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This paper develops and benchmarks an immersed peridynamics method to simulate the deformation, damage, and failure of hyperelastic materials within a fluid-structure interaction framework. The immersed peridynamics method describes an incompressible structure immersed in a viscous incompressible fluid. It expresses the momentum equation and incompressibility constraint in Eulerian form, and it describes the structural motion and resultant forces in Lagrangian form. Coupling between Eulerian and Lagrangian variables is achieved by integral transforms with Dirac delta function kernels, as in standard immersed boundary methods. The major difference between our approach and conventional immersed boundary methods is that we use peridynamics, instead of classical continuum mechanics, to determine the structural forces. We focus on non-ordinary state-based peridynamic material descriptions that allow us to use a constitutive correspondence framework that can leverage well characterized nonlinear constitutive models of soft materials. The convergence and accuracy of our approach are compared to both conventional and immersed finite element methods using widely used benchmark problems of nonlinear incompressible elasticity. We demonstrate that the immersed peridynamics method yields comparable accuracy with similar numbers of structural degrees of freedom for several choices of the size of the peridynamic horizon. We also demonstrate that the method can generate grid-converged simulations of fluid-driven material damage growth, crack formation and propagation, and rupture under large deformations.

We propose a novel learned keypoint detection method to increase the number of correct matches for the task of non-rigid image correspondence. By leveraging true correspondences acquired by matching annotated image pairs with a specified descriptor extractor, we train an end-to-end convolutional neural network (CNN) to find keypoint locations that are more appropriate to the considered descriptor. For that, we apply geometric and photometric warpings to images to generate a supervisory signal, allowing the optimization of the detector. Experiments demonstrate that our method enhances the Mean Matching Accuracy of numerous descriptors when used in conjunction with our detection method, while outperforming the state-of-the-art keypoint detectors on real images of non-rigid objects by 20 p.p. We also apply our method on the complex real-world task of object retrieval where our detector performs on par with the finest keypoint detectors currently available for this task. The source code and trained models are publicly available at //github.com/verlab/LearningToDetect_PRL_2023

A significant limitation of one-class classification anomaly detection methods is their reliance on the assumption that unlabeled training data only contains normal instances. To overcome this impractical assumption, we propose two novel classification-based anomaly detection methods. Firstly, we introduce a semi-supervised shallow anomaly detection method based on an unbiased risk estimator. Secondly, we present a semi-supervised deep anomaly detection method utilizing a nonnegative (biased) risk estimator. We establish estimation error bounds and excess risk bounds for both risk minimizers. Additionally, we propose techniques to select appropriate regularization parameters that ensure the nonnegativity of the empirical risk in the shallow model under specific loss functions. Our extensive experiments provide strong evidence of the effectiveness of the risk-based anomaly detection methods.

We introduce new control-volume finite-element discretization schemes suitable for solving the Stokes problem. Within a common framework, we present different approaches for constructing such schemes. The first and most established strategy employs a non-overlapping partitioning into control volumes. The second represents a new idea by splitting into two sets of control volumes, the first set yielding a partition of the domain and the second containing the remaining overlapping control volumes required for stability. The third represents a hybrid approach where finite volumes are combined with finite elements based on a hierarchical splitting of the ansatz space. All approaches are based on typical finite element function spaces but yield locally mass and momentum conservative discretization schemes that can be interpreted as finite volume schemes. We apply all strategies to the inf-sub stable MINI finite-element pair. Various test cases, including convergence tests and the numerical observation of the boundedness of the number of preconditioned Krylov solver iterations, as well as more complex scenarios of flow around obstacles or through a three-dimensional vessel bifurcation, demonstrate the stability and robustness of the schemes.

Fair calibration is a widely desirable fairness criteria in risk prediction contexts. One way to measure and achieve fair calibration is with multicalibration. Multicalibration constrains calibration error among flexibly-defined subpopulations while maintaining overall calibration. However, multicalibrated models can exhibit a higher percent calibration error among groups with lower base rates than groups with higher base rates. As a result, it is possible for a decision-maker to learn to trust or distrust model predictions for specific groups. To alleviate this, we propose \emph{proportional multicalibration}, a criteria that constrains the percent calibration error among groups and within prediction bins. We prove that satisfying proportional multicalibration bounds a model's multicalibration as well its \emph{differential calibration}, a fairness criteria that directly measures how closely a model approximates sufficiency. Therefore, proportionally calibrated models limit the ability of decision makers to distinguish between model performance on different patient groups, which may make the models more trustworthy in practice. We provide an efficient algorithm for post-processing risk prediction models for proportional multicalibration and evaluate it empirically. We conduct simulation studies and investigate a real-world application of PMC-postprocessing to prediction of emergency department patient admissions. We observe that proportional multicalibration is a promising criteria for controlling simultaneous measures of calibration fairness of a model over intersectional groups with virtually no cost in terms of classification performance.

This paper presents the error analysis of numerical methods on graded meshes for stochastic Volterra equations with weakly singular kernels. We first prove a novel regularity estimate for the exact solution via analyzing the associated convolution structure. This reveals that the exact solution exhibits an initial singularity in the sense that its H\"older continuous exponent on any neighborhood of $t=0$ is lower than that on every compact subset of $(0,T]$. Motivated by the initial singularity, we then construct the Euler--Maruyama method, fast Euler--Maruyama method, and Milstein method based on graded meshes. By establishing their pointwise-in-time error estimates, we give the grading exponents of meshes to attain the optimal uniform-in-time convergence orders, where the convergence orders improve those of the uniform mesh case. Numerical experiments are finally reported to confirm the sharpness of theoretical findings.

We propose a set of dependence measures that are non-linear, local, invariant to a wide range of transformations on the marginals, can show tail and risk asymmetries, are always well-defined, are easy to estimate and can be used on any dataset. We propose a nonparametric estimator and prove its consistency and asymptotic normality. Thereby we significantly improve on existing (extreme) dependence measures used in asset pricing and statistics. To show practical utility, we use these measures on high-frequency stock return data around market distress events such as the 2010 Flash Crash and during the GFC. Contrary to ubiquitously used correlations we find that our measures clearly show tail asymmetry, non-linearity, lack of diversification and endogenous buildup of risks present during these distress events. Additionally, our measures anticipate large (joint) losses during the Flash Crash while also anticipating the bounce back and flagging the subsequent market fragility. Our findings have implications for risk management, portfolio construction and hedging at any frequency.

Learning classification tasks of (2^nx2^n) inputs typically consist of \le n (2x2) max-pooling (MP) operators along the entire feedforward deep architecture. Here we show, using the CIFAR-10 database, that pooling decisions adjacent to the last convolutional layer significantly enhance accuracies. In particular, average accuracies of the advanced-VGG with m layers (A-VGGm) architectures are 0.936, 0.940, 0.954, 0.955, and 0.955 for m=6, 8, 14, 13, and 16, respectively. The results indicate A-VGG8s' accuracy is superior to VGG16s', and that the accuracies of A-VGG13 and A-VGG16 are equal, and comparable to that of Wide-ResNet16. In addition, replacing the three fully connected (FC) layers with one FC layer, A-VGG6 and A-VGG14, or with several linear activation FC layers, yielded similar accuracies. These significantly enhanced accuracies stem from training the most influential input-output routes, in comparison to the inferior routes selected following multiple MP decisions along the deep architecture. In addition, accuracies are sensitive to the order of the non-commutative MP and average pooling operators adjacent to the output layer, varying the number and location of training routes. The results call for the reexamination of previously proposed deep architectures and their accuracies by utilizing the proposed pooling strategy adjacent to the output layer.

We present a multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty. The algorithm is based on a collective smoother that at each iteration sweeps over the nodes of the computational mesh, and solves a reduced saddle-point system whose size depends on the number $N$ of samples used to discretized the probability space. We show that this reduced system can be solved with optimal $O(N)$ complexity. We test the multigrid method on three problems: a linear-quadratic problem for which the multigrid method is used to solve directly the linear optimality system; a nonsmooth problem with box constraints and $L^1$-norm penalization on the control, in which the multigrid scheme is used within a semismooth Newton iteration; a risk-adverse problem with the smoothed CVaR risk measure where the multigrid method is called within a preconditioned Newton iteration. In all cases, the multigrid algorithm exhibits very good performances and robustness with respect to all parameters of interest.

In the present paper, we formulate two versions of Frank--Wolfe algorithm or conditional gradient method to solve the DC optimization problem with an adaptive step size. The DC objective function consists of two components; the first is thought to be differentiable with a continuous Lipschitz gradient, while the second is only thought to be convex. The second version is based on the first and employs finite differences to approximate the gradient of the first component of the objective function. In contrast to past formulations that used the curvature/Lipschitz-type constant of the objective function, the step size computed does not require any constant associated with the components. For the first version, we established that the algorithm is well-defined of the algorithm and that every limit point of the generated sequence is a stationary point of the problem. We also introduce the class of weak-star-convex functions and show that, despite the fact that these functions are non-convex in general, the rate of convergence of the first version of the algorithm to minimize these functions is ${\cal O}(1/k)$. The finite difference used to approximate the gradient in the second version of the Frank-Wolfe algorithm is computed with the step-size adaptively updated using two previous iterations. Unlike previous applications of finite difference in the Frank-Wolfe algorithm, which provided approximate gradients with absolute error, the one used here provides us with a relative error, simplifying the algorithm analysis. In this case, we show that all limit points of the generated sequence for the second version of the Frank-Wolfe algorithm are stationary points for the problem under consideration, and we establish that the rate of convergence for the duality gap is ${\cal O}(1/\sqrt{k})$.