Less is often more : applied inverse problems using hp-forward models

To solve an applied inverse problem, a numerical forward model for the problem’s physics is required. Commonly, the finite element method is employed with discretizations consisting of elements with variable size h and polynomial degree p. Solutions to hp-forward models are known to converge exponentially by simultaneously decreasing h and increasing p. On the other hand, applied inverse problems are often ill-posed and their minimization rate exhibits uncertainty. Presently, the behavior of applied inverse problems incorporating hp elements of differing p, h, and geometry is not fully understood. Nonetheless, recent research suggests that employing increasingly higher-order hp-forward models (increasing mesh density and p) decreases reconstruction errors compared to inverse regimes using lower-order hp-forward models (coarser meshes and lower p). However, an affirmative or negative answer to following question has not been provided, “Does the use of higher order hp-forward models in applied inverse problems always result in lower error reconstructions than approaches using lower order hp-forward models?”

In this article we aim to reduce the current knowledge gap and answer the open question by conducting extensive numerical investigations in the context of two contemporary applied inverse problems: elasticity imaging and hydraulic tomography – nonlinear inverse problems with a PDE describing the underlying physics. Our results support a negative answer to the question – i.e. decreasing h (increasing mesh density), increasing p, or simultaneously decreasing h and increasing p does not guarantee lower error reconstructions in applied inverse problems. Rather, there is complex balance between the accuracy of the hp-forward model, noise, prior knowledge (regularization), Jacobian accuracy, and ill-conditioning of the Jacobian matrix which ultimately contribute to reconstruction errors. As demonstrated herein, it is often more advantageous to use lower-order hp-forward models than higherorder hp-forward models in applied inverse problems. These realizations and other counterintuitive behavior of applied inverse problems using hp-forward models are described in detail herein.