Sergey Korotov, Jon Eivind Vatne
In this note we present a natural generalization of the minimum angle condition, commonly used in the finite element analysis for planar triangulations, to the case of simplicial meshes in any space dimension. The equivalence of this condition with some other mesh regularity conditions is proved.
János Karátson, Balázs Kovács, Sergey Korotov
The maximum principle forms an important qualitative property of second order elliptic equations, therefore its discrete analogues, the so-called discrete maximum principles (DMPs) have drawn much attention. In this paper DMPs are established for nonlinear surface finite element problems on Riemannian manifolds, corresponding to the classical pointwise maximum principles on surfaces in the spirit of Pucci et al. Various real-life examples illustrate the scope of the results.
Ali Khademi, Sergey Korotov, Jon Eivind Vatne
In this note we present a generalization of the maximum angle condition, proposed by J. L. Synge in 1957 and M. Křížek in 1992 for triangular and tetrahedral elements, respectively, for the case of higher-dimensional simplicial finite elements. Its relations to the other angle-type conditions commonly used in finite element methods are analysed.
Jérôme Michaud, Sergey Korotov
We study a longest-edge based refinement scheme for triangulations, termed the longest-edge altitude bisection (LEAB), in which each triangle is subdivided by dropping the altitude from the vertex opposite to its longest edge. Using the normalized shape space of triangles introduced by Perdomo and Plaza in: Properties of triangulations obtained by the longest-edge bisection. \emph{Cent. Eur. J. Math.}, 12(12) (2014), 1796-1810, we show that LEAB admits a simple geometric description: the normalized left and right children of a triangle in focus are obtained by intersecting the geodesic of right triangles with rays issued from the endpoints of the longest edge and explicit formulas for the mappings are derived. This characterization implies an interesting observation that the associated refinement dynamics collapse the entire shape space onto the right-triangle geodesic in a single step and that every point on this geodesic is fixed. Two-sided bounds for the contraction of the mesh size (discretization parameter) are derived. Also, applications and limitations of the method are briefly discussed.
Leandro Farina, Sergey Korotov
This work demonstrates a methodology for using deep learning to discover simple, practical criteria for classifying matrices based on abstract algebraic properties. By combining a high-performance neural network with explainable AI (XAI) techniques, we can distill a model's learned strategy into human-interpretable rules. We apply this approach to the challenging case of monotone matrices, defined by the condition that their inverses are entrywise nonnegative. Despite their simple definition, an easy characterization in terms of the matrix elements or the derived parameters is not known. Here, we present, to the best of our knowledge, the first systematic machine-learning approach for deriving a practical criterion that distinguishes monotone from non-monotone matrices. After establishing a labelled dataset by randomly generated monotone and non-monotone matrices uniformly on $(-1,1)$, we employ deep neural network algorithms for classifying the matrices as monotone or non-monotone, using both their entries and a comprehensive set of matrix features. By saliency methods, such as integrated gradients, we identify among all features, two matrix parameters which alone provide sufficient information for the matrix classification, with $95\%$ accuracy, namely the absolute values of the two lowest-order coefficients, $c_0$ and $c_1$ of the matrix's characteristic polynomial. A data-driven study of 18,000 random $7\times7$ matrices shows that the monotone class obeys $\lvert c_{0}/c_{1}\rvert\le0.18$ with probability $>99.98\%$; because $\lvert c_{0}/c_{1}\rvert = 1/\mathrm{tr}(A^{-1})$ for monotone $A$, this is equivalent to the simple bound $\mathrm{tr}(A^{-1})\ge5.7$.