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The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle point formulation is well established since many decades. This topic was mostly studied for variational formulations defined upon the same product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever either these spaces or the two bilinear forms involving the multiplier are distinct, the saddle point problem is asymmetric. The three inf-sup conditions to be satisfied by the product spaces stipulated in work on the subject, in order to guarantee well-posedness, are well known. However, the material encountered in the literature addressing the approximation of this class of problems left room for improvement and clarifications. After making a brief review of the existing contributions to the topic that justifies such an assertion, in this paper we set up finer global error bounds for the pair primal variable-multiplier solving an asymmetric saddle point problem. Besides well-posedness, the three constants in the aforementioned inf-sup conditions are identified as all that is needed for determining the stability constant appearing therein, whose expression is exhibited. As a complement, refined error bounds depending only on these three constants are given for both unknowns separately.

This paper investigates extremal quantiles under two-way cluster dependence. We demonstrate that the limiting distribution of the unconditional intermediate order quantiles in the tails converges to a Gaussian distribution. This is remarkable as two-way cluster dependence entails potential non-Gaussianity in general, but extremal quantiles do not suffer from this issue. Building upon this result, we extend our analysis to extremal quantile regressions of intermediate order.

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in [43]. Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number $\kappa\geq 1$ of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter $\lambda$ and on $\kappa$. Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as $\lambda\downarrow-\infty$, in the regions where the weight vanishes. Our result leads us to conjecture the existence of $2^{\kappa+1}-1$ solutions for sufficiently negative $\lambda$. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in $\lambda$ and the profiles of positive solutions. Finally, we give additional numerical results suggesting that the same high multiplicity result holds true for a much larger class of weights, also arbitrarily close to situations where there is uniqueness of positive solutions.

Exceptionally elegant formulae exist for the fractional Laplacian operator applied to weighted classical orthogonal polynomials. We utilize these results to construct a solver, based on frame properties, for equations involving the fractional Laplacian of any power, $s \in (0,1)$, on an unbounded domain in one or two dimensions. The numerical method represents solutions in an expansion of weighted classical orthogonal polynomials as well as their unweighted counterparts with a specific extension to $\mathbb{R}^d$, $d \in \{1,2\}$. We examine the frame properties of this family of functions for the solution expansion and, under standard frame conditions, derive an a priori estimate for the stationary equation. Moreover, we prove one achieves the expected order of convergence when considering an implicit Euler discretization in time for the fractional heat equation. We apply our solver to numerous examples including the fractional heat equation (utilizing up to a $6^\text{th}$-order Runge--Kutta time discretization), a fractional heat equation with a time-dependent exponent $s(t)$, and a two-dimensional problem, observing spectral convergence in the spatial dimension for sufficiently smooth data.

We present a new method to compute the solution to a nonlinear tensor differential equation with dynamical low-rank approximation. The idea of dynamical low-rank approximation is to project the differential equation onto the tangent space of a low-rank tensor manifold at each time. Traditionally, an orthogonal projection onto the tangent space is employed, which is challenging to compute for nonlinear differential equations. We introduce a novel interpolatory projection onto the tangent space that is easily computed for many nonlinear differential equations and satisfies the differential equation at a set of carefully selected indices. To select these indices, we devise a new algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes any tensor train and its tangent space with tensor cross interpolants. We demonstrate the proposed method with applications to tensor differential equations arising from the discretization of partial differential equations.

Fully-strict fork-join parallelism is a powerful model for shared-memory programming due to its optimal time scaling and strong bounds on memory scaling. The latter is rarely achieved due to the difficulty of implementing continuation stealing in traditional High Performance Computing (HPC) languages -- where it is often impossible without modifying the compiler or resorting to non-portable techniques. We demonstrate how stackless coroutines (a new feature in C++20) can enable fully-portable continuation stealing and present libfork a lock-free fine-grained parallelism library, combining coroutines with user-space, geometric segmented-stacks. We show our approach is able to achieve optimal time/memory scaling, both theoretically and empirically, across a variety of benchmarks. Compared to openMP (libomp), libfork is on average 7.2x faster and consumes 10x less memory. Similarly, compared to Intel's TBB, libfork is on average 2.7x faster and consumes 6.2x less memory. Additionally, we introduce non-uniform memory access (NUMA) optimizations for schedulers that demonstrate performance matching busy-waiting schedulers.

This paper presents a scalable physics-based block preconditioner for mixed-dimensional models in beam-solid interaction and their application in engineering. In particular, it studies the linear systems arising from a regularized mortar-type approach for embedding geometrically exact beams into solid continua. Due to the lack of block diagonal dominance of the arising 2 x 2 block system, an approximate block factorization preconditioner is used. It exploits the sparsity structure of the beam sub-block to construct a sparse approximate inverse, which is then not only used to explicitly form an approximation of the Schur complement, but also acts as a smoother within the prediction step of the arising SIMPLE-type preconditioner. The correction step utilizes an algebraic multigrid method. Although, for now, the beam sub-block is tackled by a one-level method only, the multi-level nature of the computationally demanding correction step delivers a scalable preconditioner in practice. In numerical test cases, the influence of different algorithmic parameters on the quality of the sparse approximate inverse is studied and the weak scaling behavior of the proposed preconditioner on up to 1000 MPI ranks is demonstrated, before the proposed preconditioner is finally applied for the analysis of steel-reinforced concrete structures in civil engineering.

With advances in scientific computing and mathematical modeling, complex scientific phenomena such as galaxy formations and rocket propulsion can now be reliably simulated. Such simulations can however be very time-intensive, requiring millions of CPU hours to perform. One solution is multi-fidelity emulation, which uses data of different fidelities to train an efficient predictive model which emulates the expensive simulator. For complex scientific problems and with careful elicitation from scientists, such multi-fidelity data may often be linked by a directed acyclic graph (DAG) representing its scientific model dependencies. We thus propose a new Graphical Multi-fidelity Gaussian Process (GMGP) model, which embeds this DAG structure (capturing scientific dependencies) within a Gaussian process framework. We show that the GMGP has desirable modeling traits via two Markov properties, and admits a scalable algorithm for recursive computation of the posterior mean and variance along at each depth level of the DAG. We also present a novel experimental design methodology over the DAG given an experimental budget, and propose a nonlinear extension of the GMGP via deep Gaussian processes. The advantages of the GMGP are then demonstrated via a suite of numerical experiments and an application to emulation of heavy-ion collisions, which can be used to study the conditions of matter in the Universe shortly after the Big Bang. The proposed model has broader uses in data fusion applications with graphical structure, which we further discuss.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.

Deep learning is usually described as an experiment-driven field under continuous criticizes of lacking theoretical foundations. This problem has been partially fixed by a large volume of literature which has so far not been well organized. This paper reviews and organizes the recent advances in deep learning theory. The literature is categorized in six groups: (1) complexity and capacity-based approaches for analyzing the generalizability of deep learning; (2) stochastic differential equations and their dynamic systems for modelling stochastic gradient descent and its variants, which characterize the optimization and generalization of deep learning, partially inspired by Bayesian inference; (3) the geometrical structures of the loss landscape that drives the trajectories of the dynamic systems; (4) the roles of over-parameterization of deep neural networks from both positive and negative perspectives; (5) theoretical foundations of several special structures in network architectures; and (6) the increasingly intensive concerns in ethics and security and their relationships with generalizability.