Simulations of the localization of certain small electromagnetic inhomogeneities, in a three-dimensional bounded domain, are performed by making use of a framework recently introduced by the author and of a non-standard discretization process of this domain. This framework is based on a limit model in electric field and on the combination of a limit perturbation model in the tangential boundary trace of the curl of the electric field, with a Current Projection method or an Inverse Fourier method. As opposed to our recent paper relating to this framework and to experiments requiring the usual discretization process of the domain, inhomogeneities that are one order of magnitude smaller are numerically localized here.

This work is concerned with the analysis on a numerical reconstruction of the magnetic permeability. The ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by an edge element method, whose convergence is established.

This paper describes an algorithm for "direct numerical integration" of the initial value Differential-Algebraic Inequalities (DAI) in a time stepping fashion using a sequential quadratic programming (SQP) method solver for detecting and satisfying active path constraints at each time step. The activation of a path constraint generally increases the condition number of the active discretized differential algebraic equation’s (DAE) Jacobian and this difficulty is addressed by a regularization property of the $α$ method. The algorithm is locally stable when index 1 and index 2 active path constraints and bounds are active. Subject to available regularization it is seen to be stable for active index 3 active path constraints in the numerical examples. For the high index active path constraints, the algorithm uses a user-selectable parameter to perturb the smaller singular values of the Jacobian with a view to reducing the condition number so that the simulation can proceed. The algorithm can be used as a relatively cheaper estimation tool for trajectory and control planning and in the context of model predictive control solutions. It can also be used to generate initial guess values of optimization variables used as input to inequality path constrained dynamic optimization problems. The method is illustrated with examples from space vehicle trajectory and robot path planning.

A nonlinear ternary 4-point interpolatory subdivision scheme is presented. It is based on a nonlinear perturbation of the ternary subdivision scheme studied in Hassan M.F., Ivrissimtzis I.P., Dodgson N.A. and Sabin M.A. (2002): "An interpolating 4-point ternary stationary subdivision scheme", Comput. Aided Geom. Design, 19, 1-18. The convergence of the scheme and the regularity of the limit function are analyzed. It is shown that the Gibbs phenomenon, classical in linear schemes, is eliminated. The stability of the associated nonlinear multiresolution scheme is established. Up to our knowledge, this is the first interpolatory scheme of regularity larger than one, avoiding Gibbs oscillations and for which the stability of the associated multiresolution analysis is established. All these properties are very important for real applications.

A new finite volume method for solving the Stokes equations is developed in this paper. The finite volume method makes use of the $BDM_1$ mixed element in approximating the velocity unknown, and consequently, the finite volume solution features a full satisfaction of the divergence-free constraint as required for the exact solution. Optimal-order error estimates are established for the corresponding finite volume solutions in various Sobolev norms. Some preliminary numerical experiments are conducted and presented in the paper. In particular, a post-processing procedure was numerically investigated for the pressure approximation. The result shows a superconvergence for a local averaging post-processing method.

This paper deals with the numerical solution of the Heston partial differential equation (PDE) that plays an important role in financial option pricing theory, Heston (1993). A feature of this time-dependent, two-dimensional convection-diffusion-reaction equation is the presence of a mixed spatial-derivative term, which stems from the correlation between the two underlying stochastic processes for the asset price and its variance.
Semi-discretization of the Heston PDE, using finite difference schemes on non-uniform grids, gives rise to large systems of stiff ordinary differential equations. For the effective numerical solution of these systems, standard implicit time-stepping methods are often not suitable anymore, and tailored time-discretization methods are required. In the present paper, we investigate four splitting schemes of the Alternating Direction Implicit (ADI) type: the Douglas scheme, the Craig-Sneyd scheme, the Modified Craig-Sneyd scheme, and the Hundsdorfer–Verwer scheme, each of which contains a free parameter.
ADI schemes were not originally developed to deal with mixed spatial-derivative terms. Accordingly, we first discuss the adaptation of the above four ADI schemes to the Heston PDE. Subsequently, we present various numerical examples with realistic data sets from the literature, where we consider European call options as well as down-and-out barrier options. Combined with ample theoretical stability results for ADI schemes that have recently been obtained in In ’t Hout & Welfert (2007, 2009) we arrive at three ADI schemes that all prove to be very effective in the numerical solution of the Heston PDE with a mixed derivative term. It is expected that these schemes will be useful also for general two-dimensional convection-diffusion-reaction equations with mixed derivative terms.

In the present article, a new methodology has been developed to solve two-dimensional (2D) Navier-Stokes equations (NSEs) in new form proposed by Pukhnachev (J. Appl. Mech. Tech. Phys., 45:2 (2004), 167-171) who introduces a new unknown function that is related to the pressure and the stream function. The important distinguish of this formulation from vorticity-stream function form of NSEs is that stream function satisfies to the transport equation and the new unknown function satisfies to the elliptic equation. The scheme and algorithm treat the equations as a coupled system which allows one to satisfy two conditions for stream function with no condition on the new function. The numerical algorithm is applied to the lid-driven cavity flow as the benchmark problem. The characteristics of this flow are adequately represented by the new numerical model.

A spectral method using fully tensorial rational basis functions on tetrahedron, obtained from the polynomials on the reference cube through a collapsed coordinate transform, is proposed and analyzed. Theoretical and numerical results show that the rational approximation is as accurate as the polynomial approximation, but with a more effective implementation.

We propose a new method to reduce the cost of computing nonlinear terms in projection based reduced order models with global basis functions. We develop this method by extending ideas from the group finite element (GFE) method to proper orthogonal decomposition (POD) and call it the group POD method. Here, a scalar two-dimensional Burgers' equation is used as a model problem for the group POD method. Numerical results show that group POD models of Burgers' equation are as accurate and are computationally more efficient than standard POD models of Burgers' equation.

Three-dimensional wave equation prestack depth imaging is an important tool in reconstructing images of complex subsurface structures, and it has become a technique gaining wide popularity in oil and gas industry. This is a large-scale scientific computing problem and can be considered a process of data continuation downward with the surface data or the boundary data, such as the shot-gather data. In this paper, we first discuss the decomposition of a two-way wave equation and investigate four different approaches to approximate the square-root operator. Using the known shot-gather data as input, an unconditional stable hybrid method for the wavefield extrapolation is presented. The most attractive feature of the proposed method is that it has a natural parallel characteristic and can be effectively implemented using a cluster of PCs, in which each processor performs its own shot-gather imaging independently. To demonstrate the computational efficiency and the power of the parallel hybrid algorithm, we present two case studies: one is the well-known SEG/EAEG subsalt model which has been commonly used for validation of the prestack depth imaging algorithms, and the other is the application to a 3D wavefield extrapolation problem with real data provided by the China National Petroleum Corporation. The results clearly show the capability of the proposed method, and it demonstrates that the algorithm can be effectively implemented as a practical engineering tool for 3D prestack depth imaging.