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Isogeometric analysis with the boundary element method (IGABEM) has recently gained interest. In this paper, the approximability of IGABEM on 3D acoustic scattering problems will be investigated and a new improved BeTSSi submarine will be presented as a benchmark example. Both Galerkin and collocation are considered in combination with several boundary integral equations (BIE). In addition to the conventional BIE, regularized versions of this BIE will be considered. Moreover, the hyper-singular BIE and the Burton--Miller formulation are also considered. A new adaptive integration routine is presented, and the numerical examples show the importance of the integration procedure in the boundary element method. The numerical examples also include comparison between standard BEM and IGABEM, which again verifies the higher accuracy obtained from the increased inter-element continuity of the spline basis functions. One of the main objectives in this paper is benchmarking acoustic scattering problems, and the method of manufactured solution will be used frequently in this regard.

The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has been well-studied for domains in R2. Here we study the analogous problem on polyhedral surfaces. In particular, we provide a convergence proof of discrete solutions to the corresponding smooth solution of the biharmonic equation. We obtain convergence rates that are identical to the ones known for the planar setting. Our proof focuses on three different problems: solving the biharmonic equation on the surface, solving the biharmonic equation in a discrete space in the metric of the surface, and solving the biharmonic equation in a discrete space in the metric of the polyhedral approximation of the surface. We employ inverse discrete Laplacians to bound the error between the solutions of the two discrete problems, and generalize a flat strategy to bound the remaining error between the discrete solutions and the exact solution on the curved surface.

Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.

In this paper we propose a methodology to accelerate the resolution of the so-called "Sorted L-One Penalized Estimation" (SLOPE) problem. Our method leverages the concept of "safe screening", well-studied in the literature for \textit{group-separable} sparsity-inducing norms, and aims at identifying the zeros in the solution of SLOPE. More specifically, we derive a set of \(\tfrac{n(n+1)}{2}\) inequalities for each element of the \(n\)-dimensional primal vector and prove that the latter can be safely screened if some subsets of these inequalities are verified. We propose moreover an efficient algorithm to jointly apply the proposed procedure to all the primal variables. Our procedure has a complexity \(\mathcal{O}(n\log n + LT)\) where \(T\leq n\) is a problem-dependent constant and \(L\) is the number of zeros identified by the tests. Numerical experiments confirm that, for a prescribed computational budget, the proposed methodology leads to significant improvements of the solving precision.

In this article we suggest two discretization methods based on isogeometric analysis (IGA) for planar linear elasticity. On the one hand, we apply the well-known ansatz of weakly imposed symmetry for the stress tensor and obtain a well-posed mixed formulation. Such modified mixed problems have been already studied by different authors. But we concentrate on the exploitation of IGA results to handle also curved boundary geometries. On the other hand, we consider the more complicated situation of strong symmetry, i.e. we discretize the mixed weak form determined by the so-called Hellinger-Reissner variational principle. We show the existence of suitable approximate fields leading to an inf-sup stable saddle-point problem. For both discretization approaches we prove convergence statements and in case of weak symmetry we illustrate the approximation behavior by means of several numerical experiments.

We describe a cognitive architecture intended to solve a wide range of problems based on the five identified principles of brain activity, with their implementation in three subsystems: logical-probabilistic inference, probabilistic formal concepts, and functional systems theory. Building an architecture involves the implementation of a task-driven approach that allows defining the target functions of applied applications as tasks formulated in terms of the operating environment corresponding to the task, expressed in the applied ontology. We provide a basic ontology for a number of practical applications as well as for the subject domain ontologies based upon it, describe the proposed architecture, and give possible examples of the execution of these applications in this architecture.

Refractive freeform components are becoming increasingly relevant for generating controlled patterns of light, because of their capability to spatially-modulate optical signals with high efficiency and low background. However, the use of these devices is still limited by difficulties in manufacturing macroscopic elements with complex, 3-dimensional (3D) surface reliefs. Here, 3D-printed and stretchable magic windows generating light patterns by refraction are introduced. The shape and, consequently, the light texture achieved can be changed through controlled device strain. Cryptographic magic windows are demonstrated through exemplary light patterns, including micro-QR-codes, that are correctly projected and recognized upon strain gating while remaining cryptic for as-produced devices. The light pattern of micro-QR-codes can also be projected by two coupled magic windows, with one of them acting as the decryption key. Such novel, freeform elements with 3D shape and tailored functionalities is relevant for applications in illumination design, smart labels, anti-counterfeiting systems, and cryptographic communication.

Convection-diffusion-reaction equations model the conservation of scalar quantities. From the analytic point of view, solution of these equations satisfy under certain conditions maximum principles, which represent physical bounds of the solution. That the same bounds are respected by numerical approximations of the solution is often of utmost importance in practice. The mathematical formulation of this property, which contributes to the physical consistency of a method, is called Discrete Maximum Principle (DMP). In many applications, convection dominates diffusion by several orders of magnitude. It is well known that standard discretizations typically do not satisfy the DMP in this convection-dominated regime. In fact, in this case, it turns out to be a challenging problem to construct discretizations that, on the one hand, respect the DMP and, on the other hand, compute accurate solutions. This paper presents a survey on finite element methods, with a main focus on the convection-dominated regime, that satisfy a local or a global DMP. The concepts of the underlying numerical analysis are discussed. The survey reveals that for the steady-state problem there are only a few discretizations, all of them nonlinear, that at the same time satisfy the DMP and compute reasonably accurate solutions, e.g., algebraically stabilized schemes. Moreover, most of these discretizations have been developed in recent years, showing the enormous progress that has been achieved lately. Methods based on algebraic stabilization, nonlinear and linear ones, are currently as well the only finite element methods that combine the satisfaction of the global DMP and accurate numerical results for the evolutionary equations in the convection-dominated situation.

For a given nonnegative matrix $A=(A_{ij})$, the matrix scaling problem asks whether $A$ can be scaled to a doubly stochastic matrix $XAY$ for some positive diagonal matrices $X,Y$. The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization $A_{ij} \leftarrow A_{ij}/\sum_{j}A_{ij}$ and column-normalization $A_{ij} \leftarrow A_{ij}/\sum_{i}A_{ij}$ alternatively. By this algorithm, $A$ converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with $A$ has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph $G$, which is identified with the $0,1$-matrix $A_G$. Linial, Samorodnitsky, and Wigderson showed that a polynomial number of the Sinkhorn iterations for $A_G$ decides whether $G$ has a perfect matching. In this paper, we show an extension of this result: If $G$ has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker -- a certificate of the nonexistence of a perfect matching. Our analysis is based on an interpretation of the Sinkhorn algorithm as alternating KL-divergence minimization (Csisz\'{a}r and Tusn\'{a}dy 1984, Gietl and Reffel 2013) and its limiting behavior for a nonscalable matrix (Aas 2014). We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage-Mendelsohn decomposition of a bipartite graph.

The single shortest path algorithm is undefined for weighted finite-state automata over non-idempotent semirings because such semirings do not guarantee the existence of a shortest path. However, in non-idempotent semirings admitting an order satisfying a monotonicity condition (such as the plus-times or log semirings), the notion of shortest string is well-defined. We describe an algorithm which finds the shortest string for a weighted non-deterministic automaton over such semirings using the backwards shortest distance of an equivalent deterministic automaton (DFA) as a heuristic for A* search performed over a companion idempotent semiring, which is proven to return the shortest string. While there may be exponentially more states in the DFA, this algorithm needs to visit only a small fraction of them if determinization is performed "on the fly".