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 $p p_t$. 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 $f(p)\,p_t$ and $f$ is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consists of time trace observations of the acoustic pressure at a single point or on a one dimensional set $\Sigma$ representing the receiving transducer array at a fixed time. Additionally to an analysis of well-posedness of the resulting {\sc 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.
翻译:我们认为,在医疗成像和治疗中广泛使用的描述高强度超声波传播的非线性部分差异方程式中,一个不确定的系数反向问题是一个未确定的问题。在使用Westervelt方程式的压力配方标准模型中,通常的非线性术语是1美元p_t美元的形式。然而,这应被视为一个更复杂的物理模型的低顺序近似值,需要更高的顺序条件。这里我们假设了一个比较一般的情况,即表使用的是1美元(p)\\,p_t美元和1美元,从数据测量中无法找到,而且必须从数据测量中回收。与典型的测量设置相对应的是,过多的数据包括单点或一维数对声压进行的时间跟踪观测,代表固定时间接收转导器阵列的美元/Sigma美元。除了对结果 ~c pde} 的精度分析外,我们还将显示线性远方图从$f美元到过量数据的输入率,并将它用作若干迭代机制恢复$f美元的积极性。还将显示各种方法的效率。