The approximation of the Maxwell eigenvalue problem using a least-squares method

In this paper we consider an approximation to the Maxwell's eigenvalue problem based on a very weak formulation of two div-curl systems with complementary boundary conditions. We formulate each of these div-curl systems as a general variational problem with different test and trial spaces, i.e., the solution space is and components in the test spaces are in subspaces of , the Sobolev space of order one on the computational domain . A finite-element least-squares approximation to these variational problems is used as a basis for the approximation. Using the structure of the continuous eigenvalue problem, a discrete approximation to the eigenvalues is set up involving only the approximation to either of the div-curl systems. We give some theorems that guarantee the convergence of the eigenvalues to those of the continuous problem without the occurrence of spurious values. Finally, some results of numerical experiments are given.

     

Discrete linearized least-squares rational approximation on the unit circle

We already generalized the Rutishauser—Gragg—Harrod—Reichel algorithm for discrete least-squares polynomial approximation on the real axis to the rational case. In this paper, a new method for discrete least-squares linearized rational approximation on the unit circle is presented. It generalizes the algorithms of Reichel—Ammar—Gragg for discrete least-squares polynomial approximation on the unit circle to the rationale case. The algorithm is fast in the sense that it requires order m computation time where m is the number of data points and is the degree of the approximant. We describe how this algorithm can be implemented in parallel. Examples illustrate the numerical behavior of the algorithm.     

Scattering by moving bodies: The quasi stationary approximation

The scattered field produced by a rotating or vibrating object in the presence of a time harmonic plane wave is usually approximated by a time dependent sequence of stationary fields, called a quasi stationary field. Recently the mathematical theory of scattering of scalar fields of Lax and Phillips has been extended to cover moving bodies. Within this framework the quasi stationary approximation is shown to be valid in each sphere of finite radius which contains the body. Far fields for the exact field and the quasi stationary field are defined and the difference is estimated in terms of a small parameter.     

On numerical methods for discrete least-squares approximation by trigonometric polynomials

Fast, efficient and reliable algorithms for discrete least-squares approximation of a real-valued function given at arbitrary distinct nodes in by trigonometric polynomials are presented. The algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only arithmetic operations as compared to operations needed for algorithms that ignore the structure of the problem. An algorithm which solves this problem with real-valued data and real-valued solution using only real arithmetic is given. Numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general QR decomposition methods for the least-squares problem.

     

Geometrical inverse matrix approximation for least-squares problems and acceleration strategies

Numerical Algorithms - We extend the geometrical inverse approximation approach to the linear least-squares scenario. For that, we focus on the minimization of $1-cos limits (X(A^{T}A),I)$ ,...     

Local polynomial reproduction and moving least squares approximation

Local polynomial reproduction is a key ingredient in providingerror estimates for several approximation methods. To boundthe Lebesgue constants is a hard task especially in a multivariatesetting. We provide a result which allows us to bound the Lebesgueconstants uniformly and independently of the space dimensionby oversampling. We get explicit and small bounds for the Lebesgueconstants. Moreover, we use these results to establish errorestimates for the moving least squares approximation scheme,also with special emphasis on the involved constants. We discussthe numerical treatment of the method and analyse its effort.Finally, we give large scale examples.     

Truncated stochastic approximation with moving bounds: convergence

In this paper we consider a wide class of truncated stochastic approximation procedures. These procedures have three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and a dynamically changing random regression function. We establish convergence and consider several examples to illustrate the results.     

On the least-squares conjugate-gradient solution of the finite element approximation of Burgers' equation

A finite element approximation of the two-dimensional steady Burgers' equation is presented and a conjugate gradient approach is taken to solve the resulting finite element equations. The scheme is computationally efficient and is relatively easy to implement. An optimal error bound is established and a set of test problems with known analytic solutions is given to demonstrate the efficiency of the method.