A diagonal preconditioner for singularly perturbed problems
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Abstract
Using wavelet discretization with a standard wavelet diagonal preconditioning for singularly perturbed twopoint boundary value problems, one can observe that condition numbers of arising stiffness matrices are growing with decreasing parameter ϵ when a nonsymmetric part starts to dominate. We propose here a simple diagonal preconditioning which significantly improves condition numbers of the stiffness matrices with a dominating nonsymmetric part and compare it with a standard wavelet preconditioning. Further, we prove that the condition numbers of diagonally preconditioned stiffness matrices are bounded independent of the matrix size. Numerical examples are given.
Keywords
wavelets singularly perturbed problems boundary value problems preconditioningMSC
65L11 65L10 65T60 65F081 Introduction
Many problems in science and technology can be modeled by boundary value problems for singularly perturbed differential equations. Singularly perturbed problems arise for example in the modeling of fluid flow at high Reynolds numbers, water quality problems in river networks, convective heat transport problems with large Peclet numbers, drift diffusion equations of semiconductor device modeling, simulation of oil extraction from underground reservoirs, theory of plates and shells, atmospheric pollution, groundwater transport, and chemical reactor theory. For a detailed survey of different applications, we refer to [1]. Recently also singularly perturbed semilinear boundary value problems with discontinuous coefficients and nonlinear reactiondiffusion equations attracted some attention; see [2, 3] and the references therein. A vast majority of these problems cannot be solved analytically and therefore it is necessary to solve them approximately. In the modeling of the above processes, one can observe boundary and interior layers whose width can be arbitrarily small.
In recent years, there have appeared some promising results in using wavelets to solve singularly perturbed problems. In [5], a nonadaptive numerical method based on wavelets of Hermite cubic splines was presented and improved results were obtained in comparison with other techniques. In [6], the authors constructed wavelets of order 5 with five vanishing wavelet moments with respect to which stiffness matrices for ordinary differential equations with constant coefficients are very sparse (in comparison with other kinds of wavelets) and their condition numbers are similarly small as in [7] for lower order wavelets. Then they applied tensor product wavelets to the adaptive solution of a reactiondiffusion equation in two space dimensions.
In this contribution, we focus on methods based on wavelets. Using wavelet discretization with a standard wavelet diagonal preconditioning for singularly perturbed twopoint boundary value problems, one can observe that the condition numbers of arising stiffness matrices are growing with decreasing parameter ϵ when a nonsymmetric part starts to dominate. In the wavelet methods, the continuous problem is transformed into a wellconditioned discrete problem. And once a wellconditioned nonsymmetric problem is given, squaring will yield a symmetric positive definite formulation [8]. Therefore an efficient preconditioning is very important since the rate of convergence for most iterative linear solvers depends on the condition number of a preconditioned matrix. We propose here a simple diagonal preconditioning which significantly improves the condition numbers of the stiffness matrices with a dominating nonsymmetric part. A diagonal preconditioning is optimal for adaptive wavelet methods in which often stiffness matrices are not explicitly assembled and not stored in a computer memory. This paper is organized as follows: First, we briefly introduce wavelet bases and their properties. Then we propose a new diagonal preconditioning and we prove that the condition numbers of the infinite diagonally preconditioned stiffness matrices are finite. At the end, we provide some numerical examples.
2 Wavelet bases
At last, for \(s \geq0\) the space \(H^{s}\) will denote a closed subspace of the Sobolev space \(H^{s} ( 0,1) \), defined e.g. by imposing homogeneous boundary conditions at one or both endpoints, and for \(s < 0\) the space \(H^{s}\) will denote the dual space \(H^{s} := (H ^{s})'\). \(\Vert \cdot \Vert _{H^{s}}\) will denote the corresponding norm. Further \(l_{2}(\mathcal{J})\) will denote the space consisting of the power summable sequences and \(\Vert \cdot \Vert _{l_{2}( \mathcal{J})}\) will denote the corresponding norm.
 Ψ is a Riesz basis of \(H^{s}\), which means Ψ forms a basis of \(H^{s}\) and there exist constants \(c_{s},C_{s} > 0\) such that for all \(\mathbf{b}= \{ b_{\lambda} \} _{\lambda\in\mathcal{J}} \in l_{2} ( \mathcal{J} ) \) we havewhere \(\sup c_{s}\), \(\inf C_{s}\) are called Riesz bounds and \(\operatorname{cond}\Psi:= \frac{\inf C_{s}}{\sup c_{s}}\) is called the condition number of Ψ.$$ c_{s} \Vert \mathbf{b} \Vert _{l_{2} ( \mathcal{J} ) } \leq \bigl\Vert \mathbf{b}^{T} \mathbf{D}^{s} \Psi\bigr\Vert _{H^{s}} \leq C_{s} \Vert \mathbf{b} \Vert _{l_{2} ( \mathcal{J} ) }, $$(3)

Functions are local in the sense that \(\operatorname{diam}( \operatorname{supp}\psi_{\lambda} ) \leq C 2^{\vert \lambda \vert }\) for all \(\lambda\in\mathcal{J}\), where C is a constant independent of λ.
 Functions \(\psi_{\lambda}, \lambda\in\mathcal{J}_{\Psi}\), have cancellation properties of order m, i.e.It means that integration against wavelets eliminates smooth parts of functions and it is equivalent with vanishing wavelet moments of order m.$$\biggl\vert \int_{0}^{1} v(x) \psi_{\lambda}(x)\,dx \biggr\vert \leq2^{m \vert \lambda \vert } \vert v \vert _{H^{m} ( 0,1) },\quad \forall v \in H^{m} ( 0,1). $$
Norm equivalences (3) have the following important consequence which will be used later.
Theorem 1
For the proof, we refer to [8].

Vanishing wavelet moments (the cancellation property) lead to sparse representations of functions and operators.

In the case that an original continuous problem is well conditioned, the Riesz property (3) leads also to wellconditioned stiffness matrices [9]. Moreover, it implies that every function in \(H^{s}\) has a unique expansion in the scaled wavelet basis and that there is a tight relation between the function norms and wavelet coefficients. It means that small changes in wavelet coefficients can cause only small changes in the function and the other way around. Wavelet bases with small Riesz bounds were constructed for example in [10, 11, 12, 13, 14].

There are waveletbased asymptotically optimal algorithms for solving elliptic PDEs. See for example [15, 16, 17]. It means that the number of floating point operations depends linearly on the number of nonzero wavelet coefficients.
3 Wavelet discretization
Theorem 2
Proof
Then the operator A is bounded and coercive and an application of the LaxMilgram lemma implies the existence of the unique solution for any \(f \in H^{1} ( 0,1) \). Further, we prove that a diagonally scaled wavelet basis is a Riesz basis when it is scaled with the proposed diagonal scaling.
Theorem 3
Proof
And finally we prove that the condition numbers of the infinite diagonally preconditioned stiffness matrices are finite.
Theorem 4
Proof
Remark 1
From the previous theorem it immediately follows that, for \(b=0\), we obtain symmetric stiffness matrices with small condition numbers which are independent of ϵ and are dependent only on Riesz constants of the used wavelet basis.
4 Numerical tests
In adaptive settings the discrete infinitedimensional problem (10) is solved approximately up to the given target accuracy. Adaptive wavelet methods are meshless methods. An adaptive algorithm usually starts with the zero function and gradually adds new (biggest) elements to approximate (10) and then only a part of the whole stiffness matrix is used. While in nonadaptive settings, we compute with all wavelets up to the given level. In numerical experiments, we use the second approach because it is more appropriate for comparison of the properties of the wavelet stiffness matrices.
Four types of wavelets, \(\psi_{1}\), \(\psi_{2}\), \(\psi_{3}\), and \(\psi_{4}\), are constructed and the wavelet basis is then formed by translations and dilations of these four types of wavelets. Wavelets from the space \(W_{n+1}\) are orthogonal to the scaling functions from the space \(V_{n}\) for \(n \geq1\). The first two wavelets have supports in \([1,1]\) and are uniquely determined by the above orthogonality condition and by imposing that the first one is odd and the second one is even. The second two wavelets have supports in \([2,2]\). And we impose again the orthogonality condition and one of them should be odd and the second one even. This property ensures that both the mass and the stiffness matrix corresponding to the onedimensional Laplacian have at most three wavelet blocks of nonzero elements in any column and then the number of nonzero elements in any column is bounded independent of matrix size.
In the following part, we assume that all basis functions are normalized with respect to the \(L_{2}\) norm, such that their norm is equal to one. Now, we look at the spectral properties of different stiffness matrices \(\mathbf{A}_{n}\). We will use the standard wavelet preconditioning with the diagonal preconditioner \(\mathbf{D}_{n}^{S} = \sqrt{ \operatorname{diag}(\mathbf{A}_{n})}\) [6] and then we will compare it with the proposed diagonal preconditioning \(\mathbf{D}_{n} ^{\mathrm{new}} := ( \sqrt{ ( \epsilon2^{2\vert \lambda \vert }+b2^{\vert \lambda \vert }+c) } \delta_{\lambda,\mu} ) \), where the constants ϵ, b, c are constants from (7).
Example 1
We start with the equation \( \epsilon u'' + u' + u/16=f \) with the Dirichlet boundary conditions \(u(0) = u(1) = 0\) and with small positive parameter ϵ. The corresponding discrete problem is the following:
The condition numbers of the stiffness matrices for \(\pmb{n=9}\)
ϵ  \(\boldsymbol{\operatorname{cond}(\mathbf{D}_{n}^{S} \mathbf{A}_{n} \mathbf{D}_{n}^{S})}\)  \(\boldsymbol{\operatorname{cond}(\mathbf{D}_{n}^{\mathrm{new}} \mathbf{A}_{n} \mathbf{D}_{n}^{\mathrm{new}})}\)  \(\boldsymbol{\operatorname{cond}(\mathbf{A}_{n})}\) 

10^{−0}  2.69 × 10^{1}  1.45 × 10^{2}  2.67 × 10^{6} 
10^{−1}  7.40 × 10^{1}  1.32 × 10^{2}  3.31 × 10^{5} 
10^{−2}  2.20 × 10^{2}  7.07 × 10^{1}  1.10 × 10^{5} 
10^{−3}  6.68 × 10^{2}  1.66 × 10^{2}  1.11 × 10^{5} 
10^{−4}  1.23 × 10^{3}  3.31 × 10^{2}  1.18 × 10^{5} 
10^{−5}  1.37 × 10^{3}  3.72 × 10^{2}  1.22 × 10^{5} 
10^{−6}  1.39 × 10^{3}  3.77 × 10^{2}  1.22 × 10^{5} 
10^{−7}…10^{−12}  1.39 × 10^{3}  3.78 × 10^{2}  1.23 × 10^{5} 
Example 2
The second equation will be \( \epsilon u'' + u'=f \) with the Dirichlet boundary conditions \(u(0) = u(1) = 0\) and with small positive parameter ϵ. The corresponding discrete problem is the following:
The condition numbers of the stiffness matrices for \(\pmb{n=9}\)
ϵ  \(\boldsymbol{\operatorname{cond}(\mathbf{D}_{n}^{S} \mathbf {A}_{n} \mathbf{D}_{n}^{S})}\)  \(\boldsymbol{\operatorname{cond}(\mathbf{D}_{n}^{\mathrm{new}} \mathbf {A}_{n} \mathbf{D}_{n}^{\mathrm{new}})}\)  \(\boldsymbol{\operatorname{cond}(\mathbf{A}_{n})}\) 

10^{−0}  1.47 × 10^{2}  3.67 × 10^{1}  6.27 × 10^{6} 
10^{−1}  1.12 × 10^{3}  7.27 × 10^{1}  8.17 × 10^{5} 
10^{−2}  3.95 × 10^{3}  6.46 × 10^{1}  2.75 × 10^{5} 
10^{−3}  2.94 × 10^{4}  1.24 × 10^{2}  2.78 × 10^{5} 
10^{−4}  6.30 × 10^{4}  2.07 × 10^{2}  1.87 × 10^{5} 
10^{−5}  6.41 × 10^{4}  2.03 × 10^{2}  1.57 × 10^{5} 
10^{−6}  6.41 × 10^{4}  2.03 × 10^{2}  1.55 × 10^{5} 
10^{−7}…10^{−12}  6.41 × 10^{4}  2.03 × 10^{2}  1.54 × 10^{5} 
Example 3
The third equation will be \( \epsilon u'' + u' + x^{2} u = f \) with the Dirichlet boundary conditions \(u(0) = u(1) = 0\) and with small positive parameter ϵ. The corresponding discrete problem is the following:
The condition numbers of the stiffness matrices for \(\pmb{n=9}\)
ϵ  \(\boldsymbol{\operatorname{cond}(\mathbf{D}_{n}^{S} \mathbf {A}_{n} \mathbf{D}_{n}^{S})}\)  \(\boldsymbol{\operatorname{cond}(\mathbf{D}_{n}^{\mathrm{new}} \mathbf {A}_{n} \mathbf{D}_{n}^{\mathrm{new}})}\)  \(\boldsymbol{\operatorname{cond}(\mathbf{A}_{n})}\) 

10^{−0}  7.93 × 10^{1}  8.25 × 10^{2}  3.80 × 10^{6} 
10^{−1}  4.26 × 10^{2}  6.69 × 10^{2}  4.97 × 10^{5} 
10^{−2}  2.31 × 10^{3}  4.37 × 10^{2}  1.67 × 10^{5} 
10^{−3}  1.28 × 10^{4}  7.74 × 10^{2}  1.62 × 10^{5} 
10^{−4}  7.25 × 10^{4}  1.29 × 10^{3}  1.69 × 10^{5} 
10^{−5}  4.12 × 10^{5}  1.39 × 10^{3}  1.71 × 10^{5} 
10^{−6}  2.23 × 10^{6}  1.40 × 10^{3}  1.72 × 10^{5} 
10^{−7}  7.81 × 10^{6}  1.40 × 10^{3}  1.72 × 10^{5} 
10^{−8}  1.20 × 10^{7}  1.40 × 10^{3}  1.72 × 10^{5} 
10^{−9}  1.28 × 10^{7}  1.40 × 10^{3}  1.72 × 10^{5} 
10^{−10}…10^{−12}  1.29 × 10^{7}  1.40 × 10^{3}  1.72 × 10^{5} 
5 Conclusion
We have proposed a new diagonal preconditioning for singularly perturbed problems discretized by wavelets. Numerical experiments show that the proposed preconditioning leads to significantly smaller condition numbers of stiffness matrices with a dominating nonsymmetric part in comparison with a standard wavelet diagonal preconditioning. Furthermore, we proved that the condition numbers of diagonally preconditioned stiffness matrices are bounded independent of the matrix size.
Notes
Acknowledgements
This work was supported by grant No. GA1609541S of the Czech Science Foundation.
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