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We present a general family of subcell limiting strategies to construct robust high-order accurate nodal discontinuous Galerkin (DG) schemes. The main strategy is to construct compatible low order finite volume (FV) type discretizations that allow for convex blending with the high-order variant with the goal of guaranteeing additional properties, such as bounds on physical quantities and/or guaranteed entropy dissipation. For an implementation of this main strategy, four main ingredients are identified that may be combined in a flexible manner: (i) a nodal high-order DG method on Legendre-Gauss-Lobatto nodes, (ii) a compatible robust subcell FV scheme, (iii) a convex combination strategy for the two schemes, which can be element-wise or subcell-wise, and (iv) a strategy to compute the convex blending factors, which can be either based on heuristic troubled-cell indicators, or using ideas from flux-corrected transport methods. By carefully designing the metric terms of the subcell FV method, the resulting methods can be used on unstructured curvilinear meshes, are locally conservative, can handle strong shocks efficiently while directly guaranteeing physical bounds on quantities such as density, pressure or entropy. We further show that it is possible to choose the four ingredients to recover existing methods such as a provably entropy dissipative subcell shock-capturing approach or a sparse invariant domain preserving approach. We test the versatility of the presented strategies and mix and match the four ingredients to solve challenging simulation setups, such as the KPP problem (a hyperbolic conservation law with non-convex flux function), turbulent and hypersonic Euler simulations, and MHD problems featuring shocks and turbulence.

Iterative regularization exploits the implicit bias of an optimization algorithm to regularize ill-posed problems. Constructing algorithms with such built-in regularization mechanisms is a classic challenge in inverse problems but also in modern machine learning, where it provides both a new perspective on algorithms analysis, and significant speed-ups compared to explicit regularization. In this work, we propose and study the first iterative regularization procedure able to handle biases described by non smooth and non strongly convex functionals, prominent in low-complexity regularization. Our approach is based on a primal-dual algorithm of which we analyze convergence and stability properties, even in the case where the original problem is unfeasible. The general results are illustrated considering the special case of sparse recovery with the $\ell_1$ penalty. Our theoretical results are complemented by experiments showing the computational benefits of our approach.

This article discusses a particular case of the data clustering problem, where it is necessary to find groups of adjacent text segments of the appropriate length that match a fuzzy pattern represented as a sequence of fuzzy properties. To solve this problem, a heuristic algorithm for finding a sufficiently large number of solutions is proposed. The key idea of the proposed algorithm is the use of the prefix structure to track the process of mapping text segments to fuzzy properties. An important special case of the text segmentation problem is the fuzzy string matching problem, when adjacent text segments have unit length and, accordingly, the fuzzy pattern is a sequence of fuzzy properties of text characters. It is proven that the heuristic segmentation algorithm in this case finds all text segments that match the fuzzy pattern. Finally, we consider the problem of a best segmentation of the entire text based on a fuzzy pattern, which is solved using the dynamic programming method. Keywords: fuzzy clustering, fuzzy string matching, approximate string matching

Random field models are mathematical structures used in the study of stochastic complex systems. In this paper, we compute the shape operator of Gaussian random field manifolds using the first and second fundamental forms (Fisher information matrices). Using Markov Chain Monte Carlo techniques, we simulate the dynamics of these random fields and compute the Gaussian curvature of the parametric space, analyzing how this quantity changes along phase transitions. During the simulation, we have observed an unexpected phenomenon that we called the \emph{curvature effect}, which indicates that a highly asymmetric geometric deformation happens in the underlying parametric space when there are significant increase/decrease in the system's entropy. This asymmetric pattern relates to the emergence of hysteresis, leading to an intrinsic arrow of time along the dynamics.

Trefftz methods are numerical methods for the approximation of solutions to boundary and/or initial value problems. They are Galerkin methods with particular test and trial functions, which solve locally the governing partial differential equation (PDE). This property is called the Trefftz property. Quasi-Trefftz methods were introduced to leverage the advantages of Trefftz methods for problems governed by variable coefficient PDEs, by relaxing the Trefftz property into a so-called quasi-Trefftz property: test and trial functions are not exact solutions but rather local approximate solutions to the governing PDE. In order to develop quassi-Trefftz methods for aero-acoustics problems governed by the convected Helmholtz equation, the present work tackles the question of the definition, construction and approximation properties of three families of quasi-Trefftz functions: two based on generalizations on plane wave solutions, and one polynomial.

Classical interior penalty discontinuous Galerkin (IPDG) methods for diffusion problems require a number of assumptions on the local variation of mesh-size, polynomial degree, and of the diffusion coefficient to determine the values of the, so-called, discontinuity-penalization parameter and/or to perform error analysis. Variants of IPDG methods involving weighted averages of the gradient of the approximate solution have been proposed in the context of high-contrast diffusion coefficients to mitigate the dependence of the contrast in the stability and in the error analysis. Here, we present a new IPDG method, involving carefully constructed weighted averages of the gradient of the approximate solution, which is shown to be robust even for the most extreme simultaneous local mesh, polynomial degree and diffusion coefficient variation scenarios, without resulting in unreasonably large penalization. The new method, henceforth termed as \emph{robust IPDG} (RIPDG), offers typically significantly better conditioning than the standard IPDG method when applied to scenarios with strong mesh/polynomial degree/diffusion local variation. On the other hand, when using uniform meshes, constant polynomial degree and for problems with constant diffusion coefficients, the RIPDG method is identical to the classical IPDG. Numerical experiments indicate the favourable performance of the new RIPDG method over the classical version in terms of conditioning and error.

We consider a class of structured fractional minimization problems, in which the numerator part of the objective is the sum of a differentiable convex function and a convex nonsmooth function, while the denominator part is a concave or convex function. This problem is difficult to solve since it is nonconvex. By exploiting the structure of the problem, we propose two Coordinate Descent (CD) methods for solving this problem. One is applied to the original fractional function, the other is based on the associated parametric problem. The proposed methods iteratively solve a one-dimensional subproblem \textit{globally}, and they are guaranteed to converge to coordinate-wise stationary points. In the case of a convex denominator, we prove that the proposed CD methods using sequential nonconvex approximation find stronger stationary points than existing methods. Under suitable conditions, CD methods with an appropriate initialization converge linearly to the optimal point (also the coordinate-wise stationary point). In the case of a concave denominator, we show that the resulting problem is quasi-convex, and any critical point is a global minimum. We prove that the algorithms converge to the global optimal solution with a sublinear convergence rate. We demonstrate the applicability of the proposed methods to some machine learning and signal processing models. Our experiments on real-world data have shown that our method significantly and consistently outperforms existing methods in terms of accuracy.

We establish some limit theorems for quasi-arithmetic means of random variables. This class of means contains the arithmetic, geometric and harmonic means. Our feature is that the generators of quasi-arithmetic means are allowed to be complex-valued, which makes considerations for quasi-arithmetic means of random variables which could take negative values possible. Our motivation for the limit theorems is finding simple estimators of the parameters of the Cauchy distribution. By applying the limit theorems, we obtain some closed-form unbiased strongly-consistent estimators for the joint of the location and scale parameters of the Cauchy distribution, which are easy to compute and analyze.

We prove two theorems related to the Central Limit Theorem (CLT) for Martin-L\"of Random (MLR) sequences. Martin-L\"of randomness attempts to capture what it means for a sequence of bits to be "truly random". By contrast, CLTs do not make assertions about the behavior of a single random sequence, but only on the distributional behavior of a sequence of random variables. Semantically, we usually interpret CLTs as assertions about the collective behavior of infinitely many sequences. Yet, our intuition is that if a sequence of bits is "truly random", then it should provide a "source of randomness" for which CLT-type results should hold. We tackle this difficulty by using a sampling scheme that generates an infinite number of samples from a single binary sequence. We show that when we apply this scheme to a Martin-L\"of random sequence, the empirical moments and cumulative density functions (CDF) of these samples tend to their corresponding counterparts for the normal distribution. We also prove the well known almost sure central limit theorem (ASCLT), which provides an alternative, albeit less intuitive, answer to this question. Both results are also generalized for Schnorr random sequences.

This paper investigates the interference nulling capability of reconfigurable intelligent surface (RIS) in a multiuser environment where multiple single-antenna transceivers communicate simultaneously in a shared spectrum. From a theoretical perspective, we show that when the channels between the RIS and the transceivers have line-of-sight and the direct paths are blocked, it is possible to adjust the phases of the RIS elements to null out all the interference completely and to achieve the maximum $K$ degrees-of-freedom (DoF) in the overall $K$-user interference channel, provided that the number of RIS elements exceeds some finite value that depends on $K$. Algorithmically, for any fixed channel realization we formulate the interference nulling problem as a feasibility problem, and propose an alternating projection algorithm to efficiently solve the resulting nonconvex problem with local convergence guarantee. Numerical results show that the proposed alternating projection algorithm can null all the interference if the number of RIS elements is only slightly larger than a threshold of $2K(K-1)$. For the practical sum-rate maximization objective, this paper proposes to use the zero-forcing solution obtained from alternating projection as an initial point for subsequent Riemannian conjugate gradient optimization and shows that it has a significant performance advantage over random initializations. For the objective of maximizing the minimum rate, this paper proposes a subgradient projection method which is capable of achieving excellent performance at low complexity.