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In this paper, efficient numerical methods are developed for the pricing of American options governed by the Black–Scholes equation. The front-fixing technique is first employed to transform the free boundary of optimal exercise prices to some a priori known temporal line for a one-dimensional parabolic problem via the change of variables. The perfectly matched layer (PML) technique is then applied to the pricing problem for the effective truncation of the semi-infinite domain. Finite element methods using the respective continuous and discontinuous Galerkin discretization are proposed for the resulting truncated PML problems related to the options and Greeks. The free boundary is determined by Newton’s method coupled with the discrete truncated PML problem. Stability and nonnegativeness are established for the approximate solution to the truncated PML problem. Under some weak assumptions on the PML medium parameters, it is also proved that the solution of the truncated PML problem converges to that of the unbounded Black–Scholes equation in the computational domain and decays exponentially in the perfectly matched layer. Numerical experiments are conducted to test the performance of the proposed methods and to compare them with some existing methods. [ABSTRACT FROM PUBLISHER]

In this paper, we develop a novel method of reconstructing acoustic obstacles in R2, which follows a similar spirit of the linear sampling method originated by Colton and Kirsch. The reconstruction scheme makes use of the near-field measurements encoded into the boundary Dirichlet-to-Neumann map or the Neumann-to-Dirichlet map. Both the plane waves and cylindrical waves are shown to meet the reconstruction purpose. Rigorous mathematical justification of the reconstruction scheme is established. The mapping properties of the newly introduced function operators involved in the reconstruction scheme are established. These results are of significant mathematical interests for their own sake. Moreover, due to the distinct properties of the function operators, the indictor function in the proposed reconstruction scheme exhibits completely different behaviors from those having been established for the indictor function in the original linear sampling method for inverse scattering problems. Numerical experiments are presented to illustrate the effectiveness of the proposed reconstruction scheme. [ABSTRACT FROM AUTHOR]

Abstract: We analyze higher order finite elements applied to second order elliptic interface problems. Our a priori error estimates in the - and -norm are expressed in terms of the approximation order p and a parameter δ that quantifies how well the interface is resolved by the finite element mesh. The optimal p-th order convergence in the -norm is only achieved under stringent assumptions on δ, namely, . Under weaker conditions on δ, optimal a priori estimates can be established in the - and in the -norm, where is a subdomain that excludes a tubular neighborhood of the interface of width . In particular, if the interface is approximated by an interpolation spline of order p and if full regularity is assumed, then optimal convergence orders and p for the approximation in the - and the -norm can be expected but not order p for the approximation in the -norm. Numerical examples in 2D and 3D illustrate and confirm our theoretical results. [Copyright &y& Elsevier]

We present and analyze a first order least squares method for convection dominated diffusion problems, which provides robust L 2 a priori error estimate for the scalar variable even if the given data f ∈ L 2 ( Ω ) . The novel theoretical approach is to rewrite the method in the framework of discontinuous Petrov–Galerkin (DPG) method, and then show numerical stability by using a key equation discovered by Gopalakrishnan and Qiu (2014). This new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations. We also show that the condition number of the global matrix is independent of the diffusion coefficient. A key feature of the method is that there is no stabilization parameter chosen empirically. In addition, Dirichlet boundary condition is weakly imposed. Numerical experiments verify our theoretical results and, in particular, show our way of weakly imposing Dirichlet boundary condition is essential to the design of least squares methods—numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes, which are unstructured quasi-uniform. [ABSTRACT FROM AUTHOR]