Nathan Albin

Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams

On the basis of recently developed Fourier continuation (FC) methods and associated efficient parallelization techniques, this text introduces numerical algorithms that, due to very low dispersive errors, can accurately and efficiently solve the types of nonlinear partial differential equation (PDE) models of nonlinear acoustics in hundred-wavelength domains as arise in the simulation of focused medical ultrasound. As demonstrated in the examples presented in this text, the FC approach can be used to produce solutions to nonlinear acoustics PDEs models with significantly reduced discretization requirements over those associated with finite-difference, finite-element and finite-volume methods, especially in cases involving waves that travel distances that are orders of magnitude longer than their respective wavelengths. In these examples, the FC methodology is shown to lead to improvements in computing times by factors of hundreds and even thousands over those required by the standard approaches. A variety of one-and two-dimensional examples presented in this text demonstrate the power and capabilities of the proposed methodology, including an example containing a number of scattering centers and nonlinear multiple-scattering events.

Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws

We introduce a multi-domain Fourier-continuation/WENO hybrid method (FC-WENO) that enables high-order and non-oscillatory solution of systems of nonlinear conservation laws, and which enjoys essentially dispersionless, spectral character away from discontinuities, as well as mild CFL constraints (comparable to those of finite difference methods). The hybrid scheme employs the expensive, shock-capturing WENO method in small regions containing discontinuities and the efficient FC method in the rest of the computational domain, yielding a highly effective overall scheme for applications with a mix of discontinuities and complex smooth structures. The smooth and discontinuous solution regions are distinguished using the multi-resolution procedure of Harten [J. Comput. Phys. 115 (1994) 319-338]. We consider WENO schemes of formal orders five and nine and a FC method of order five. The accuracy, stability and efficiency of the new hybrid method for conservation laws is investigated for problems with both smooth and non-smooth solutions. In the latter case, we solve the Euler equations for gas dynamics for the standard test case of a Mach three shock wave interacting with an entropy wave, as well as a shock wave (with Mach 1.25, three or six) interacting with a very small entropy wave and evaluate the efficiency of the hybrid FC-WENO method as compared to a purely WENO-based approach as well as alternative hybrid based techniques. We demonstrate considerable computational advantages of the new FC-based method, suggesting a potential of an order of magnitude acceleration over alternatives when extended to fully three-dimensional problems.

Sensitivity of microwave ablation models to tissue biophysical properties: A first step toward probabilistic modeling and treatment planning.

PURPOSE Computational models of microwave ablation (MWA) are widely used during the design optimization of novel devices and are under consideration for patient-specific treatment planning. The objective of this study was to assess the sensitivity of computational models of MWA to tissue biophysical properties. METHODS The Morris method was employed to assess the global sensitivity of the coupled electromagnetic-thermal model, which was implemented with the finite element method (FEM). The FEM model incorporated temperature dependencies of tissue physical properties. The variability of the model was studied using six different outputs to characterize the size and shape of the ablation zone, as well as impedance matching of the ablation antenna. Furthermore, the sensitivity results were statistically analyzed and absolute influence of each input parameter was quantified. A framework for systematically incorporating model uncertainties for treatment planning was suggested. RESULTS A total of 1221 simulations, incorporating 111 randomly sampled starting points, were performed. Tissue dielectric parameters, specifically relative permittivity, effective conductivity, and the threshold temperature at which they transitioned to lower values (i.e., signifying desiccation), were identified as the most influential parameters for the shape of the ablation zone and antenna impedance matching. Of the thermal parameters considered in this study, the nominal blood perfusion rate and the temperature interval across which the tissue changes phase were identified as the most influential. The latent heat of tissue water vaporization and the volumetric heat capacity of the vaporized tissue were recognized as the least influential parameters. Based on the evaluation of absolute changes, the most important parameter (perfusion) had approximately 40.23 times greater influence on ablation area than the least important parameter (volumetric heat capacity of vaporized tissue). Another significant input parameter (permittivity) had 22.26 times higher influence on the deviation of ablation edge shape from a sphere than one of the less important parameters (latent heat of liver tissue vaporization). CONCLUSIONS Dielectric parameters, blood perfusion rate, and the temperature interval across which the tissue changes phase were found to have the most significant impact on MWA model outputs. The latent heat of tissue water vaporization and the volumetric heat capacity of the vaporized tissue were recognized as the least influential parameters. Uncertainties in model outputs identified in this study can be incorporated to provide probabilistic maps of expected ablation outcome for patient-specific treatment planning.

Minimal subfamilies and the probabilistic interpretation for modulus on graphs

AbstractThe notion of p-modulus of a family of objects on a graph is a measure of the richness of such families. We develop the notion of minimal subfamilies using the method of Lagrangian duality for p-modulus. We show that minimal subfamilies have at most |E| elements and that these elements carry a weight related to their “importance” in relation to the corresponding p-modulus problem. When

Numerical investigation of metrics for epidemic processes on graphs

p=2

Maximizing algebraic connectivity in interconnected networks

p=2, this measure of importance is in fact a probability measure and modulus can be thought as trying to minimize the expected overlap in the family.

Multiple-antenna microwave ablation: analysis of non-parallel antenna implants

The modulus of a family of walks quantifies the richness of the family by favoring many short walks over fewer longer ones. In this paper we investigate various families of walks in order to introduce new measures for quantifying network properties using modulus. The proposed new measures are compared to other known quantities such as current-flow closeness centrality, out-degree centrality, and current-flow betweenness centrality. Our proposed method is based on walks on a network, and therefore will work in great generality. For instance, the networks we consider can be directed, multi-edged, weighted, and even contain disconnected parts. Examples are provided to show the effectiveness of our measures.

Numerical simulation of focused nonlinear acoustic beams: A Fourier continuation direct solver and comparisons to a paraxial approximation.

This study develops the epidemic hitting time (EHT) metric on graphs measuring the expected time an epidemic starting at node a in a fully susceptible network takes to propagate and reach node b. An associated EHT centrality measure is then compared to degree, betweenness, spectral, and effective resistance centrality measures through exhaustive numerical simulations on several real-world network data-sets. We find two surprising observations: first, EHT centrality is highly correlated with effective resistance centrality; second, the EHT centrality measure is much more delocalized compared to degree and spectral centrality, highlighting the role of peripheral nodes in epidemic spreading on graphs.