Mathematics

A new numerical approach is proposed for the simulation of coupled three-dimensional and one-dimensional elliptic equations (3D-1D coupling) arising from dimensionality reduction of 3D-3D problems with thin inclusions. The method is based on a well posed mathematical formulation and results in a numerical scheme with high robustness and flexibility in handling geometrical complexities. This is achieved by means of a three-field approach to split the 1D problems from the bulk 3D problem, and then resorting to the minimization of a properly designed functional to impose matching conditions at the interfaces. Thanks to the structure of the functional, the method allows the use of independent meshes for the various subdomains.

A P k+2 polynomial lifting operator is defined on polygons and polyhedrons. It lifts discontinuous polynomials inside the polygon/polyhedron and on the faces to a one-piece P k+2 polynomial. With this lifting operator, we prove that the weak Galerkin finite element solution, after this lifting, converges at two orders higher than the optimal order, in both L 2 and H 1 norms. The theory is confirmed by numerical solutions of 2D and 3D Poisson equations.

In this work, we are interested in the determination of the shape of the scatterer for the two dimensional time harmonic inverse medium scattering problems in acoustics. The scatterer is assumed to be a piecewise constant function with a known value inside inhomogeneities, and its shape is represented by the level set functions for which we investigate the information using the Bayesian method. In the Bayesian framework, the solution of the geometric inverse problem is defined as a posterior probability distribution. The well-posedness of the posterior distribution would be discussed, and the Markov chain Monte Carlo (MCMC) methods will be applied to generate samples from the arising posterior distribution. Numerical experiments will be presented to demonstrate the effectiveness of the proposed method.

We present randUBV, a randomized algorithm for matrix sketching based on the block Lanzcos bidiagonalization process. Given a matrix A , it produces a low-rank approximation of the form UBV T , where U and V have orthonormal columns in exact arithmetic and B is block bidiagonal. In finite precision, the columns of both U and V will be close to orthonormal. Our algorithm is closely related to the randQB algorithms of Yu, Gu, and Li (2018) in that the entries of B are incrementally generated and the Frobenius norm approximation error may be efficiently estimated. Our algorithm is therefore suitable for the fixed-accuracy problem, and so is designed to terminate as soon as a user input error tolerance is reached. Numerical experiments suggest that the block Lanczos method is generally competitive with or superior to algorithms that use power iteration, even when A has significant clusters of singular values.

We present an alternative relatively easy way to understand and determine the zeros of a quintic whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in his famous \textit{Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade} are here shortened via Heymann's theory of transformations. Also, we give a complete explanation of the so-called icosahedral equation and its solution in terms of Gaussian hypergeometric functions. As an innovative element, we construct this solution by using algebraic transformations of hypergeometric series. Within this framework, we develop a practical algorithm to compute the zeros of the quintic.

We propose a method for tracing implicit real algebraic curves defined by polynomials with rank-deficient Jacobians. For a given curve f ?? (0) , it first utilizes a regularization technique to compute at least one witness point per connected component of the curve. We improve this step by establishing a sufficient condition for testing the emptiness of f ?? (0) . We also analyze the convergence rate and carry out an error analysis for refining the witness points. The witness points are obtained by computing the minimum distance of a random point to a smooth manifold embedding the curve while at the same time penalizing the residual of f at the local minima. To trace the curve starting from these witness points, we prove that if one drags the random point along a trajectory inside a tubular neighborhood of the embedded manifold of the curve, the projection of the trajectory on the manifold is unique and can be computed by numerical continuation. We then show how to choose such a trajectory to approximate the curve by computing eigenvectors of certain matrices. Effectiveness of the method is illustrated by examples.

We present a high-order method for flow simulation on unstructured curved nonconforming sliding meshes. This method utilizes dynamic transfinite mortar elements to exchange flow information between the two sides of a sliding interface. The method is arbitrarily high-order accurate in space, provably conservative, and satisfies outflow condition. Moreover, it retains the accuracy of a time marching scheme, and thus allows substantial reduction of rotational speed effects when equipped with a high-order temporal scheme. The method's capability of simultaneously handling multiple rotational objects is also explored. Details on the implementation are provided as well.

In this paper, a deep collocation method (DCM) for thin plate bending problems is proposed. This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning. Besides, the proposed DCM is based on a feedforward deep neural network (DNN) and differs from most previous applications of deep learning for mechanical problems. First, batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries. A loss function is built with the aim that the governing partial differential equations (PDEs) of Kirchhoff plate bending problems, and the boundary/initial conditions are minimised at those collocation points. A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters. In Kirchhoff plate bending problems, the C1 continuity requirement poses significant difficulties in traditional mesh-based methods. This can be solved by the proposed DCM, which uses a deep neural network to approximate the continuous transversal deflection, and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.

In a recent work, two of the authors have formulated the non-linear space-time Hasegawa-Mima plasma equation as a coupled system of two linear PDEs, a solution of which is a pair (u,w) , with w=(I?��?u . The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa-Mima problem with periodic boundary conditions were also derived. Using the Fourier basis in the space variables, existence of solutions were obtained. Implementation of algorithms based on Fourier series leads to systems of dense matrices. In this paper, we use a finite element space-domain approach to semi-discretize the coupled variational Hasegawa-Mima model, obtaining global existence of solutions in H 2 on any time interval [0,T] for all T. In the sequel, full-discretization using an implicit time scheme on the semi-discretized system leads to a nonlinear full space-time discrete system with a nonrestrictive condition on the time step. Tests on a semi-linear version of the implicit nonlinear full-discrete system are conducted for several initial data, assessing the efficiency of our approach.

We propose a flexible power method for computing the leftmost, i.e., algebraically smallest, eigenvalue of an infinite dimensional tensor eigenvalue problem, Hx=λx , where the infinite dimensional symmetric matrix H exhibits a translational invariant structure. We assume the smallest eigenvalue of H is simple and apply a power iteration of e ?�H with the eigenvector represented in a compact way as a translational invariant infinite Tensor Ring (iTR). Hence, the infinite dimensional eigenvector can be represented by a finite number of iTR cores of finite rank. In order to implement this power iteration, we use a small parameter t so that the infinite matrix-vector operation e ?�Ht x can efficiently be approximated by the Lie product formula, also known as Suzuki--Trotter splitting, and we employ a low rank approximation through a truncated singular value decomposition on the iTR cores in order to keep the cost of subsequent power iterations bounded. We also use an efficient way for computing the iTR Rayleigh quotient and introduce a finite size iTR residual which is used to monitor the convergence of the Rayleigh quotient and to modify the timestep t . In this paper, we discuss 2 different implementations of the flexible power algorithm and illustrate the automatic timestep adaption approach for several numerical examples.