We develop a controllability strategy for the computation of frequency-domain solutions of the 3D visco-elastic wave equation, in the perspective of seismic imaging applications. We generalize the controllability results for such equations beyond the sound-soft scattering (obstacle) problem. We detail the conjugate gradient implementation and show how an inner elliptic problem needs to be solved to compute the Riesz representative of the gradient at each iteration. We select a spectral-element spatial discretization and a fourth-order Runge-Kutta time discretization. We implement the controllability method in the framework of the SEM46 full waveform modeling and inversion software, to inherit for its excellent scalability which relies on an efficient domain decomposition algorithm. We perform a series of numerical experiments to validate the approach and illustrate its scalability up to more than fifteen hundred cores. In this case, with an elapsed time of less than 50 minutes, we solve a problem on a cubic domain containing up to 160 wavelengths in each direction, involving more than 1.7 billion unknowns.

从地震成像应用的角度，我们开发了一种可控性策略，用于计算 3D 粘弹性波方程的频域解。我们将这些方程的可控性结果推广到声音软散射（障碍物）问题之外。我们详细介绍了共轭梯度的实现，并展示了如何解决内椭圆问题来计算每次迭代中梯度的 Riesz 代表。我们选择谱元空间离散化和四阶龙格-库塔时间离散化。我们在SEM46全波形建模与反演软件的框架下实现了可控性方法，以继承其依赖于高效域分解算法的卓越可扩展性。我们进行了一系列数值实验来验证该方法并说明其可扩展到超过 1500 个内核。在这种情况下，我们用不到 50 分钟的时间解决了每个方向上包含多达 160 个波长的三次域上的问题，涉及超过 17 亿个未知数。