Determining the nonlinearity in an acoustic wave equation
Determining the nonlinearity in an acoustic wave equation
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DOI：
10.1002/mma.8001
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发表时间：
2021-07
影响因子：
2.9
通讯作者：
B. Kaltenbacher;W. Rundell
中科院分区：
文献类型：
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作者：
B. Kaltenbacher;W. Rundell
我们考虑一个描述高强度超声传播的非线性偏微分方程的待定系数反问题，该方程广泛用于医学成像和治疗。在压力公式中使用韦斯特丙酮方程的标准模型中，通常的非线性项的形式为ppt。然而，这应该被认为是对更复杂的物理模型的低阶近似，其中需要高阶项。在这里，我们假设一个更一般的情况下，采取的形式是f（p）pt和f是未知的，必须从数据测量恢复。对应于典型的测量设置，叠加数据由单个点处或代表接收换能器阵列的一维集合上的声压的时间跟踪观测值组成。除了分析所得到的偏微分方程的适定性之外，我们还展示了从f到叠加数据的线性化正向映射的内射性，并将其用作几个迭代方案的动机来恢复f。数值模拟也将被示出，以说明该方法的效率。
We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high‐intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form ppt. However, this should be considered as a low‐order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is f(p) pt and f is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consist of time trace observations of the acoustic pressure at a single point or on a one‐dimensional set Σ representing the receiving transducer array at a fixed time. Additionally to an analysis of well‐posedness of the resulting pde, we show injectivity of the linearized forward map from f to the overposed data and use this as motivation for several iterative schemes to recover f. Numerical simulations will also be shown to illustrate the efficiency of the methods.