A Numerical Study of Conjugate Gradient Directions for an Ultrasound Inverse Problem

In ultrasound inverse problems, the integral equation can be nonlinear, ill-posed, and computationally expensive. One approach to solving such problems is the conjugate gradient (CG) method. A key parameter in the CG method is the conjugate gradient direction. In this paper, we investigate the CG directions proposed by Polyak et al. (PPR), Hestenes and Stiefel (HS), Fletcher and Reeves (FR), Dai and Yuan (YD), and the two-parameter family generalization proposed by Nazareth (TPF). Each direction is applied to three test cases with different contrasts and phase shifts. Test case 1 has low contrast with a phase shift of 0.2π. Reconstruction of the object is obtained for all directions. The performances of the PPR, HS, YD, and TPF directions are comparable, while the FR direction gives the poorest performance. Test case 2 has medium contrast with a phase shift of 0.75π. Reconstruction is obtained for all but the FR direction. The PPR, HS, YD, and TPF directions have similar mean square error; the YD direction takes the least amount of CPU time. Test case 3 has the highest contrast with a phase shift of 1.003π. Only the YD direction gives reasonably accurate results.