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We present and implement an algorithm for computing the invariant circle and the corresponding stable manifolds for 2-dimensional maps. The algorithm is based on the parameterization method, and it is backed up by an a-posteriori theorem established in [YdlL21]. The algorithm works irrespective of whether the internal dynamics in the invariant circle is a rotation or it is phase-locked. The algorithm converges quadratically and the number of operations and memory requirements for each step of the iteration is linear with respect to the size of the discretization. We also report on the result of running the implementation in some standard models to uncover new phenomena. In particular, we explored a bundle merging scenario in which the invariant circle loses hyperbolicity because the angle between the stable directions and the tangent becomes zero even if the rates of contraction are separated. We also discuss and implement a generalization of the algorithm to 3 dimensions, and implement it on the 3-dimensional Fattened Arnold Family (3D-FAF) map with non-resonant eigenvalues and present numerical results.

We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and $\mathbb{P}_1$ or $\mathbb{Q}_1$ finite elements. Time integration is performed using the Crank-Nicolson method or an explicit strong stability preserving Runge-Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time-dependent and stationary convection-diffusion-reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

This paper deals with unconstrained optimization problems based on numerical analysis of ordinary differential equations (ODEs). Although it has been known for a long time that there is a relation between optimization methods and discretization of ODEs, research in this direction has recently been gaining attention. In recent studies, the dissipation laws of ODEs have often played an important role. By contrast, in the context of numerical analysis, a technique called geometric numerical integration, which explores discretization to maintain geometrical properties such as the dissipation law, is actively studied. However, in research investigating the relationship between optimization and ODEs, techniques of geometric numerical integration have not been sufficiently investigated. In this paper, we show that a recent geometric numerical integration technique for gradient flow reads a new step-size criterion for the steepest descent method. Consequently, owing to the discrete dissipation law, convergence rates can be proved in a form similar to the discussion in ODEs. Although the proposed method is a variant of the existing steepest descent method, it is suggested that various analyses of the optimization methods via ODEs can be performed in the same way after discretization using geometric numerical integration.

In this paper, we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation. The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system, which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system. Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem. Under the consistent initial condition, the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation. In addition, the Fourier pseudo-spectral method is used for spatial discretization, resulting in fully discrete energy-preserving schemes. To implement the proposed methods effectively, we present a very efficient iterative technique, which not only greatly saves the calculation cost, but also achieves the purpose of practically preserving structure. Ample numerical results are addressed to confirm the expected order of accuracy, conservative property and efficiency of the proposed algorithms.

The main two algorithms for computing the numerical radius are the level-set method of Mengi and Overton and the cutting-plane method of Uhlig. Via new analyses, we explain why the cutting-plane approach is sometimes much faster or much slower than the level-set one and then propose a new hybrid algorithm that remains efficient in all cases. For matrices whose fields of values are a circular disk centered at the origin, we show that the cost of Uhlig's method blows up with respect to the desired relative accuracy. More generally, we also analyze the local behavior of Uhlig's cutting procedure at outermost points in the field of values, showing that it often has a fast Q-linear rate of convergence and is Q-superlinear at corners. Finally, we identify and address inefficiencies in both the level-set and cutting-plane approaches and propose refined versions of these techniques.

Stokes flows are a type of fluid flow where convective forces are small in comparison with viscous forces, and momentum transport is entirely due to viscous diffusion. Besides being routinely used as benchmark test cases in numerical fluid dynamics, Stokes flows are relevant in several applications in science and engineering including porous media flow, biological flows, microfluidics, microrobotics, and hydrodynamic lubrication. The present study concerns the discretization of the equations of motion of Stokes flows in three dimensions utilizing the MINI mixed finite element, focusing on the superconvergence of the method which was investigated with numerical experiments using five purpose-made benchmark test cases with analytical solution. Despite the fact that the MINI element is only linearly convergent according to standard mixed finite element theory, a recent theoretical development proves that, for structured meshes in two dimensions, the pressure superconverges with order 1.5, as well as the linear part of the computed velocity with respect to the piecewise-linear nodal interpolation of the exact velocity. The numerical experiments documented herein suggest a more general validity of the superconvergence in pressure, possibly to unstructured tetrahedral meshes and even up to quadratic convergence which was observed with one test problem, thereby indicating that there is scope to further extend the available theoretical results on convergence.

We consider the deformation of a geological structure with non-intersecting faults that can be represented by a layered system of viscoelastic bodies satisfying rate- and state-depending friction conditions along the common interfaces. We derive a mathematical model that contains classical Dieterich- and Ruina-type friction as special cases and accounts for possibly large tangential displacements. Semi-discretization in time by a Newmark scheme leads to a coupled system of non-smooth, convex minimization problems for rate and state to be solved in each time step. Additional spatial discretization by a mortar method and piecewise constant finite elements allows for the decoupling of rate and state by a fixed point iteration and efficient algebraic solution of the rate problem by truncated non-smooth Newton methods. Numerical experiments with a spring slider and a layered multiscale system illustrate the behavior of our model as well as the efficiency and reliability of the numerical solver.

In this paper we propose a modified Lie-type spectral splitting approximation where the external potential is of quadratic type. It is proved that we can approximate the solution to a nonlinear Schroedinger equation by solving the linear problem and treating the nonlinear term separately, with a rigorous estimate of the remainder term. Furthermore, we show by means of numerical experiments that such a modified approximation is more efficient than the standard one.

In this paper we are concerned with energy-conserving methods for Poisson problems, which are effectively solved by defining a suitable generalization of HBVMs, a class of energy-conserving methods for Hamiltonian problems. The actual implementation of the methods is fully discussed, with a particular emphasis on the conservation of Casimirs. Some numerical tests are reported, in order to assess the theoretical findings.

Generative adversarial nets (GANs) have generated a lot of excitement. Despite their popularity, they exhibit a number of well-documented issues in practice, which apparently contradict theoretical guarantees. A number of enlightening papers have pointed out that these issues arise from unjustified assumptions that are commonly made, but the message seems to have been lost amid the optimism of recent years. We believe the identified problems deserve more attention, and highlight the implications on both the properties of GANs and the trajectory of research on probabilistic models. We recently proposed an alternative method that sidesteps these problems.