Basic ideas of operator preconditioning
Contents
2 Operator preconditioning based on pseudodifferential operator of opposite orders
3 Boundary integral operators considered as pseudodifferential operators
1 Pseudodifferential operator
Derivative properties of Fourier transform
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Fourier transform on a rapidly decreasing function \begin{align} D^{\alpha }(\mathcal {F}\varphi )(\xi ) &= (-\rmi )^{\abs {\alpha }}\mathcal {F}(x^{\alpha }\varphi )(\xi ) \\ \mathcal {F}(D^{\alpha }\varphi )(\xi )&=\rmi ^{\abs {\alpha }}\xi ^{\alpha }(\mathcal {F}\varphi )(\xi ) \end{align}
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Fourier transform on a tempered distribution \begin{align} D^{\alpha }(\mathcal {F}T)&=(-\rmi )^{\abs {\alpha }}\mathcal {F}(\xi ^{\alpha }T)\\ \mathcal {F}(D^{\alpha }T)&=\rmi ^{\abs {\alpha }}x^{\alpha }(\mathcal {F}T) \end{align}
We can see that when the partial differential operator \(D^{\alpha }\) is applied to a function \(\varphi \) or a distribution \(T\), it is equivalent to multiply a factor \(\xi ^{\alpha }\) or \(x^{\alpha }\) along with the complex constant \(\mathrm {i}^{\lvert \alpha \rvert }\), i.e. a monomial with respect to the variable in the reciprocal space 1, to the Fourier transform of the original \(\varphi \) or \(T\).
Therefore, for a general linear partial differential operator (Folland, Chapter 8) \begin{equation} L = \sum _{\lvert \alpha \rvert \leq k} a_{\alpha }(x) D^{\alpha }, a_{\alpha }(x)\in C^{\infty }(\Omega ), \end{equation} Its influence on a rapidly decreasing function \(u\) in the frequency domain is multiplying a polynomial with respect to \(\xi \) \(\sigma _L(x,\xi )\) to \(\hat {u}(\xi )\), i.e. \begin{equation} \mathcal {F}(Lu)(\xi ) = \sigma _L(x,\xi ) \hat {u}(\xi ) = \left ( \sum _{\lvert \alpha \rvert \leq k} a_{\alpha }(x)\mathrm {i}^{\lvert \alpha \rvert }\xi ^{\alpha } \right ) \hat {u}(\xi ). \end{equation} If we generalize the polynomial \(\sigma _L(x,\xi )\) to a larger class of functions \(p(x,\xi )\), more types of linear operators, such as integral operators, can be considered in a same unified framework. This brings about the concept of pseudodifferential operators.
Definition 1 (Space of symbols) The space of symbols of order \(m\) is a set containing all \(p(x,\xi )\in C^{\infty }(\Omega \times \mathbb {R}^d)\) such that for all multi-indices \(\alpha \) and \(\beta \) and any compact set \(K\subset \Omega \), there exists \(C_{\alpha ,\beta ,K}\) satisfying \begin{equation} \sup _{x\in K} \lvert D_x^{\beta }D_{\xi }^{\alpha }p(x,\xi ) \rvert \leq C_{\alpha ,\beta ,K}(1+\lvert \xi \rvert )^{m-\lvert \alpha \rvert }. \end{equation}
Definition 2 (Pseudodifferential operator) A pseudodifferential operator \(p(x,D)\) of order \(m\) is a linear map from \(\mathcal {D}(\Omega )\) to \(C^{\infty }(\Omega )\) such that \begin{equation} \mathcal {F}(p(x,D)u) = p(x,\xi )\hat {u}(\xi ), \end{equation} where \(p(x,\xi )\) is a symbol of order \(m\).
The ingenious idea here is converting a differential operator to an integral representation in the frequency domain via Fourier transform. This is not only equivalent to the original form, but also makes it possible to handle more classes of operators.
2 Operator preconditioning based on pseudodifferential operator of opposite orders
The construction of a preconditioner \(B\) for an operator \(A\) is equivalent to looking for an approximate inverse operator of \(A\). In the discrete case, the preconditioning matrix \(\underline {B}^{-1}\) should be spectrally equivalent to \(A\) and \(\underline {B}\) should be an approximate inverse matrix of \(\mathcal {A}\). Therefore, the behavior of the composite operator \(BA\) should be similar to an identity operator. Being a pseudodifferential operator of order \(m\), \(A\) has the symbol \(p(x,\xi )\) whose mixed partial derivatives (with respect to both \(x\) and \(\xi \)) are controlled by \(C_{\alpha ,\beta ,K}(1+\lvert \xi \rvert )^{m-\lvert \alpha \rvert }\). For partial derivatives with respect to only \(\xi \), they are comparable to the polynomial of \(1+\lvert \xi \rvert ^m\) of order \(m\). If the symbol of the preconditioner \(B\) is \(\frac {1}{1+\lvert \xi \rvert ^m}\) of order \(-m\), the symbol of the composite operator \(BA\) is comparable to \(1\). Therefore, the operator preconditioning method looks for a pseudodifferential operator having the opposite order.
3 Boundary integral operators considered as pseudodifferential operators
According to the definition in (Steinbach and Wendland), the pseudodifferential operator \(A: H^s(\Gamma ) \rightarrow H^{s-2\alpha }(\Gamma )\) has an order \(2\alpha \). Therefore, we have the orders for boundary integral operators which are treated as pseudodifferential operators.
| Boundary integral operator | Order |
| \(V: H^{-1/2}(\Gamma ) \rightarrow H^{1/2}(\Gamma )\) | -1 |
| \(K: H^{1/2}(\Gamma ) \rightarrow H^{1/2}(\Gamma )\) | 0 |
| \(K': H^{-1/2}(\Gamma ) \rightarrow H^{-1/2}(\Gamma )\) | 0 |
| \(D: H^{1/2}(\Gamma ) \rightarrow H^{-1/2}(\Gamma )\) | 1 |
Among these operators, \(V\) and \(D\) are self-adjoint and appear as the main operators in PDEs. They happen to have opposite orders. In addition, according to (Steinbach, Corollary 6.19 P138), \(V\) and \(D\) have the following relations: \begin{equation} \begin{aligned} VD &= (\sigma I+K)((1-\sigma )I-K) \\ DV &= (\sigma I+K')((1-\sigma )I-K') \end{aligned}. \end{equation} The terms on the right hand sides of these two equations involve \(K\) and \(K'\). Both of them are zero order pseudodifferential operators. So are \(VD\) and \(DV\). This is consistent with the fact that \(V\) and \(D\) have opposite orders. Therefore, \(V\) is inherently the preconditioner of \(D\) and vice versa.
References
Gerald B. Folland. Introduction to Partial Differential Equations. Second Edition. Princeton University Press, second edition edition. ISBN 978-0-691-04361-6.
Olaf Steinbach. Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer Science & Business Media. ISBN 978-0-387-31312-2.
Olaf Steinbach and Wolfgang L. Wendland. The construction of some efficient preconditioners in the boundary element method. 9(1-2):191–216. URL http://link.springer.com/article/10.1023/A:1018937506719.
1As a convention, we use \(x\) as the variable for \(\varphi \) in the “time” domain and \(\xi \) in the reciprocal, i.e. “frequency”, domain. According to the definition of the Fourier transform of a distribution, the variable adopted for \(T\) should be \(\xi \) instead and the variable for its Fourier transform should be \(x\).