Parameter-robust preconditioning for the optimal control of the wave equation

Abstract

In this paper, we propose and analyze a new matching-type Schur complement preconditioner for solving the discretized first-order necessary optimality conditions that characterize the optimal control of wave equations. Coupled with this is a recently developed second-order implicit finite difference scheme used for the full space-time discretization of the optimality system of PDEs. Eigenvalue bounds for the preconditioned system are derived, which provide insights into the convergence rates of the preconditioned Krylov subspace method applied. Numerical examples are presented to validate our theoretical analysis and demonstrate the effectiveness of the proposed preconditioner, in particular its robustness with respect to very small regularization parameters, and all mesh sizes in the spatial variables.

Appendix A

Lemma A.1

LetZ_N = pendiag([1,− 2, 1, 0, 0]) be a squarematrix with dimensionN × N,then its inverse is a lower triangular Toeplitz matrix with the followingexpression:

$$ Z_{N}^{-1}=\text{LTToeplitz}([1,2,\cdots,N-1,N]^{{\intercal}}), $$

in which $[1,2,\cdots ,N-1,N]^{{\intercal }}$ gives the first column of $Z_{N}^{-1}$.

Proof

We prove the above statement by mathematical induction on the dimension N. When N = 3, the result holds, since it is straightforward to verify that:

$$ Z_{3}^{-1}=\left[\begin{array}{ccc} 1 & 0 & 0\\ -2 & 1 & 0\\ 1 & -2 & 1 \end{array}\right]^{-1} =\left[\begin{array}{ccc} 1 & 0 & 0\\ 2 & 1 & 0\\ 3 & 2 & 1 \end{array}\right] =\text{LTToeplitz}([1,2,3]^{{\intercal}}). $$

Now, we assume the conclusion is true for N = k and proceed to show the conclusion also holds for N = k + 1. When N = k + 1, we have the following recursive block structure:

$$ \begin{array}{@{}rcl@{}} Z_{k+1}&=\left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1 & 0 &0 &0& {\cdots} &0 \\ -2 & 1&0&0 &{\cdots} & 0 \\ 1&-2&1&0&\ddots& {\vdots} \\ 0&\ddots&{\ddots} &{\ddots} &{\ddots} & \vdots\\ 0& 0& {\ddots} &-2&1 &0 \\ 0&0&{\cdots} &1& -2 &1 \end{array}\right] = \left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1&0\\ z&Z_{k} \end{array}\right], \end{array} $$

where $z=[-2,1,0,\cdots ,0]^{{\intercal }}$. A simple step of block Gaussian elimination leads to

$$ \left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1&0\\ -z&I \end{array}\right] \left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1&0\\ z&Z_{k} \end{array}\right]= \left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1&0\\ 0&Z_{k} \end{array}\right], $$

which hence implies:

$$ Z_{k+1}^{-1}=\left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1&0\\ z&Z_{k} \end{array}\right]^{-1} = \left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1&0\\ 0&Z_{k}^{-1} \end{array}\right] \left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1&0\\ -z&I \end{array}\right] = \left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1&0\\ -Z_{k}^{-1}z& Z_{k}^{-1} \end{array}\right]. $$

(A.1)

By the inductive assumption, it holds that:

$$ Z_{k}^{-1}=\text{LTToeplitz}([1,2,\cdots,k-1,k]^{{\intercal}}). $$

Based on this, we can also easily obtain (noting that $z=[-2,1,0,\cdots ,0]^{{\intercal }}$ only contains nonzeros in the first two entries) that:

$$ -Z_{k}^{-1}z=[2,3,\cdots,k+1]^{{\intercal}}. $$

Inserting both blocks back into (A.1), we arrive at the following desired lower triangular Toeplitz matrix:

$$ Z_{k+1}^{-1}= \left[ \begin{array}{ccccccccccccccccccccccccccccccccccccc} 1 & 0 &0 &0& {\cdots} &0 \\ 2 & 1&0&0 &{\cdots} & 0 \\ 3& 2&1&0&\ddots& {\vdots} \\ \vdots&\ddots&{\ddots} &{\ddots} &{\ddots} & \vdots\\ k& \ddots& {\ddots} &2&1 &0 \\ k+1&k&{\cdots} &3& 2 &1 \end{array}\right] =\text{LTToeplitz}([1,2,\cdots,k-1,k,k+1]^{{\intercal}}), $$

which therefore completes the proof by the principle of mathematical induction. □