The objective of this paper is to review recent developments in numerical reconstruction methods for inverse transport problems in imaging applications, mainly optical tomography, fluorescence tomography and bioluminescence tomography. In those inverse problems, one aims at reconstructing physical parameters, such as the absorption coefficient, the scattering coefficient and the fluorescence light source, inside heterogeneous media, from partial knowledge of transport solutions on the boundaries of the media. The physical parameters recovered can be used for diagnostic purpose. Numerical reconstruction techniques for those inverse transport problems can be roughly classified into two categories: linear reconstruction methods and nonlinear reconstruction methods. In the first type of methods, the inverse problems are linearized around some known background to obtain linear inverse problems. Classical regularization techniques are then applied to solve those inverse problems. The second type of methods are either based on regularized nonlinear least-square techniques or based on gradient-driven iterative methods for nonlinear operator equations. In either case, the unknown parameters are iteratively updated until the solutions of the transport equations with the those parameters match the measurements to a certain extent. We review linear and nonlinear reconstruction methods for inverse transport problems in medical imaging with stationary, frequency-domain and time-dependent data. The materials presented include both existing and new results. Meanwhile, we attempt to present similar algorithms for different problems in the same framework to make it more straightforward to generalize those algorithms to other inverse (transport) problems.

We present a variable time step, fully adaptive in space, hybrid method for the accurate simulation of incompressible two-phase flows in the presence of surface tension in two dimensions. The method is based on the hybrid level set/front-tracking approach proposed in [H. D. Ceniceros and A. M. Roma, J. Comput. Phys., 205, 391-400, 2005]. Geometric, interfacial quantities are computed from front-tracking via the immersed-boundary setting while the signed distance (level set) function, which is evaluated fast and to machine precision, is used as a fluid indicator. The surface tension force is obtained by employing the mixed Eulerian/Lagrangian representation introduced in [S. Shin, S. I. Abdel-Khalik, V. Daru and D. Juric, J. Comput. Phys., 203, 493-516, 2005] whose success for greatly reducing parasitic currents has been demonstrated. The use of our accurate fluid indicator together with effective Lagrangian marker control enhance this parasitic current reduction by several orders of magnitude. To resolve accurately and efficiently sharp gradients and salient flow features we employ dynamic, adaptive mesh refinements. This spatial adaption is used in concert with a dynamic control of the distribution of the Lagrangian nodes along the fluid interface and a variable time step, linearly implicit time integration scheme. We present numerical examples designed to test the capabilities and performance of the proposed approach as well as three applications: the long-time evolution of a fluid interface undergoing Rayleigh-Taylor instability, an example of bubble ascending dynamics, and a drop impacting on a free interface whose dynamics we compare with both existing numerical and experimental data.

It is known that the submerged seamount/ridge is a source for the generation of internal tides. In this paper, a three-dimensional two-layer model is set up to study the generation of internal tides by tidal flows over a submerged seamount/ridge in the channel. Several numerical experiments with different topographic features, upper layer depths, tidal flows and background currents are carried out to study the variations of the induced internal tides. It is shown that, for the specific stratification, the seamount feature, the slope, the initial upper layer depth and the imposing driven force determine the Froude number near the seamount peak. Once when the Froude number is supercritical, the associated maximum amplitude of the induced internal tide is so large that the internal tide begins to disintegrate, which brings about severe variations of the current velocity and the water elevation fields, and the associated induced baroclinic tidal energy around the seamount peak is much larger than the barotropic one. The Richardson number greater than 1/4 is a criterion for stability of shear flow. Since the maximum tidal velocity changes within 0 ∼ 360^◦ with time in a period around the seamount peak, the induced internal tide does not stride the seamount peak before it disintegrates, which is different from the two-dimensional modeled results. The asymmetrical slope of the submerged seamount is a mechanism for the asymmetrical internal tide generation.

This paper presents a lumped particle model for simulating a large number of particles. The lumped particle model is a flexible framework in modeling particle flows, embodying fundamental features that are intrinsic in particle laden flow, including advection, diffusion and dispersion. In this paper, the particles obey a simplified version of the Bassinet-Boussinesq-Oseen equation for a single spherical particle. However, instead of tracking the individual dynamics of each particle, a weighted spatial averaging procedure is used where the external forces are applied to a “lump” of particles, from which an average position and velocity is derived. The temporal evolution of the particles is computed by partitioning the lumped particle into smaller entities, which are then transported throughout the physical domain. These smaller entities recombine into new particle lumps at their target destinations. For particles prone to the effects of Brownian motion or similar phenomena, a symmetric spreading of the particles is included as well. Numerical experiments show that the lumped particle model reproduces the effects of Brownian diffusion and uniform particle transport by a fluid and gravity. The late time scale diffusive nature of particle motion is also reproduced.

In this paper, we investigate the stability for a finite harmonic lattice under a certain class of boundary conditions. A rigorous eigenvalue study clarifies that the invalidity of Fourier modes as the basis results in the deficiency of standard reflection coefficient approach for stability analysis. In a certain parameter range, unstable surface modes exist in the form of exponential decay in space, and exponential growth in time. An approximate eigen-polynomial is proposed to ease the stability analysis. Moreover, the eigenvalues with small positive real part quantitatively explain the long time instability in wave propagation computations. Numerical results verify the analysis.

This paper presents the development of parallel direct Vlasov solvers using the Spectral Element Method (SEM). Instead of the standard Particle-In-Cell (PIC) approach for kinetic space plasma simulation, i.e. solving the Vlasov-Maxwell equations, the direct method has been used in this paper. There are several benefits to solve the Vlasov equation directly, such as avoiding noise associated with the finite number of particles and the capability to capture the fine structure in the plasma, etc. The most challenging part of direct Vlasov solver comes from high dimension, as the computational cost increases as N^2d, where d is the dimension of the physical space. Recently due to fast development of supercomputers, the possibility of high dimensions becomes more realistic. A significant effort has been devoted to solve the Vlasov equation in low dimensions so far, now more interests focus on higher dimensions. Different numerical methods have been tried so far, such as finite difference method, Fourier spectral method, finite volume method, etc. In this paper SEM has been successfully applied to construct these solvers. SEM has shown several advantages, such as easy interpolation due to local element structure and long time integration due to its high order accuracy. Domain decomposition in high dimensions have been used for parallelization, these include scalable parallel 1D and 2D Poisson solvers. Benchmark results have been shown and simulation results have been reported for two different cases: one dimension (1P1V), and two dimensions (2P2V) in both physical and velocity spaces.

Non-local thermodynamic equilibrium (NLTE) conditions are universal in laboratory and astrophysical plasmas and, for this reason, the theory of NLTE plasmas is nowadays a very active subject. The populations of atomic levels and radiative properties are essential magnitudes in the study of these plasmas and the calculation of those properties relies on the so-called collisional-radiative (CR) models. However, the complexity of these models has led to the development of numerous collisional-radiative codes and this is a current research topic in plasmas. In this work is presented a versatile computational package, named ABAKO/RAPCAL, to calculate the populations of atomic levels and radiative properties of optically thin and thick, low-to-high Z, NLTE plasmas. ABAKO/RAPCAL combines a set of analytical approximations which yield substantial savings in computing running time, still comparing well with more elaborated codes and experimental data. In order to show the capabilities of the code and the accuracy of its results, calculations of several relevant plasma magnitudes for various plasma situations are shown and compared.

A set of 2D steady state finite element numerical simulations of electromagnetic fields, eddy currents distribution and induction heating pattern has been done for different thicknesses of a metal workpiece. Comparison between the calculation results show the importance of workpiece thickness on induction heating process including electromagnetic field distribution, eddy currents pattern, heating structure and coil efficiency.

For a simple model of a scalar wave equation with a random wave speed, Gottlieb and Xiu [Commun. Comput. Phys., 3 (2008), pp. 505-518] employed the generalized polynomial chaos (gPC) method and demonstrated that when uncertainty causes the change of characteristic directions, the resulting deterministic system of equations is a symmetric hyperbolic system with both positive and negative eigenvalues. Consequently, a consistent method of imposing the boundary conditions is proposed and its convergence is established under the assumption that the expansion coefficients decay fast asymptotically. In this work, we investigate stochastic collocation methods for the same type of scalar wave equation with random wave speed. It will be demonstrated that the rate of convergence depends on the regularity of the solutions; and the regularity is determined by the random wave speed and the initial and boundary data. Numerical examples are presented to support the analysis and also to show the sharpness of the assumptions on the relationship between the random wave speed and the initial and boundary data. An accuracy enhancement technique is investigated following the multi-element collocation method proposed by Foo, Wan and Karniadakis [J. Comput. Phys., 227 (2008), pp. 9572-9595].