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For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-$p$ setting, and answer a question of Talagrand on the analogous result for general positive empirical processes.

It is well known that for singular inconsistent range-symmetric linear systems, the generalized minimal residual (GMRES) method determines a least squares solution without breakdown. The reached least squares solution may be or not be the pseudoinverse solution. We show that a lift strategy can be used to obtain the pseudoinverse solution. In addition, we propose a new iterative method named RSMAR (minimum $\mathbf A$-residual) for range-symmetric linear systems $\mathbf A\mathbf x=\mathbf b$. At step $k$ RSMAR minimizes $\|\mathbf A\mathbf r_k\|$ in the $k$th Krylov subspace generated with $\{\mathbf A, \mathbf r_0\}$ rather than $\|\mathbf r_k\|$, where $\mathbf r_k$ is the $k$th residual vector and $\|\cdot\|$ denotes the Euclidean vector norm. We show that RSMAR and GMRES terminate with the same least squares solution when applied to range-symmetric linear systems. We provide two implementations for RSMAR. Our numerical experiments show that RSMAR is the most suitable method among GMRES-type methods for singular inconsistent range-symmetric linear systems.

We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon (FRS) codes can be made to run in nearly-linear time. This yields, to our knowledge, the first known family of codes that can be decoded in nearly linear time, even as they approach the list decoding capacity. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list-decoding. It is known that for every $\epsilon >0$, and rate $R \in (0,1)$, there exist explicit families of these codes that have rate $R$ and can be list-decoded from a $(1-R-\epsilon)$ fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory, 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above tasks in nearly linear time. Our algorithms have two main components. The first builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a nearly linear time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design nearly-linear time algorithms for two natural algebraic problems. The first algorithm solves linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ where $Q$ has the form $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i = 0}^m Q_i(x)\cdot y_i$. The second solves functional equations of the form $Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$ where $\gamma$ is a high-order field element. These algorithms can be viewed as generalizations of classical algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.

The Boundary Element Method (BEM) is implemented using piecewise linear elements to solve the two-dimensional Dirichlet problem for Laplace's equation posed on a disk. A benefit of the BEM as opposed to many other numerical solution techniques is that discretization only occurs on the boundary, i.e., the complete domain does not need to be discretized. This provides an advantage in terms of time and cost. The algorithm's performance is illustrated through sample test problems with known solutions. A comparison between the exact solution and the BEM numerical solution is done, and error analysis is performed on the results.

The Concordance Index (C-index) is a commonly used metric in Survival Analysis for evaluating the performance of a prediction model. In this paper, we propose a decomposition of the C-index into a weighted harmonic mean of two quantities: one for ranking observed events versus other observed events, and the other for ranking observed events versus censored cases. This decomposition enables a finer-grained analysis of the relative strengths and weaknesses between different survival prediction methods. The usefulness of this decomposition is demonstrated through benchmark comparisons against classical models and state-of-the-art methods, together with the new variational generative neural-network-based method (SurVED) proposed in this paper. The performance of the models is assessed using four publicly available datasets with varying levels of censoring. Using the C-index decomposition and synthetic censoring, the analysis shows that deep learning models utilize the observed events more effectively than other models. This allows them to keep a stable C-index in different censoring levels. In contrast to such deep learning methods, classical machine learning models deteriorate when the censoring level decreases due to their inability to improve on ranking the events versus other events.

The need to Fourier transform data sets with irregular sampling is shared by various domains of science. This is the case for example in astronomy or sismology. Iterative methods have been developed that allow to reach approximate solutions. Here an exact solution to the problem for band-limited periodic signals is presented. The exact spectrum can be deduced from the spectrum of the non-equispaced data through the inversion of a Toeplitz matrix. The result applies to data of any dimension. This method also provides an excellent approximation for non-periodic band-limit signals. The method allows to reach very high dynamic ranges ($10^{13}$ with double-float precision) which depend on the regularity of the samples.

Dynamical systems across the sciences, from electrical circuits to ecological networks, undergo qualitative and often catastrophic changes in behavior, called bifurcations, when their underlying parameters cross a threshold. Existing methods predict oncoming catastrophes in individual systems but are primarily time-series-based and struggle both to categorize qualitative dynamical regimes across diverse systems and to generalize to real data. To address this challenge, we propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries based on the extraction of topologically invariant features. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a wide range of applications. Our convolutional attention method is trained with data augmentations that encourage the learning of topological invariants which can be used to detect bifurcation boundaries in unseen systems and to design models of biological systems like oscillatory gene regulatory networks. We further demonstrate our method's use in analyzing real data by recovering distinct proliferation and differentiation dynamics along pancreatic endocrinogenesis trajectory in gene expression space based on single-cell data. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.

A deep learning initialized iterative (Int-Deep) method is developed for numerically solving Navier-Stokes Darcy model. For this purpose, Newton iterative method is mentioned for solving the relative finite element discretized problem. It is proved that this method converges quadratically with the convergence rate independent of the finite element mesh size under certain standard conditions. Later on, a deep learning algorithm is proposed for solving this nonlinear coupled problem. Following the ideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithm is constructed for the previous problem in order to further improve the computational efficiency. A series of numerical examples are reported to confirm that the Int-Deep algorithm converges to the true solution rapidly and is robust with respect to the physical parameters in the model.

This paper analyzes the stability of the class of Time-Accurate and Highly-Stable Explicit Runge-Kutta (TASE-RK) methods, introduced in 2021 by Bassenne et al. (J. Comput. Phys.) for the numerical solution of stiff Initial Value Problems (IVPs). Such numerical methods are easy to implement and require the solution of a limited number of linear systems per step, whose coefficient matrices involve the exact Jacobian $J$ of the problem. To significantly reduce the computational cost of TASE-RK methods without altering their consistency properties, it is possible to replace $J$ with a matrix $A$ (not necessarily tied to $J$) in their formulation, for instance fixed for a certain number of consecutive steps or even constant. However, the stability properties of TASE-RK methods strongly depend on this choice, and so far have been studied assuming $A=J$. In this manuscript, we theoretically investigate the conditional and unconditional stability of TASE-RK methods by considering arbitrary $A$. To this end, we first split the Jacobian as $J=A+B$. Then, through the use of stability diagrams and their connections with the field of values, we analyze both the case in which $A$ and $B$ are simultaneously diagonalizable and not. Numerical experiments, conducted on Partial Differential Equations (PDEs) arising from applications, show the correctness and utility of the theoretical results derived in the paper, as well as the good stability and efficiency of TASE-RK methods when $A$ is suitably chosen.

Anderson acceleration (AA) is a technique for accelerating the convergence of an underlying fixed-point iteration. AA is widely used within computational science, with applications ranging from electronic structure calculation to the training of neural networks. Despite AA's widespread use, relatively little is understood about it theoretically. An important and unanswered question in this context is: To what extent can AA actually accelerate convergence of the underlying fixed-point iteration? While simple enough to state, this question appears rather difficult to answer. For example, it is unanswered even in the simplest (non-trivial) case where the underlying fixed-point iteration consists of applying a two-dimensional affine function. In this note we consider a restarted variant of AA applied to solve symmetric linear systems with restart window of size one. Several results are derived from the analytical solution of a nonlinear eigenvalue problem characterizing residual propagation of the AA iteration. This includes a complete characterization of the method to solve $2 \times 2$ linear systems, rigorously quantifying how the asymptotic convergence factor depends on the initial iterate, and quantifying by how much AA accelerates the underlying fixed-point iteration. We also prove that even if the underlying fixed-point iteration diverges, the associated AA iteration may still converge.