Several applications in medical imaging and non-destructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly non-linear ill-posed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex non-linear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimization-based approaches for inverse coefficient problems.
翻译:在医学成像和非破坏性材料测试中,有几个应用导致逆向椭圆系数问题,在这种情况下,对椭圆形PDE中未知的系数功能要根据对其解决办法的部分了解来确定,这通常是一个高度非线性不良的反向问题,对于这个问题,很难实现独特的可重建性结果、稳定性估计和数字方法的全球趋同。本说明的目的是指出反向系数问题与有助于应对这些挑战的半无穷编程之间的新联系。我们表明,以有限数量测量的反向椭圆形罗宾传播问题,可以等同于一个独特的可溶性可调和的非线性半线性优化问题。这样可以明确估计实现预期解决方案所需的测量数量,为噪音数据得出误差估计,并克服通常出现在以优化为基础的办法中用于反向系数问题的当地微型问题。