Preconditioning rectangular spectral collocation
Rectangular spectral collocation (RSC) methods have recently been proposed to solve linear and nonlinear differential equations with general boundary conditions and/or other constraints. The involved linear systems in RSC become extremely ill-conditioned as the number of collocation points increase. By introducing suitable Birkhoff-type interpolation problems, we present pseudospectral integration preconditioning matrices for the ill-conditioned linear systems in RSC. The condition numbers of the preconditioned linear systems are independent of the number of collocation points. Numerical examples are given.
agrange interpolation, Birkhoff-type interpolation, rectangular spectral collocation, integration preconditioning
65L60, 41A05, 41A10
Rectangular spectral collocation methods  have recently been demonstrated to be a convenient means of solving the problems when the row replacement or ‘boundary bordering’ strategy of standard spectral collocation methods [4, 11, 1, 2, 9] becomes ambiguous. Specifically, an th-order differential operator is discretized by a rectangular matrix directly, allowing constraints to be appended to form an invertible square system. However, the involved linear systems become extremely ill-conditioned as the number of collocation points increases. Typically, the condition number grows like . Efficient preconditioners are highly required when solving the linear systems by an iterative method.
Recently, Wang, Samson, and Zhao  proposed a well-conditioned collocation method to solve linear differential equations with various types of boundary conditions. By introducing a suitable Birkhoff interpolation problem , they constructed a pseudospectral integration preconditioning matrix, which is the exact inverse of the pseudospectral discretization matrix of the th-order derivative operator together with boundary conditions. In this paper, we employ the similar idea to construct a pseudospectral integration matrix, which is the exact inverse of the discretization matrix arising in the rectangular spectral collocation method for th-order derivative operator together with general linear constraints. The condition number of the resulting linear system is independent of the number of collocation points when the new pseudospectral integration matrix is used as a right preconditioner for an th-order linear differential operator together with the same constraints.
The rest of the paper is organized as follows. In §2, we review several topics required in the following sections. In §3, we introduce the new pseudospectral integration matrix by a suitable Birkhoff-type interpolation problem. In §4, we present the preconditioning rectangular spectral collocation method. Numerical examples are reported in §5. We present brief concluding remarks in §6.
2.1 Barycentric resampling matrix
Let be a set of distinct interpolation points satisfying
The associated barycentric weights are defined by
Let be another set of distinct interpolation points satisfying
The barycentric resampling matrix , , which interpolates between the points and , is defined by
If , then
2.2 Pseudospectral differentiation matrices
The Lagrange interpolation basis polynomials of degree associated with the points are defined by
where is the barycentric weight (2.2). Define the pseudospectral differentiation matrices:
The matrix is called a rectangular th-order differentiation matrix, which maps values of a polynomial defined on to the values of its th-order derivative on . Explicit formulae and recurrences for rectangular differentiation matrices are given in .
2.3 Chebyshev polynomials and Chebyshev points
The most widely used spectral methods for non-periodic problems are those based on Chebyshev polynomials and Chebyshev points. In this paper, we focus on these polynomials and points. However, everything we discuss can be easily generalized to the case of Jacobi polynomials and corresponding points.
The Chebyshev points of the first kind (also known as Gauss-Chebyshev points) are given by
In this case, the Gauss-Chebyshev quadrature weights are given by 
and the barycentric weights are given by 
Let be the set of all algebraic polynomials of degree at most . We have
The Chebyshev points of the second kind (also known as Gauss-Chebyshev-Lobatto points) are given by
In this case, the Gauss-Chebyshev-Lobatto quadrature weights are given by 
and the barycentric weights are given by 
Let be the Chebyshev polynomials (see, for example, ) given by
They are mutually orthogonal:
Let denote the Lagrange interpolation basis polynomials of degree associated with the points . The polynomial can be rewritten as
If is a subset of or , can be obtained with ease. For example, suppose that is a proper subset of . Let denote the map such that . Let denote the set such that if then . By (2.4) and (2.5), we have, for
Here , , can be obtained by solving the following linear system
In particular, if denote the Lagrange interpolation basis polynomials of degree associated with , we have
If denote the Lagrange interpolation basis polynomials of degree associated with , we have
Define the integral operators:
3 Pseudospectral integration matrices
Given and with , we consider the Birkhoff-type interpolation problem:
where each is a linear functional. Let be the Lagrange interpolation basis polynomials of degree associated with the points . Then the Birkhoff-type interpolation polynomial takes the form
where can be determined by the linear constraints Obviously, the existence and uniqueness of the Birkhoff-type interpolation polynomial is equivalent to that of . After obtaining , we can rewrite (3.1) as
Let and be the points as in (2.1). Define the th-order pseudospectral integration matrix (PSIM) as:
Define the matrices
It is easy to show that
Let be the discretization of the linear constraints , . We have the following theorem.
If for any ,
The result follows from
and Lemma 2.1.
Now we give concrete examples. Consider the non-separable linear constraint
and the global linear constraint
where , and are given constants. They are straightforward to discretize: for (3.3),
and for (3.4),
and is a column vector of Clenshaw-Curtis quadrature weights .
The first-order Birkhoff-type interpolation problem takes the form:
Given with , we have
Given , we have
4 Preconditioning rectangular spectral collocation
Consider the th-order differential equations of the form
together with linear constraints
Here we use boldface letters to indicate a column vector obtained by discretizing at the points except for the unknown . For example,
The global collocation system is given by
5 Numerical results
In this section, we compare the rectangular spectral collocation (RSC) scheme (4.3) and the preconditioned rectangular spectral collocation (P-RSC) scheme (4.4). In all computations, the Chebyshev points of the second kind are chosen as and the Chebyshev points of the first kind are chosen as .
We consider the equation
with the linear constraint
or the linear constraint
where is a given constant. We report in Table 1 the condition numbers of the linear systems in RSC and P-RSC with , and various . We observe that the condition numbers of P-RSC are independent of , while those of RSC behave like .
|Constraint (5.2)||Constraint (5.3)||Constraint (5.2)||Constraint (5.3)|
In Figure 1 (a) we plot the exact solution against the numerical solutions obtained by RSC and P-RSC with . In Figure 1 (b) we plot the maximum point-wise errors of RSC and P-RSC. It indicates that for this example, even for very large , both RSC and P-RSC are very stable.
We consider the equation
with the linear constraints
The function , and are chosen such that the exact solution of (5.4) is
In Tables 2-4, we present the condition numbers, the maximum point-wise errors, and the number of iterations via the GMRES algorithm  with the relative tolerance equal to and the restart number equal to , for the cases , , and , respectively. We observe that the condition numbers of P-RSC are independent of , while those of RSC behave like .
6 Concluding remarks
We have proposed a preconditioning rectangular spectral collocation scheme for th-order ordinary differential equations together with general linear constraints. The condition number of the resulting linear system is typically independent of the number of collocation points. And the linear system can be solved by an iterative solver within a few iterations. The application of the preconditioning scheme to nonlinear problems is straightforward.
We thank Prof. Li-Lian Wang (Nanyang Technological University, Singapore) for providing the MATLAB codes used in .
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