-the-art libraries such as Intel MKL. Our performance measurements show a performance that is on par or better than Intel MKL (up to ~26%) for a wide range of matrix bandwidths on two different Intel CPU systems.

Defect prediction is crucial for software quality assurance and has been extensively researched over recent decades. However, prior studies rarely focus on data complexity in defect prediction tasks, and even less on understanding the difficulties of these tasks from the perspective of data complexity. In this paper, we conduct an empirical study to estimate the hardness of over 33,000 instances, employing a set of measures to characterize the inherent difficulty of instances and the characteristics of defect datasets. Our findings indicate that: (1) instance hardness in both classes displays a right-skewed distribution, with the defective class exhibiting a more scattered distribution; (2) class overlap is the primary factor influencing instance hardness and can be characterized through feature, structural, instance, and multiresolution overlap; (3) no universal preprocessing technique is applicable to all datasets, and it may not consistently reduce data complexity, fortunately, dataset complexity measures can help identify suitable techniques for specific datasets; (4) integrating data complexity information into the learning process can enhance an algorithm's learning capacity. In summary, this empirical study highlights the crucial role of data complexity in defect prediction tasks, and provides a novel perspective for advancing research in defect prediction techniques.

We study the computational scalability of a Gaussian process (GP) framework for solving general nonlinear partial differential equations (PDEs). This framework transforms solving PDEs to solving quadratic optimization problem with nonlinear constraints. Its complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel of the GP and its partial derivatives at collocation points. We present a sparse Cholesky factorization algorithm for such kernel matrices based on the near-sparsity of the Cholesky factor under a new ordering of Diracs and derivative measurements. We rigorously identify the sparsity pattern and quantify the exponentially convergent accuracy of the corresponding Vecchia approximation of the GP, which is optimal in the Kullback-Leibler divergence. This enables us to compute $\epsilon$-approximate inverse Cholesky factors of the kernel matrices with complexity $O(N\log^d(N/\epsilon))$ in space and $O(N\log^{2d}(N/\epsilon))$ in time. With the sparse factors, gradient-based optimization methods become scalable. Furthermore, we can use the oftentimes more efficient Gauss-Newton method, for which we apply the conjugate gradient algorithm with the sparse factor of a reduced kernel matrix as a preconditioner to solve the linear system. We numerically illustrate our algorithm's near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Amp\`ere equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs.

The aim of this paper is to study the shape optimization method for solving the Bernoulli free boundary problem, a well-known ill-posed problem that seeks the unknown free boundary through Cauchy data. Different formulations have been proposed in the literature that differ in the choice of the objective functional. Specifically, it was shown respectively in [14] and [16] that tracking Neumann data is well-posed but tracking Dirichlet data is not. In this paper we propose a new well-posed objective functional that tracks Dirichlet data at the free boundary. By calculating the Euler derivative and the shape Hessian of the objective functional we show that the new formulation is well-posed, i.e., the shape Hessian is coercive at the minimizers. The coercivity of the shape Hessian may ensure the existence of optimal solutions for the nonlinear Ritz-Galerkin approximation method and its convergence, thus is crucial for the formulation. As a summary, we conclude that tracking Dirichlet or Neumann data in its energy norm is not sufficient, but tracking it in a half an order higher norm will be well-posed. To support our theoretical results we carry out extensive numerical experiments.