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It is known that standard stochastic Galerkin methods encounter challenges when solving partial differential equations with high-dimensional random inputs, which are typically caused by the large number of stochastic basis functions required. It becomes crucial to properly choose effective basis functions, such that the dimension of the stochastic approximation space can be reduced. In this work, we focus on the stochastic Galerkin approximation associated with generalized polynomial chaos (gPC), and explore the gPC expansion based on the analysis of variance (ANOVA) decomposition. A concise form of the gPC expansion is presented for each component function of the ANOVA expansion, and an adaptive ANOVA procedure is proposed to construct the overall stochastic Galerkin system. Numerical results demonstrate the efficiency of our proposed adaptive ANOVA stochastic Galerkin method for both diffusion and Helmholtz problems.

For finite element approximations of transport phenomena, it is often necessary to apply a form of limiting to ensure that the discrete solution remains well-behaved and satisfies physical constraints. However, these limiting procedures are typically performed at discrete nodal locations, which is not sufficient to ensure the robustness of the scheme when the solution must be evaluated at arbitrary locations (e.g., for adaptive mesh refinement, remapping in arbitrary Lagragian--Eulerian solvers, overset meshes, etc.). In this work, a novel limiting approach for discontinuous Galerkin methods is presented which ensures that the solution is continuously bounds-preserving (i.e., across the entire solution polynomial) for any arbitrary choice of basis, approximation order, and mesh element type. Through a modified formulation for the constraint functionals, the proposed approach requires only the solution of a single spatial scalar minimization problem per element for which a highly efficient numerical optimization procedure is presented. The efficacy of this approach is shown in numerical experiments by enforcing continuous constraints in high-order unstructured discontinuous Galerkin discretizations of hyperbolic conservation laws, ranging from scalar transport with maximum principle preserving constraints to compressible gas dynamics with positivity-preserving constraints.

We study least-squares trace regression when the parameter is the sum of a $r$-low-rank matrix and a $s$-sparse matrix and a fraction $\epsilon$ of the labels is corrupted. For subgaussian distributions and feature-dependent noise, we highlight three needed design properties, each one derived from a different process inequality: a "product process inequality", "Chevet's inequality" and a "multiplier process inequality". These properties handle, simultaneously, additive decomposition, label contamination and design-noise interaction. They imply the near-optimality of a tractable estimator with respect to the effective dimensions $d_{eff,r}$ and $d_{eff,s}$ of the low-rank and sparse components, $\epsilon$ and the failure probability $\delta$. The near-optimal rate is $\mathsf{r}(n,d_{eff,r}) + \mathsf{r}(n,d_{eff,s}) + \sqrt{(1+\log(1/\delta))/n} + \epsilon\log(1/\epsilon)$, where $\mathsf{r}(n,d_{eff,r})+\mathsf{r}(n,d_{eff,s})$ is the optimal rate in average with no contamination. Our estimator is adaptive to $(s,r,\epsilon,\delta)$ and, for fixed absolute constant $c>0$, it attains the mentioned rate with probability $1-\delta$ uniformly over all $\delta\ge\exp(-cn)$. Without matrix decomposition, our analysis also entails optimal bounds for a robust estimator adapted to the noise variance. Our estimators are based on "sorted" versions of Huber's loss. We present simulations matching the theory. In particular, it reveals the superiority of "sorted" Huber's losses over the classical Huber's loss.

Microaggregation is a method to coarsen a dataset, by optimally clustering data points in groups of at least $k$ points, thereby providing a $k$-anonymity type disclosure guarantee for each point in the dataset. Previous algorithms for univariate microaggregation had a $O(k n)$ time complexity. By rephrasing microaggregation as an instance of the concave least weight subsequence problem, in this work we provide improved algorithms that provide an optimal univariate microaggregation on sorted data in $O(n)$ time and space. We further show that our algorithms work not only for sum of squares cost functions, as typically considered, but seamlessly extend to many other cost functions used for univariate microaggregation tasks. In experiments we show that the presented algorithms lead to real world performance improvements.

In the Bayes paradigm and for a given loss function, we propose the construction of a new type of posterior distributions, that extends the classical Bayes one, for estimating the law of an $n$-sample. The loss functions we have in mind are based on the total variation and Hellinger distances as well as some $\mathbb{L}_{j}$-ones. We prove that, with a probability close to one, this new posterior distribution concentrates its mass in a neighbourhood of the law of the data, for the chosen loss function, provided that this law belongs to the support of the prior or, at least, lies close enough to it. We therefore establish that the new posterior distribution enjoys some robustness properties with respect to a possible misspecification of the prior, or more precisely, its support. For the total variation and squared Hellinger losses, we also show that the posterior distribution keeps its concentration properties when the data are only independent, hence not necessarily i.i.d., provided that most of their marginals or the average of these are close enough to some probability distribution around which the prior puts enough mass. The posterior distribution is therefore also stable with respect to the equidistribution assumption. We illustrate these results by several applications. We consider the problems of estimating a location parameter or both the location and the scale of a density in a nonparametric framework. Finally, we also tackle the problem of estimating a density, with the squared Hellinger loss, in a high-dimensional parametric model under some sparsity conditions. The results established in this paper are non-asymptotic and provide, as much as possible, explicit constants.

The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In settings with large coarsest-level problems, approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using these results, we derive coarsest-level stopping criterion through which we may control the difference between the approximation computed by a V-cycle method with approximate coarsest-level solver and the approximation which would be computed if the coarsest-level problems were solved exactly. The coarsest-level stopping criterion may thus be set up such that the V-cycle method converges to a chosen finest-level accuracy in (nearly) the same number of V-cycle iterations as the V-cycle method with exact coarsest-level solver. We also utilize the theoretical results to discuss how the convergence of the V-cycle method may be affected by the choice of a tolerance in a coarsest-level stopping criterion based on the relative residual norm.

There is a folkloric belief that a depth-$\Theta(m)$ quantum circuit is needed to estimate the trace of the product of $m$ density matrices (i.e., a multivariate trace), a subroutine crucial to applications in condensed matter and quantum information science. We prove that this belief is overly conservative by constructing a constant quantum-depth circuit for the task, inspired by the method of Shor error correction. Furthermore, our circuit demands only local gates in a two dimensional circuit -- we show how to implement it in a highly parallelized way on an architecture similar to that of Google's Sycamore processor. With these features, our algorithm brings the central task of multivariate trace estimation closer to the capabilities of near-term quantum processors. We instantiate the latter application with a theorem on estimating nonlinear functions of quantum states with "well-behaved" polynomial approximations.

We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Gr\"onwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have any restrictions on the step size ratio. The Gr\"onwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Gr\"onwall inequalities are then applied to subdiffusion problems and the time fractional Allen-Cahn equations for illustration.

The Ising model is important in statistical modeling and inference in many applications, however its normalizing constant, mean number of active vertices and mean spin interaction -- quantities needed in inference -- are computationally intractable. We provide accurate approximations that make it possible to numerically calculate these quantities in the homogeneous case. Simulation studies indicate good performance of our approximation formulae that are scalable and unfazed by the size (number of nodes, degree of graph) of the Markov Random Field. The practical import of our approximation formulae is illustrated in performing Bayesian inference in a functional Magnetic Resonance Imaging activation detection experiment, and also in likelihood ratio testing for anisotropy in the spatial patterns of yearly increases in pistachio tree yields.

In this contribution we deal with Gaussian quadrature rules based on orthogonal polynomials associated with a weight function $w(x)= x^{\alpha} e^{-x}$ supported on an interval $(0,z)$, $z>0.$ The modified Chebyshev algorithm is used in order to test the accuracy in the computation of the coefficients of the three-term recurrence relation, the zeros and weights, as well as the dependence on the parameter $z.$