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The calibration of constitutive models from full-field data has recently gained increasing interest due to improvements in full-field measurement capabilities. In addition to the experimental characterization of novel materials, continuous structural health monitoring is another application that is of great interest. However, monitoring is usually associated with severe time constraints, difficult to meet with standard numerical approaches. Therefore, parametric physics-informed neural networks (PINNs) for constitutive model calibration from full-field displacement data are investigated. In an offline stage, a parametric PINN can be trained to learn a parameterized solution of the underlying partial differential equation. In the subsequent online stage, the parametric PINN then acts as a surrogate for the parameters-to-state map in calibration. We test the proposed approach for the deterministic least-squares calibration of a linear elastic as well as a hyperelastic constitutive model from noisy synthetic displacement data. We further carry out Markov chain Monte Carlo-based Bayesian inference to quantify the uncertainty. A proper statistical evaluation of the results underlines the high accuracy of the deterministic calibration and that the estimated uncertainty is valid. Finally, we consider experimental data and show that the results are in good agreement with a Finite Element Method-based calibration. Due to the fast evaluation of PINNs, calibration can be performed in near real-time. This advantage is particularly evident in many-query applications such as Markov chain Monte Carlo-based Bayesian inference.

Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate such notions from Turing machines to uncountable spaces. Since these machines are used as a baseline for computability in these approaches, countability restrictions on the ordered structures are fundamental. Here, we show several relations between the usual countability restrictions in order-theoretic theories of computability and some more common order-theoretic countability constraints, like order density properties and functional characterizations of the order structure in terms of multi-utilities. As a result, we show how computability can be introduced in some order structures via countability order density and multi-utility constraints.

We consider stochastic gradient descents on the space of large symmetric matrices of suitable functions that are invariant under permuting the rows and columns using the same permutation. We establish deterministic limits of these random curves as the dimensions of the matrices go to infinity while the entries remain bounded. Under a ``small noise'' assumption the limit is shown to be the gradient flow of functions on graphons whose existence was established in~\cite{oh2021gradient}. We also consider limits of stochastic gradient descents with added properly scaled reflected Brownian noise. The limiting curve of graphons is characterized by a family of stochastic differential equations with reflections and can be thought of as an extension of the classical McKean-Vlasov limit for interacting diffusions to the graphon setting. The proofs introduce a family of infinite-dimensional exchangeable arrays of reflected diffusions and a novel notion of propagation of chaos for large matrices of diffusions converging to such arrays in a suitable sense.

We propose a coefficient that measures dependence in paired samples of functions. It has properties similar to the Pearson correlation, but differs in significant ways: 1) it is designed to measure dependence between curves, 2) it focuses only on extreme curves. The new coefficient is derived within the framework of regular variation in Banach spaces. A consistent estimator is proposed and justified by an asymptotic analysis and a simulation study. The usefulness of the new coefficient is illustrated on financial and and climate functional data.

Delamination is a critical mode of failure that occurs between plies in a composite laminate. The cohesive element, developed based on the cohesive zone model, is widely used for modeling delamination. However, standard cohesive elements suffer from a well-known limit on the mesh density-the element size must be much smaller than the cohesive zone size. This work develops a new set of elements for modelling composite plies and their interfaces in 3D. A triangular Kirchhoff-Love shell element is developed for orthotropic materials to model the plies. A structural cohesive element, conforming to the shell elements of the plies, is developed to model the interface delamination. The proposed method is verified and validated on the classical benchmark problems of Mode I, Mode II, and mixed-mode delamination of unidirectional laminates, as well as on the single-leg bending problem of a multi-directional laminate. All the results show that the element size in the proposed models can be ten times larger than that in the standard cohesive element models, with more than 90% reduction in CPU time, while retaining prediction accuracy. This would then allow more effective and efficient modeling of delamination in composites without worrying about the cohesive zone limit on the mesh density.

The normal-inverse-Wishart (NIW) distribution is commonly used as a prior distribution for the mean and covariance parameters of a multivariate normal distribution. The family of NIW distributions is also a minimal exponential family. In this short note we describe a convergent procedure for converting from mean parameters to natural parameters in the NIW family, or -- equivalently -- for performing maximum likelihood estimation of the natural parameters given observed sufficient statistics. This is needed, for example, when using a NIW base family in expectation propagation

This research conducts a thorough reevaluation of seismic fragility curves by utilizing ordinal regression models, moving away from the commonly used log-normal distribution function known for its simplicity. It explores the nuanced differences and interrelations among various ordinal regression approaches, including Cumulative, Sequential, and Adjacent Category models, alongside their enhanced versions that incorporate category-specific effects and variance heterogeneity. The study applies these methodologies to empirical bridge damage data from the 2008 Wenchuan earthquake, using both frequentist and Bayesian inference methods, and conducts model diagnostics using surrogate residuals. The analysis covers eleven models, from basic to those with heteroscedastic extensions and category-specific effects. Through rigorous leave-one-out cross-validation, the Sequential model with category-specific effects emerges as the most effective. The findings underscore a notable divergence in damage probability predictions between this model and conventional Cumulative probit models, advocating for a substantial transition towards more adaptable fragility curve modeling techniques that enhance the precision of seismic risk assessments. In conclusion, this research not only readdresses the challenge of fitting seismic fragility curves but also advances methodological standards and expands the scope of seismic fragility analysis. It advocates for ongoing innovation and critical reevaluation of conventional methods to advance the predictive accuracy and applicability of seismic fragility models within the performance-based earthquake engineering domain.

Approximating invariant subspaces of generalized eigenvalue problems (GEPs) is a fundamental computational problem at the core of machine learning and scientific computing. It is, for example, the root of Principal Component Analysis (PCA) for dimensionality reduction, data visualization, and noise filtering, and of Density Functional Theory (DFT), arguably the most popular method to calculate the electronic structure of materials. For a Hermitian definite GEP $HC=SC\Lambda$, let $\Pi_k$ be the true spectral projector on the invariant subspace that is associated with the $k$ smallest (or largest) eigenvalues. Given $H,$ $S$, an integer $k$, and accuracy $\varepsilon\in(0,1)$, we show that we can compute a matrix $\widetilde\Pi_k$ such that $\lVert\Pi_k-\widetilde\Pi_k\rVert_2\leq \varepsilon$, in $O\left( n^{\omega+\eta}\mathrm{polylog}(n,\varepsilon^{-1},\kappa(S),\mathrm{gap}_k^{-1}) \right)$ bit operations in the floating point model with probability $1-1/n$. Here, $\eta>0$ is arbitrarily small, $\omega\lesssim 2.372$ is the matrix multiplication exponent, $\kappa(S)=\lVert S\rVert_2\lVert S^{-1}\rVert_2$, and $\mathrm{gap}_k$ is the gap between eigenvalues $k$ and $k+1$. To the best of our knowledge, this is the first end-to-end analysis achieving such "forward-error" approximation guarantees with nearly $O(n^{\omega+\eta})$ bit complexity, improving classical $\widetilde O(n^3)$ eigensolvers, even for the regular case $(S=I)$. Our methods rely on a new $O(n^{\omega+\eta})$ stability analysis for the Cholesky factorization, and a new smoothed analysis for computing spectral gaps, which can be of independent interest. Ultimately, we obtain new matrix multiplication-type bit complexity upper bounds for PCA problems, including classical PCA and (randomized) low-rank approximation.

Lattice structures have been widely used in applications due to their superior mechanical properties. To fabricate such structures, a geometric processing step called triangulation is often employed to transform them into the STL format before sending them to 3D printers. Because lattice structures tend to have high geometric complexity, this step usually generates a large amount of triangles, a memory and compute-intensive task. This problem manifests itself clearly through large-scale lattice structures that have millions or billions of struts. To address this problem, this paper proposes to transform a lattice structure into an intermediate model called meta-mesh before undergoing real triangulation. Compared to triangular meshes, meta-meshes are very lightweight and much less compute-demanding. The meta-mesh can also work as a base mesh reusable for conveniently and efficiently triangulating lattice structures with arbitrary resolutions. A CPU+GPU asynchronous meta-meshing pipeline has been developed to efficiently generate meta-meshes from lattice structures. It shifts from the thread-centric GPU algorithm design paradigm commonly used in CAD to the recent warp-centric design paradigm to achieve high performance. This is achieved by a new data compression method, a GPU cache-aware data structure, and a workload-balanced scheduling method that can significantly reduce memory divergence and branch divergence. Experimenting with various billion-scale lattice structures, the proposed method is seen to be two orders of magnitude faster than previously achievable.

Inference for functional linear models in the presence of heteroscedastic errors has received insufficient attention given its practical importance; in fact, even a central limit theorem has not been studied in this case. At issue, conditional mean estimates have complicated sampling distributions due to the infinite dimensional regressors, where truncation bias and scaling issues are compounded by non-constant variance under heteroscedasticity. As a foundation for distributional inference, we establish a central limit theorem for the estimated conditional mean under general dependent errors, and subsequently we develop a paired bootstrap method to provide better approximations of sampling distributions. The proposed paired bootstrap does not follow the standard bootstrap algorithm for finite dimensional regressors, as this version fails outside of a narrow window for implementation with functional regressors. The reason owes to a bias with functional regressors in a naive bootstrap construction. Our bootstrap proposal incorporates debiasing and thereby attains much broader validity and flexibility with truncation parameters for inference under heteroscedasticity; even when the naive approach may be valid, the proposed bootstrap method performs better numerically. The bootstrap is applied to construct confidence intervals for centered projections and for conducting hypothesis tests for the multiple conditional means. Our theoretical results on bootstrap consistency are demonstrated through simulation studies and also illustrated with a real data example.