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Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations. Solving such equations numerically is both complicated and computationally intensive, and numerical methods often prove to be unstable. Collocation schemes are easier to implement than full finite element formulations, but little appears to be known about their stability and convergence. Here we derive and analyse some new stable collocation schemes for the single layer equation for transient acoustic scattering, and use (spatial) Fourier and (temporal) Laplace transform techniques to demonstrate that such stable schemes are second order convergent. 1 2. Preliminaries. We now describe the notation and some basic results used in the manuscript. The stability and convergence analysis in § §4-5 is for the scalar RPIE (1.1) posed on an infinite flat surface, i.e. for R 2 u(x , t−|x −x|) |x −x| dx = a(x, t) on R 2 × (0, T ) , (2.1)where u and a satisfy (1.2).

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A new viscoelastic wave inversion method for MRE, called Heterogeneous Multifrequency Direct Inversion (HMDI), was developed which accommodates heterogeneous elasticity within a direct inversion (DI) by incorporating first-order gradients and combining results from a narrow band of multiple frequencies. The method is compared with a Helmholtz-type DI, Multifrequency Dual Elasto-Visco inversion (MDEV), both on ground-truth Finite Element Method simulations at varied noise levels and a prospective in vivo brain cohort of 48 subjects ages 18-65. In simulated data, MDEV recovered background material within 5% and HMDI within 1% of prescribed up to SNR of 20 dB. In vivo HMDI and MDEV were then combined with segmentation from SPM to create a fully automated "brain palpation" exam for both whole brain (WB), and brain white matter (WM), measuring two parameters, the complex modulus magnitude |G*| , which measures tissue "stiffness", and the slope of |G*| values across frequencies, a measure of viscous dispersion. |G*| values for MDEV and HMDI were comparable to the literature (for a 3-frequency set centered at 50 Hz, WB means were 2.17 and 2.15 kPa respectively, and WM means were 2.47 and 2.49 kPa respectively). Both methods showed moderate correlation to age in both WB and WM, for both |G*| and |G*| slope, with Pearson's r ≥ 0.4 in the most sensitive frequency sets. In comparison to MDEV, HMDI showed better preservation of recovered target shapes, more noise-robustness, and stabler recovery values in regions with rapid property change, however summary statistics for both methods were quite similar. By eliminating homogeneity assumptions within a fast, fully automatic, regularization-free direct inversion, HMDI appears to be a worthwhile addition to the MRE image reconstruction repertoire. In addition to supporting the literature showing decrease in brain viscoelasticity with age, our work supports a wide range of inter-individual variation in brain MRE results.

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