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We introduce a class of high order accurate, semi-implicit Runge-Kutta schemes in the general setting of evolution equations that arise as gradient flow for a cost function, possibly with respect to an inner product that depends on the solution, and we establish their energy stability. This class includes as a special case high order, unconditionally stable schemes obtained via convexity splitting. The new schemes are demonstrated on a variety of gradient flows, including partial differential equations that are gradient flow with respect to the Wasserstein (mass transport) distance., arXiv admin note: text overlap with arXiv:1908.10246

We show how auction algorithms, originally developed for the assignment problem, can be utilized in Merriman, Bence, and Osher's threshold dynamics scheme to simulate multi-phase motion by mean curvature in the presence of equality and inequality volume constraints on the individual phases. The resulting algorithms are highly efficient and robust, and can be used in simulations ranging from minimal partition problems in Euclidean space to semi-supervised machine learning via clustering on graphs. In the case of the latter application, numerous experimental results on benchmark machine learning datasets show that our approach exceeds the performance of current state-of-the-art methods, while requiring a fraction of the computation time.

We study extensions of Merriman, Bence, and Osher’s threshold dynamics scheme to weighted mean curvature flow, which arises as gradient descent for anisotropic (normal dependent) surface energies. In particular, we investigate, in both two and three dimensions, those anisotropies for which the convolution kernel in the scheme can be chosen to be positive and/or to possess a positive Fourier transform. We provide a complete, geometric characterization of such anisotropies. This has implications for the unconditional stability and, in the two-phase setting, the monotonicity, of the scheme. We also revisit previous constructions of convolution kernels from a variational perspective, and propose a new one. The variational perspective differentiates between the normal dependent mobility and surface tension factors (both of which contribute to the normal speed) that results from a given convolution kernel. This more granular understanding is particularly useful in the multiphase setting, where junctions are present.

We show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces, including networks of them, in both two and three dimensions. Moreover, we discuss necessary and sufficient conditions for the positivity of the kernel which, in the case of two-phase flow, ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow. In particular, we show, in a barrier-type statement, that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions, and give a complete characterization. Among other results is a threshold dynamics scheme that is guaranteed to dissipate a non-local approximation to the interfacial energy in the fully anisotropic, multiphase setting, using the new kernel construction.

The threshold dynamics algorithm of Merriman, Bence, and Osher is only first order accurate in the two-phase setting. Its accuracy degrades further to half order in the multi-phase setting, a shortcoming it has in common with other related, more recent algorithms such as the equal surface tension version of the Voronoi implicit interface method. As a first, rigorous step in addressing this shortcoming, we present two different second order accurate versions of two-phase threshold dynamics. Unlike in previous efforts in this direction, we present careful consistency calculations for both of our algorithms. The first algorithm is consistent with its limit (motion by mean curvature) up to second order in any space dimension. The second achieves second order accuracy only in dimension two, but comes with a rigorous stability guarantee (unconditional energy stability) in any dimension – a first for high order schemes of its type.

We introduce a class of unconditionally energy stable, high order accurate schemes for gradient flows in a very general setting. The new schemes are a high order analogue of the minimizing movements approach for generating a time discrete approximation to a gradient flow by solving a sequence of optimization problems. In particular, each step entails minimizing the associated energy of the gradient flow plus a movement limiter term that is, in the classical context of steepest descent with respect to an inner product, simply quadratic. A variety of existing unconditionally stable numerical methods can be recognized as (typically just first order accurate in time) minimizing movement schemes for their associated evolution equations, already requiring the optimization of the energy plus a quadratic term at every time step. Therefore, our approach gives a painless way to extend these to high order accurate in time schemes while maintaining their unconditional stability. In this sense, it can be viewed as a variational analogue of Richardson extrapolation.

Graphene is a carbon molecule with the structure of a honeycomb lattice. We show how this structure can arise in two dimensions as the minimizer of an interaction energy with two-body and three-body terms. In the engineering literature, the Brenner potential is commonly used to describe the interactions between carbon atoms. We consider a potential of Stillinger-Weber type that incorporates certain characteristics of the Brenner potential: the preferred bond angles are 180 degrees and all interactions have finite range. We show that the thermodynamic limit of the ground state energy per particle is the same as that of a honeycomb lattice. We also prove that, subject to periodic boundary conditions, the minimizers are translated versions of the honeycomb lattice., Comment: Added references to Section 1, Commun. Math. Phys. (2016)

We explore the effects on grain size distribution of incorporating grain rotation into the curvature driven grain boundary migration model of Mullins. A new, extremely streamlined and efficient algorithm allows simulations with large numbers of grains. Some of these simulations yield size distributions and microstructures close to those from recent, atomistic simulations of microstructural evolution using the phase field crystal method that was shown to reproduce experimental size distributions observed in fiber textured, nanocrystalline, thin metallic films.

We present a simplified version of the threshold dynamics algorithm given in the work of Esedoglu and Otto (2015). The new version still allows specifying N-choose-2 possibly distinct surface tensions and N-choose-2 possibly distinct mobilities for a network with N phases, but achieves this level of generality without the use of retardation functions. Instead, it employs linear combinations of Gaussians in the convolution step of the algorithm. Convolutions with only two distinct Gaussians is enough for the entire network, maintaining the efficiency of the original thresholding scheme. We discuss stability and convergence of the new algorithm, including some counterexamples in which convergence fails. The apparently convergent cases include unequal surface tensions given by the Read \& Shockley model and its three dimensional extensions, along with equal mobilities, that are a very common choice in computational materials science., 21 pages, 4 figures, 4 tables

We present a detailed comparison of multiple statistical grain metrics for previously reported experimental thin film samples of aluminum with 2D simulations obtained from the Phase-Field-Crystal (PFC) model and a Mullins grain boundary motion model. For all these results, ``universality'' is observed with respect to the dynamics and initial conditions. This comparison reveals that PFC reproduces geometric metrics such as area and perimeter, but does not capture grain shape and topology as accurately. Similarly, the Mullins model captures the number of sides distribution quite well but not other metrics. Our collective comparison of such measurements underscores the critical importance of the use of multiple metrics for comparison of experiments with all present and future models of grain growth in polycrystalline materials., Comment: 17 pages, 6 figures

Nondominated sorting is a fundamental combinatorial problem in multiobjective optimization and is equivalent to the longest chain problem in combinatorics and random growth models for crystals in materials science. In a previous work [SIAM J. Math. Anal., 46 (2014), pp. 603--638], we showed that nondominated sorting has a continuum limit that corresponds to solving a Hamilton--Jacobi equation. In this work we present and analyze a fast numerical scheme for this Hamilton--Jacobi equation and show how it can be used to design a fast algorithm for approximate nondominated sorting.

We show that nondominated sorting of a sequence $X_1,\dots,X_n$ of independent and identically distributed random variables in $\mathbb{R}^d$ has a continuum limit that corresponds to solving a Hamilton--Jacobi equation involving the probability density function $f$ of $X_i$. Nondominated sorting is a fundamental problem in multiobjective optimization and is equivalent to finding the canonical antichain partition and to problems involving the longest chain among Euclidean points. As an application of this result, we show that nondominated sorting is asymptotically stable under bounded random perturbations in $X_1,\dots,X_n$. We give a numerical scheme for computing the viscosity solution of this Hamilton--Jacobi equation and present some numerical simulations for various density functions.

The representation of curves by integral invariant signatures is an important step in shape recognition and classification. Integral invariants are preferred over their differential counterparts due to their robustness with respect to noise. However, in contrast to differential invariants of curves, it is currently unknown whether integral signatures offer unique representations of curves. In this article, we prove some results on the uniqueness of the circular area signature. In particular, we study the case for graphs of periodic functions. We show that the circular area signature is unique if taken with respect to parametrization by the $x$-axis. Furthermore, we prove that the true circular area signature (parametrized by arclength) is unique in a neighborhood of constant functions. Finally, we show uniqueness in the special case that the functions of interest agree on an interval of width $2r$.

We introduce variants of the variational image denoising method proposed by Blomgren et al. (In: Numerical Analysis 1999 (Dundee), pp. 43–67. Chapman & Hall, Boca Raton, FL, 2000), which interpolates between total-variation denoising and isotropic diffusion denoising. We study how parameter choices affect results and allow tuning between TV denoising and isotropic diffusion for respecting texture on one spatial scale while denoising features assumed to be noise on finer spatial scales. Furthermore, we prove existence and (where appropriate) uniqueness of minimizers. We consider both L2 and L1 data fidelity terms.