Distributed-memory large deformation diffeomorphic 3D image registration

We present a parallel distributed-memory algorithm for large deformation diffeomorphic registration of volumetric images that produces large isochoric deformations (locally volume preserving). Image registration is a key technology in medical image analysis. Our algorithm uses a partial differential equation constrained optimal control formulation. Finding the optimal deformation map requires the solution of a highly nonlinear problem that involves pseudo-differential operators, biharmonic operators, and pure advection operators both forward and back- ward in time. A key issue is the time to solution, which poses the demand for efficient optimization methods as well as an effective utilization of high performance computing resources. To address this problem we use a preconditioned, inexact, Gauss-Newton- Krylov solver. Our algorithm integrates several components: a spectral discretization in space, a semi-Lagrangian formulation in time, analytic adjoints, different regularization functionals (including volume-preserving ones), a spectral preconditioner, a highly optimized distributed Fast Fourier Transform, and a cubic interpolation scheme for the semi-Lagrangian time-stepping. We demonstrate the scalability of our algorithm on images with resolution of up to $1024^3$ on the "Maverick" and "Stampede" systems at the Texas Advanced Computing Center (TACC). The critical problem in the medical imaging application domain is strong scaling, that is, solving registration problems of a moderate size of $256^3$---a typical resolution for medical images. We are able to solve the registration problem for images of this size in less than five seconds on 64 x86 nodes of TACC's "Maverick" system.Comment: accepted for publication at SC16 in Salt Lake City, Utah, USA; November 201

Fourth Order Dispersion in Nonlinear Media

In recent years, there has been an explosion of interest in media bearing quarticdispersion. After the experimental realization of so-called pure-quartic solitons, asignificant number of studies followed both for bright and for dark solitonic struc-tures exploring the properties of not only quartic, but also setic, octic, decic etc.dispersion, but also examining the competition between, e.g., quadratic and quarticdispersion among others.In the first chapter of this Thesis, we consider the interaction of solitary waves ina model involving the well-known φ4 Klein-Gordon theory, bearing both Laplacian and biharmonic terms with different prefactors. As a result of the competition ofthe respective linear operators, we obtain three distinct cases as we vary the modelparameters. In the first the biharmonic effect dominates, yielding an oscillatoryinter-wave interaction; in the third the harmonic effect prevails yielding exponen-tial interactions, while we find an intriguing linearly modulated exponential effectin the critical second case, separating the above two regimes. For each case, wecalculate the force between the kink and antikink when initially separated with suf-ficient distance. Being able to write the acceleration as a function of the separationdistance, and its corresponding ordinary differential equation, we test the corre-sponding predictions, finding very good agreement, where appropriate, with thecorresponding partial differential equation results. Where the two findings differ,we explain the source of disparities. Finally, we offer a first glimpse of the interplayof harmonic and biharmonic effects on the results of kink-antikink collisions andthe corresponding single- and multi-bounce windows.In the next two Chapters, we explore the competition of quadratic and quar-tic dispersion in producing kink-like solitary waves in a model of the nonlinearSchroedinger type bearing cubic nonlinearity. We present 6 families of multikink so-lutions and explore their bifurcations as a prototypical parameter is varied, namelythe strength of the quadratic dispersion. We reveal a rich bifurcation structure forthe system, connecting two-kink states with ones involving 4-, as well as 6-kinks.The stability of all of these states is explored. For each family, we discuss a “lowerbranch” adhering to the energy landscape of the 2-kink states (also discussed inthe previous Chapter). We also, however, study in detail the “upper branches”bearing higher numbers of kinks. In addition to computing the stationary statesand analyzing their stability at the PDE level, we develop an effective particle the-ory that is shown to be surprisingly efficient in capturing the kink equilibria and normal (as well as unstable) modes. Finally, the results of the bifurcation analysisare corroborated with direct numerical simulations involving the excitation of thestates in a targeted way in order to explore their instability-induced dynamics.While the previous two studies were focused on the one-dimensional problem,in the fourth and final chapter, we explore a two-dimensional realm. More specif-ically, we provide a characterization of the ground states of a higher-dimensionalquadratic-quartic model of the nonlinear Schr ̈odinger class with a combination of afocusing biharmonic operator with either an isotropic or an anisotropic defocusingLaplacian operator (at the linear level) and power-law nonlinearity. Examiningprincipally the prototypical example of dimension d = 2, we find that instabilityarises beyond a certain threshold coefficient of the Laplacian between the cubic andquintic cases, while all solutions are stable for powers below the cubic. Above thequintic, and up to a critical nonlinearity exponent p, there exists a progressivelynarrowing range of stable frequencies. Finally, above the critical p all solutionsare unstable. The picture is rather similar in the anisotropic case, with the dif-ference that even before the cubic case, the numerical computations suggest aninterval of unstable frequencies. Our analysis generalizes the relevant observationsfor arbitrary combinations of Laplacian prefactor b and nonlinearity power p.We conclude the thesis with a summary of its main findings, as well as with anoutlook towards interesting future problem

Rapid evaluation of radial basis functions

Over the past decade, the radial basis function method has been shown to produce high quality solutions to the multivariate scattered data interpolation problem. However, this method has been associated with very high computational cost, as compared to alternative methods such as finite element or multivariate spline interpolation. For example. the direct evaluation at M locations of a radial basis function interpolant with N centres requires O(M N) floating-point operations. In this paper we introduce a fast evaluation method based on the Fast Gauss Transform and suitable quadrature rules. This method has been applied to the Hardy multiquadric, the inverse multiquadric and the thin-plate spline to reduce the computational complexity of the interpolant evaluation to O(M + N) floating point operations. By using certain localisation properties of conditionally negative definite functions this method has several performance advantages against traditional hierarchical rapid summation methods which we discuss in detail

The contact line behaviour of solid-liquid-gas diffuse-interface models

A solid-liquid-gas moving contact line is considered through a diffuse-interface model with the classical boundary condition of no-slip at the solid surface. Examination of the asymptotic behaviour as the contact line is approached shows that the relaxation of the classical model of a sharp liquid-gas interface, whilst retaining the no-slip condition, resolves the stress and pressure singularities associated with the moving contact line problem while the fluid velocity is well defined (not multi-valued). The moving contact line behaviour is analysed for a general problem relevant for any density dependent dynamic viscosity and volume viscosity, and for general microscopic contact angle and double well free-energy forms. Away from the contact line, analysis of the diffuse-interface model shows that the Navier--Stokes equations and classical interfacial boundary conditions are obtained at leading order in the sharp-interface limit, justifying the creeping flow problem imposed in an intermediate region in the seminal work of Seppecher [Int. J. Eng. Sci. 34, 977--992 (1996)]. Corrections to Seppecher's work are given, as an incorrect solution form was originally used.Comment: 33 pages, 3 figure