Some new reflection principles for Maxwell's equations are first
established, which are then applied to derive two novel
identifiability results in inverse electromagnetic obstacle
scattering problems with polyhedral scatterers.
We start from a realistic half space model for thermal imaging, which we
then use to develop a mathematical asymptotic analysis well suited for the design of reconstruction algorithms. We seek to reconstruct thermal anomalies only through their
rough features. With this way our proposed algorithms are stable against measurement
noise and geometry perturbations. Based on rigorous asymptotic estimates, we first
obtain an approximation for the temperature profile which we then use to design noniterative detection algorithms. We show on numerical simulations evidence that they are
accurate and robust. Moreover, we provide a mathematical model for ultrasonic temperature imaging, which is an important technique in cancerous tissue ablation therapy.
In this paper, a generalized Laguerre-spherical harmonic spectral
method is proposed for the Cauchy problem of three-dimensional
nonlinear Klein-Gordon equation. The goal is to make the numerical
solutions to preserve the same conservation as that for
the exact solution.
The stability and convergence of the proposed
scheme are proved. Numerical results demonstrate the efficiency of
this approach. We also establish some basic results on the
generalized Laguerre-spherical harmonic orthogonal
approximation, which play an important
role in spectral methods for various
problems defined on the whole space and unbounded domains with spherical geometry.
Local mesh refinement is one of the key steps in the
implementations of adaptive finite element methods. This paper
presents a parallel algorithm for distributed memory parallel
computers for adaptive local refinement of tetrahedral meshes using
bisection. This algorithm is used in PHG, Parallel
Hierarchical Grid (http://lsec.cc.ac.cn/phg/), a toolbox
under active development for parallel adaptive finite element
solutions of partial differential equations. The algorithm proposed
is characterized by allowing simultaneous refinement of submeshes to
arbitrary levels before synchronization between submeshes and
without the need of a central coordinator process for managing new
vertices. Using the concept of canonical refinement, a simple
proof of the independence of the resulting mesh on the mesh
partitioning is given, which is useful in better understanding the
behaviour of the bisectioning refinement procedure.
Surface reconstruction from unorganized data points is a challenging
problem in Computer Aided Design and Geometric Modeling. In this
paper, we extend the mathematical model proposed by Jüttler and
Felis (Adv. Comput. Math., 17 (2002), pp. 135-152) based on tensor
product algebraic spline surfaces from fixed meshes to adaptive
meshes. We start with a tensor product algebraic B-spline surface
defined on an initial mesh to fit the given data based on an
optimization approach. By measuring the fitting errors over each
cell of the mesh, we recursively insert new knots in cells over
which the errors are larger than some given threshold, and construct
a new algebraic spline surface to better fit the given data locally.
The algorithm terminates when the error over each cell is less than
the threshold. We provide some examples to demonstrate our algorithm
and compare it with Jüttler's method. Examples suggest that our
method is effective and is able to produce reconstruction surfaces
of high quality.
In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An $\varepsilon$-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.
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