In this paper we apply and discuss a multilevel method to solve a scattering problem. The multilevel method belongs to the class of incremental unknowns method as in [10]; in this work, the best performance was obtained with a coarsest grid having roughly two points per linear wavelength. We analyze this method for a simple model problem following H. Yserentant [17]. In this case, the main limitation to multilevel methods is closely linked to the indefiniteness of the Helmholtz problem.

In this paper, we reconsider a contact problem between an elasto-viscoplastic body and a deformable obstacle. The contact is modeled by the classical normal compliance contact condition. Then, fully discrete approximations are obtained by using the finite element method to approximate the spatial variable and the forward Euler scheme to discretize time derivatives. An a posteriori error analysis is provided and upper and lower error bounds are obtained. Finally, some two-dimensional numerical simulations are presented to demonstrate the accuracy and the behavior of the error estimators.

In this paper, we consider discontinuous Galerkin approximations to the solution of Naghdi arches and show how to post-process them in an element-by-element fashion to obtain a far better approximation. Indeed, we prove that, if polynomials of degree $k$ are used, the post-processed approximation converges with order $2k+1$ in the $L^2$-norm throughout the domain. This has to be contrasted with the fact that before post-processing, the approximation converges with order $k + 1$ only. Moreover, we show that this superconvergence property does not deteriorate as the thickness of the arch becomes extremely small. Numerical experiments verifying the above-mentioned theoretical results are displayed.

The aim of this paper is to investigate the application of a semi-implicit additive operator splitting scheme based binary level set method to source reconstruction problems. We reformulate the original model to be a new constrained optimization problem under the binary level set framework and solve it by the augmented Lagrangian method. Then we propose an efficient gradient-type algorithm based on the additive operator splitting scheme. The proposed algorithm can create new holes during the evolution. Topological changes can be handled automatically and complex geometry can be recovered under a certain amount of noise in the observation data. Numerical examples are presented to show the effectiveness and efficiency of our method.

We propose a second-order accurate method for computing the solutions to the Aliev-Panfilov model of cardiac excitation. This two-variable reaction-diffusion system is due to its simplicity a popular choice for modeling important problems in electrocardiology; e.g. cardiac arrhythmias. The solutions might be very complicated in structure, and hence highly resolved numerical simulations are called for to capture the fine details. Usually the forward Euler time-integrator is applied in these computations; it is very simple to implement and can be effective for coarse grids. For fine-scale simulations, however, the forward Euler method suffers from a severe time-step restriction, rendering it less efficient for simulations where high resolution and accuracy are important.
We analyze the stability of the proposed second-order method and the forward Euler scheme when applied to the Aliev-Panfilov model. Compared to the Euler method the suggested scheme has a much weaker time-step restriction, and promises to be more efficient for computations on finer meshes.

A Robin-Robin non-overlapping domain decomposition method for an optimal boundary control problem associated with an elliptic boundary value problem is presented. The existence of the whole domain and subdomain optimal solutions is proven. The convergence of the subdomain optimal solutions to the whole domain optimal solution is shown. The optimality system is derived and a gradient-type method is defined for finding the optimal solution. A theoretic convergence result for the gradient method is established. The finite element version of the Robin-Robin non-overlapping domain decomposition method is analyzed and some numerical results by the method on both serial and parallel computers (using MPI) are presented.

In this paper, an adaptive mesh refinement technique is developed and analyzed for the non-conforming immersed finite element (IFE) method proposed in [27]. The IFE method was developed for solving the second order elliptic boundary value problem with interfaces across which the coefficient may be discontinuous. The IFE method was based on a triangulation that does not need to fit the interface. One of the key ideas of IFE method is to modify the basis functions so that the natural jump conditions are satisfied across the interface. The IFE method has shown to be order of $O(h^2)$ and $O(h)$ in $L^2$ norm and $H^1$ norm, respectively. In order to develop the adaptive mesh refinement technique, additional priori and posterior error estimations are derived in this paper. Our new a-priori error estimation shows that the generic constant is only linearly proportional to ratio of the diffusion coefficient $\beta^-$ and $\beta^+$, which improves the corresponding result in [27]. We also show that a-posteriori error estimate similar to the one obtained by Bernardi and Verfürth [4] holds for the IFE solutions. Numerical examples support our theoretical results and show that the adaptive mesh refinement strategy is effective for the IFE approximation.

The deterministic convergence for a Double Parallel Feedforward Neural Network (DPFNN) is studied. DPFNN is a parallel connection of a multi-layer feedforward neural network and a single layer feedforward neural network. Gradient method is used for training DPFNN with finite training sample set. The monotonicity of the error function in the training iteration is proved. Then, some weak and strong convergence results are obtained, indicating that the gradient of the error function tends to zero and the weight sequence goes to a fixed point, respectively. Numerical examples are provided, which support our theoretical findings and demonstrate that DPFNN has faster convergence speed and better generalization capability than the common feedforward neural network.

We propose a penalty-function method for constrained molecular dynamics simulation by defining a quadratic penalty function for the constraints. The simulation with such a method can be done by using a conventional, unconstrained solver only with the penalty parameter increased in an appropriate manner as the simulation proceeds. More specifically, we scale the constraints with their force constants when forming the penalty terms. The resulting force function can then be viewed as a smooth continuation of the original force field as the penalty parameter increases. The penalty function method is easy to implement and costs less than a Lagrange multiplier method, which requires the solution of a nonlinear system of equations in every time step. We have first implemented a penalty function method in CHARMM and applied it to protein Bovine Pancreatic Trypsin Inhibitor (BPTI). We compared the simulation results with Verlet and Shake, and found that the penalty function method had high correlations with Shake and outperformed Verlet. In particular, the RMSD fluctuations of backbone and non-backbone atoms and the velocity auto correlations of $C_{\alpha}$ atoms of the protein calculated by the penalty function method agreed well with those by Shake. We have also tested the method on a group of argon clusters constrained with a set of interatomic distances in their global energy minimum states. The results showed that the method was able to impose the constraints effectively and the clusters tended to converge to their energy minima more rapidly than not confined by the constraints.

In this article we are interested in the study of the errors introduced by different finite volume discretizations in the study of the wave frequencies. This study is made in the context of hyperbolic systems arising in meteorology and oceanography. We show the existence of significant errors in the dispersion relation, only the long waves being computed accurately; this conclusion is similar to the finite differences case described in the article of Grotjahn and O’Brien quoted below. For the case of inertia-gravity waves, we study three often used schemes which are based on the upwind, centered or Lax-Wendroff fluxes. Moreover, the Total Variation Diminishing method (TVD) made from these fluxes (which usually provides an efficient way to eliminate spurious numerical oscillations) will give the same errors in the dispersion relation.