In traditional models for valuation of mortgages with a stochastic interest rate, one parabolic equation starting from the maturity is assumed to govern the whole life of a mortgage. Following the valuation of zero-coupon bond, a new model is proposed, where an initial value problem is restarted after a mortgage payment each month. In addition, the low and high limits on the interest rate are incorporated into the initial-boundary value problems, so that the partial differential equation remains regular and the solution better approximates the real value. We show the existence and uniqueness of the solution and the free boundary (which determines early prepayment). A finite element method is introduced with a convergence analysis. Numerical tests are presented and the results are interpreted for guiding mortgage practice.

This work introduces novel unconditionally stable operator splitting methods for solving the time dependent nonlinear Poisson-Boltzmann (NPB) equation for the electrostatic analysis of solvated biomolecules. In a pseudo-transient continuation solution of the NPB equation, the nonlinear term is analytically integrated, so that the difficulties in direct treatment of the strong nonlinearity can be bypassed. However, in a pseudo-time NPB computation, the use of large time increments is necessary to reach the steady state efficiently. The existing alternating direction implicit (ADI) methods for the transient NPB equation are known to be conditionally stable, although being fully implicit. To overcome this difficulty, we propose several new operator splitting schemes, in both multiplicative and additive styles, including locally one-dimensional (LOD) schemes and additive operator splitting (AOS) schemes. The proposed schemes become much more stable than the ADI methods, and some of them are indeed unconditionally stable in dealing with solvated proteins with source singularities and non-smooth solutions. By using finite differences in space and implicit integrations in time, the numerical orders of the proposed schemes are found to be one in both space and time. Nevertheless, the precision in calculating the electrostatic free energy is low, unless a small time increment is used. Further accuracy improvements are thus considered, through constructing a Richardson extrapolation procedure and a tailored recovery scheme in treating the vacuum case. After acceleration, the optimized LOD method can produce a reliable energy estimate by integrating for a small and fixed number of time steps. Since one only needs to solve a tridiagonal matrix in each one dimensional subsystem, the overall computation is very efficient. The unconditionally stable LOD method scales linearly with respect to the number of atoms in the protein studies, and is over 20 times faster than the conditionally stable ADI methods.

With the explosive growth of mobile video applications, analysis of video quality becomes increasingly important because it is an important Key Performance Indicator (KPI) for Quality of Experience (QoE). In this paper, a framework for non-reference video quality analysis is proposed and applied to Video Telephony (VT) in LTE networks. Three metrics, blockiness, blur and freezing, are used to estimate the MOS. Blockiness is detected by taking the H.264 codec features into account, blur is estimated by utilizing the percentage of noticeable blurred edges in each frame, and freezing is evaluated by using a sigmoid function to mimic the effect of different freezing duration on the Human Visual System (HVS). Furthermore, the three metrics are combined into one objective MOS by considering different weighting factors and using the linear curve fitting. Above 90% correlation is achieved between the objective MOS score and subjective MOS score.

In this paper we study the behaviour of a micro-sized semiconductor device by means of a hybrid model of hydrodynamic equations. First of all, taking into account the quantum effects in the semiconductor device, we derive a new model called the hybrid quantum hydrodynamic model (H-QHD) coupled with the Poisson equation for electric potential. In particular, we write the Bohm potential in a revised form. This new potential is derived heuristically by assuming that the energy of the electrons depends on the charge density $n$ and on $\bigtriangledown n$ just in a restricted part of the device domain, whereas the remaining parts are modeled classically. Namely, the device is designed with some parts that feel the quantum effects and some parts do not. The main target is to investigate the existence of the stationary solutions for the hybrid quantum hydrodynamic model. Since the quantum effect is regionally degenerate, this will also makes the working equation regionally degenerate regarding its ellipticity, and the corresponding solutions are weak only. This paper seems the first framework to treat the equation with regionally degenerate ellipticity. In order to prove the existence of such weak solutions, we first construct a sequence of smooth QHD solutions, then prove that such a sequence weakly converges and its limit is just our desired weak solution for the hybrid QHD problem. The Debye length limit is also studied. Indeed, we prove that the weak solutions of the hybrid QHD weakly converge to their targets as the spacial Debye length vanishes. Finally, we carry out some numerical simulations for a simple device, which also confirm our theoretical results.

In this paper, we discuss discontinuous finite volume approximations of the distributed optimal control problems governed by a class of semilinear hyperbolic partial differential equations with control constraints. The spatial discretization of the state and costate variables follows discontinuous finite volume schemes with piecewise linear elements, whereas three different strategies are used for the control approximation: variational discretization, piecewise constant and piecewise linear discretization. As the resulting semi-discrete optimal system is non-symmetric, we have employed optimize then discretize approach to approximate the control problem. A priori error estimates for control, state and costate variables are derived in suitable natural norms. The present analysis is an extension of the analysis given in Kumar and Sandilya [Int. J. Numer. Anal. Model. (2016), 13: 545-568]. Numerical experiments are presented to illustrate the performance of the proposed scheme and to confirm the predicted accuracy of the theoretical convergence rates.

This paper presents a conforming finite element semi-discretization of the streamfunction form of the one-layer unsteady quasi-geostrophic equations, which are a commonly used model for large-scale wind-driven ocean circulation. We derive optimal error estimates and present numerical results.

Recently, Chumchob-Chen-Brito (2011) proposed the so-called mean curvature model which appeared to deliver better registration results for both smooth and non-smooth deformation fields than a large class of competing methods. However, the two displacement variables in a deformation field are regularized separately in their model and so coupling between them is not present. Therefore their mean curvature model has a weakness and may not yield the best registration results in some situations such as in non-smooth registration problems with non-axis-aligned discontinuities, as expected of a high order model. To design a new model based on interdependence between components of the deformation field, suitable for smooth and non-smooth registration problems, we propose a new vectorial curvature regularizer in this paper and present an iterative method for numerical solution of the resulting variational model. Experiments using both synthetic and realistic images confirm that the proposed model is more robust than the Chumchob-Chen-Brito (2011) model in registration quality for a wide range of examples.

We consider efficient finite difference methods for solving the three-dimensional (3D) acoustic scattering by an impenetrable circular cylindrical obstacle. By using the separation of variable and other techniques, we first transform the 3D problem into a series of one-dimensional (1D) problems in this paper, and then construct some efficient and accuracy finite difference methods to solve these 1D problems instead of the 3D one. There are mainly two advantages for these methods: one is that they are pollution free for the problem to be considered in this paper; and the other is that the linear systems generated from these schemes have tri-diagonal structures. These features lead to easy implementation and much less computational cost. Numerical examples are presented to verify the efficiency and accuracy of the numerical methods, even with the wave number greater than 100.