Inverse wave problems (IWPs) amount in non-linear optimization problems where a certain distance between a state variable and some observations of a wavefield is to be minimized. Additionally, we require the state variable to be the solution of a model equation that involves a set of parameters to be optimized. Typical approaches to solve IWPs includes the adjoint method, which generates a sequence of parameters and strictly enforces the model equation at each iteration, and, the wavefield reconstruction inversion (WRI) method, which jointly generates a sequence of parameters and state variable but does not strictly enforce the model. WRI is considered to be an interesting approach because, by virtue of not enforcing the model at each iteration, it expands the search space, and can thus find solutions that may not be found by a typical adjoint method. However, WRI techniques generally requires the tuning of a penalty parameter until the model equation is considered satisfied. Alternatively, a fixed penalty parameter can be chosen but, in such case, it is impossible for the algorithm to find a solution that satisfies the model equation exactly. In the present work, we present a, to our knowledge, novel technique of WRI type which jointly generates a sequence of parameters and state variable, and which loosely enforces the model. The method is based on a TR-SQP method which aims at minimizing, at each iteration, both the residual relative to the linearized model and a quadratic approximation of the cost functional. Our method approximately solves a sequence of quadratic subproblems by using a Krylov method. The Hessian-vector product is computed using the second-order adjoint method. The method is demonstrated on a synthetic case, with a configuration relevant to medical imaging.

逆波问题 (IWP) 大量出现在非线性优化问题中，其中状态变量与波场的某些观测值之间的特定距离要最小化。此外，我们要求状态变量是涉及一组要优化的参数的模型方程的解。解决 IWP 的典型方法包括伴随方法，它生成一系列参数并在每次迭代时严格执行模型方程，以及波场重建反演 (WRI)方法，它联合生成一系列参数和状态变量，但不严格执行模型。WRI 被认为是一种有趣的方法，因为由于在每次迭代时不强制执行模型，它扩展了搜索空间，因此可以找到典型伴随方法可能无法找到的解决方案。然而，WRI 技术通常需要调整惩罚参数，直到认为满足模型方程。或者，可以选择固定的惩罚参数，但在这种情况下，算法不可能找到完全满足模型方程的解。在目前的工作中，据我们所知，我们提出了一种 WRI 类型的新技术，它联合生成参数和状态变量的序列，并且松散地强制执行模型。该方法基于 TR-SQP 方法，该方法旨在在每次迭代时最小化相对于线性化模型的残差和成本函数的二次近似。我们的方法通过使用 Krylov 方法近似解决了一系列二次子问题。Hessian 向量积是使用二阶伴随法计算的。该方法在一个合成案例上进行了演示，具有与医学成像相关的配置。