John Lowengrub

Arvind Baskaran, John Lowengrub, Cheng Wang et al.

In this paper we provide a detailed convergence analysis for an unconditionally energy stable, second-order accurate convex splitting scheme for the Modified Phase Field Crystal equation, a generalized damped wave equation for which the usual Phase Field Crystal equation is a special degenerate case. The fully discrete, fully second-order finite difference scheme in question was derived in a recent work [2]. An introduction of a new variable ψ, corresponding to the temporal derivative of the phase variable ϕ, could bring an accuracy reduction in the formal consistency estimate, because of the hyperbolic nature of the equation. A higher order consistency analysis by an asymptotic expansion is performed to overcome this difficulty. In turn, second order convergence in both time and space is established in a discrete L^\infty (0,T; H^3) norm.

Zhen Guan, John Lowengrub, Cheng Wang

In this paper we provide a detailed convergence analysis for fully discrete second order (in both time and space) numerical schemes for nonlocal Allen-Cahn (nAC) and nonlocal Cahn-Hilliard (nCH) equations. The unconditional unique solvability and energy stability ensures $\ell^4$ stability. The convergence analysis for the nAC equation follows the standard procedure of consistency and stability estimate for the numerical error function. For the nCH equation, due to the complicated form of the nonlinear term, a careful expansion of its discrete gradient is undertaken and an $H^{-1}$ inner product estimate of this nonlinear numerical error is derived to establish convergence. In addition, an a-priori $W^{1,\infty}$ bound of the numerical solution at the discrete level is needed in the error estimate. Such a bound can be obtained by performing a higher order consistency analysis by using asymptotic expansions for the numerical solution. Following the technique originally proposed by Strang (e.g., 1964), instead of the standard comparison between the exact and numerical solutions, an error estimate between the numerical solution and the constructed approximate solution yields an $O( s^3 + h^4)$ convergence in $\ell^\infty (0, T; \ell^2)$ norm, which leads to the necessary bound under a standard constraint $s \le C h$. Here, we also prove convergence of the scheme in the maximum norm under the same constraint.

Wenqiang Feng, Zhen Guan, John Lowengrub et al.

We present and analyze an unconditionally energy stable and convergent finite difference scheme for the Functionalized Cahn-Hilliard equation. One key difficulty associated with the energy stability is based on the fact that one nonlinear energy functional term in the expansion appears as non-convex, non-concave. To overcome this subtle difficulty, we add two auxiliary terms to make the combined term convex, which in turns yields a convex-concave decomposition of the physical energy. As a result, an application of the convex splitting methodology assures both the unique solvability and the unconditional energy stability of the proposed numerical scheme. To deal with a 4-Laplacian solver in an $H^{-1}$ gradient flow at each time step, we apply an efficient preconditioned steepest descent algorithm to solve the corresponding nonlinear systems. In addition, a global in time $H_{\rm per}^2$ stability of the numerical scheme is established at a theoretical level, which in turn ensures the full order convergence analysis of the scheme. A few numerical results are presented, which confirm the stability and accuracy of the proposed numerical scheme.

Patrick Bilic, Patrick Christ, Hongwei Bran Li et al.

In this work, we report the set-up and results of the Liver Tumor Segmentation Benchmark (LiTS), which was organized in conjunction with the IEEE International Symposium on Biomedical Imaging (ISBI) 2017 and the International Conferences on Medical Image Computing and Computer-Assisted Intervention (MICCAI) 2017 and 2018. The image dataset is diverse and contains primary and secondary tumors with varied sizes and appearances with various lesion-to-background levels (hyper-/hypo-dense), created in collaboration with seven hospitals and research institutions. Seventy-five submitted liver and liver tumor segmentation algorithms were trained on a set of 131 computed tomography (CT) volumes and were tested on 70 unseen test images acquired from different patients. We found that not a single algorithm performed best for both liver and liver tumors in the three events. The best liver segmentation algorithm achieved a Dice score of 0.963, whereas, for tumor segmentation, the best algorithms achieved Dices scores of 0.674 (ISBI 2017), 0.702 (MICCAI 2017), and 0.739 (MICCAI 2018). Retrospectively, we performed additional analysis on liver tumor detection and revealed that not all top-performing segmentation algorithms worked well for tumor detection. The best liver tumor detection method achieved a lesion-wise recall of 0.458 (ISBI 2017), 0.515 (MICCAI 2017), and 0.554 (MICCAI 2018), indicating the need for further research. LiTS remains an active benchmark and resource for research, e.g., contributing the liver-related segmentation tasks in \url{http://medicaldecathlon.com/}. In addition, both data and online evaluation are accessible via \url{www.lits-challenge.com}.

Mariano Franco-de-León, John Lowengrub

In this paper, we implement non-stiff interface tracking methods for the evolution of 2-D curves that follow Airy flow, a curvature-dependent dispersive geometric evolution law. The curvature of the curve satisfies the modified Korteweg-de Vries equation, a dispersive non-linear soliton equation. We present a fully discrete space-time analysis of the equations (proof of convergence) and numerical evidence that confirms the accuracy, convergence, efficiency, and stability of the methods.

Zhenlin Guo, Ping Lin, Steven Wise et al.

In this paper we describe two fully mass conservative, energy stable, finite difference methods on a staggered grid for the quasi-incompressible Navier-Stokes-Cahn-Hilliard (q-NSCH) system governing a binary incompressible fluid flow with variable density and viscosity. Both methods, namely the primitive method (finite difference method in the primitive variable formulation) and the projection method (finite difference method in a projection-type formulation), are so designed that the mass of the binary fluid is preserved, and the energy of the system equations is always non-increasing in time at the fully discrete level. We also present an efficient, practical nonlinear multigrid method - comprised of a standard FAS method for the Cahn-Hilliard equation, and a method based on the Vanka-type smoothing strategy for the Navier-Stokes equation - for solving these equations. We test the scheme in the context of Capillary Waves, rising droplets and Rayleigh-Taylor instability. Quantitative comparisons are made with existing analytical solutions or previous numerical results that validate the accuracy of our numerical schemes. Moreover, in all cases, mass of the single component and the binary fluid was conserved up to 10 to -8 and energy decreases in time.

Jana Lipková, Markus Rempfler, Patrick Christ et al.

The segmentation of liver lesions is crucial for detection, diagnosis and monitoring progression of liver cancer. However, design of accurate automated methods remains challenging due to high noise in CT scans, low contrast between liver and lesions, as well as large lesion variability. We propose a 3D automatic, unsupervised method for liver lesions segmentation using a phase separation approach. It is assumed that liver is a mixture of two phases: healthy liver and lesions, represented by different image intensities polluted by noise. The Cahn-Hilliard equation is used to remove the noise and separate the mixture into two distinct phases with well-defined interfaces. This simplifies the lesion detection and segmentation task drastically and enables to segment liver lesions by thresholding the Cahn-Hilliard solution. The method was tested on 3Dircadb and LITS dataset.

Arvind Baskaran, Zhen Guan, John Lowengrub

In this paper we present a numerical method for hydrodynamic models that arise from time dependent density functional theories of freezing. The models take the form of compressible Navier-Stokes equations whose pressure is determined by the variational derivative of a free energy, which is a functional of the density field. We present unconditionally energy stable and mass conserving implicit finite difference methods for the models. The methods are based on a convex splitting of the free energy and that ensures that a discrete energy is non-increasing for any choice of time and space step. The methods are applicable to a large class of models, including both local and non-local free energy functionals. The theoretical basis for the numerical method is presented in a general context. The method is applied to problems using two specific free energy functionals: one local and one non-local functional. A nonlinear multigrid method is used to solve the numerical method, which is nonlinear at the implicit time step. The non-local functional, which is a convolution operator, is approximated using the Discrete Fourier Transform. Numerical simulations that confirm the stability and accuracy of the numerical method are presented.

Karl Yngve Lervåg, John Lowengrub

In recent work, Li et al.\ (Comm.\ Math.\ Sci., 7:81-107, 2009) developed a diffuse-domain method (DDM) for solving partial differential equations in complex, dynamic geometries with Dirichlet, Neumann, and Robin boundary conditions. The diffuse-domain method uses an implicit representation of the geometry where the sharp boundary is replaced by a diffuse layer with thickness $ε$ that is typically proportional to the minimum grid size. The original equations are reformulated on a larger regular domain and the boundary conditions are incorporated via singular source terms. The resulting equations can be solved with standard finite difference and finite element software packages. Here, we present a matched asymptotic analysis of general diffuse-domain methods for Neumann and Robin boundary conditions. Our analysis shows that for certain choices of the boundary condition approximations, the DDM is second-order accurate in $ε$. However, for other choices the DDM is only first-order accurate. This helps to explain why the choice of boundary-condition approximation is important for rapid global convergence and high accuracy. Our analysis also suggests correction terms that may be added to yield more accurate diffuse-domain methods. Simple modifications of first-order boundary condition approximations are proposed to achieve asymptotically second-order accurate schemes. Our analytic results are confirmed numerically in the $L^2$ and $L^\infty$ norms for selected test problems.

Edward Meeds, Michael Chiang, Mary Lee et al.

In many domains, scientists build complex simulators of natural phenomena that encode their hypotheses about the underlying processes. These simulators can be deterministic or stochastic, fast or slow, constrained or unconstrained, and so on. Optimizing the simulators with respect to a set of parameter values is common practice, resulting in a single parameter setting that minimizes an objective subject to constraints. We propose a post optimization posterior analysis that computes and visualizes all the models that can generate equally good or better simulation results, subject to constraints. These optimization posteriors are desirable for a number of reasons among which easy interpretability, automatic parameter sensitivity and correlation analysis and posterior predictive analysis. We develop a new sampling framework based on approximate Bayesian computation (ABC) with one-sided kernels. In collaboration with two groups of scientists we applied POPE to two important biological simulators: a fast and stochastic simulator of stem-cell cycling and a slow and deterministic simulator of tumor growth patterns.