Andrew Christlieb

David Lawlor, Yang Wang, Andrew Christlieb

We present a new deterministic algorithm for the sparse Fourier transform problem, in which we seek to identify k << N significant Fourier coefficients from a signal of bandwidth N. Previous deterministic algorithms exhibit quadratic runtime scaling, while our algorithm scales linearly with k in the average case. Underlying our algorithm are a few simple observations relating the Fourier coefficients of time-shifted samples to unshifted samples of the input function. This allows us to detect when aliasing between two or more frequencies has occurred, as well as to determine the value of unaliased frequencies. We show that empirically our algorithm is orders of magnitude faster than competing algorithms.

Andrew Christlieb, Wei Guo, Yan Jiang

In this paper, we present a novel numerical scheme for solving a class of nonlinear degenerate parabolic equations with non-smooth solutions. The proposed method relies on a special kernel based formulation of the solutions found in our early work on the method of lines transpose and successive convolution. In such a framework, a high order weighted essentially non-oscillatory (WENO) methodology and a nonlinear filter are further employed to avoid spurious oscillations. High order accuracy in time is realized by using the high order explicit strong-stability-preserving (SSP) Runge-Kutta method. Moreover, theoretical investigations of the kernel based formulation combined with an explicit SSP method indicates that the combined scheme is unconditionally stable and up to third order accuracy. Evaluation of the kernel based approach is done with a fast $\mathcal{O}(N)$ summation algorithm. The new method allows for much larger time step evolution compared with other explicit schemes with the same order accuracy, leading to remarkable computational savings.

Andrew Christlieb, Wei Guo, Yan Jiang

In this paper, a high order implicit Method of Line Transpose (MOL$^T$ ) method based on a weighted essentially non-oscillatory (WENO) methodology is developed for one-dimensional linear transport equations and further applied to the Vlasov-Poisson (VP) simulations via dimensional splitting. In the MOL$^T$ framework, the time variable is first discretized by a diagonally implicit strong-stability-preserving Runge-Kutta method, resulting in a boundary value problem (BVP) at the discrete time levels. Then an integral formulation coupled with a high order WENO methodology is employed to solve the BVP. As a result, the proposed scheme is high order accurate in both space and time and free of oscillations even though the solution is discontinuous or has sharp gradients. Moreover, the scheme is able to take larger time step evolution compared with an explicit MOL WENO scheme with the same order of accuracy. The desired positivity-preserving (PP) property of the scheme is further attained by incorporating a newly proposed high order PP limiter. We perform numerical experiments on several benchmarks including linear advection, solid body rotation problem; and on the Landau damping, two-stream instabilities, bump-on-tail, and plasma sheath by solving the VP system. The efficacy and efficiency of the proposed scheme is numerically verified.

Andrew Christlieb, Wei Guo, Yan Jiang

In this paper, a class of high order numerical schemes is proposed for solving Hamilton-Jacobi (H-J) equations. This work is regarded as an extension of our previous work for nonlinear degenerate parabolic equations, see Christlieb et al. \emph{arXiv preprint arXiv:1707.09294},, which relies on a special kernel-based formulation of the solutions and successive convolution. When applied to the H-J equations, the newly proposed scheme attains genuinely high order accuracy in both space and time, and more importantly, it is unconditionally stable, hence allowing for much larger time step evolution compared with other explicit schemes and saving computational cost. A high order weighted essentially non-oscillatory methodology and a novel nonlinear filter are further incorporated to capture the correct viscosity solution. Furthermore, by coupling the recently proposed inverse Lax-Wendroff boundary treatment technique, this method is very flexible in handing complex geometry as well as general boundary conditions. We perform numerical experiments on a collection of numerical examples, including H-J equations with linear, nonlinear, convex or non-convex Hamiltonians. The efficacy and efficiency of the proposed scheme in approximating the viscosity solution of general H-J equations is verified.

Matthew Causley, Hana Cho, Andrew Christlieb

In this work, we develop an $\mathcal{O}(N)$ implicit real space method in 1D and 2D for the Cahn Hilliard (CH) and vector Cahn Hilliard (VCH) equations, based on the Method Of Lines Transpose (MOL$^\text{T}$) formulation. This formulation results in a semi-discrete time stepping algorithm, which we prove is gradient stable in the $H^{-1}$ norm. The spatial discretization follows from dimensional splitting, and an $\mathcal{O}(N)$ matrix-free solver, which applies fast convolution to the modified Helmholtz equation. We propose a novel factorization technique, in which fourth order spatial derivatives are incorporated into the solver. The splitting error is included in the nonlinear fixed point iteration, resulting in a high order, logically Cartesian (line-by-line) update. Our method is fast, but not restricted to periodic boundaries like the fast Fourier transform (FFT). The basic solver is implemented using the Backward Euler formulation, and we extend this to both backward difference (BDF) stencils, implicit Runge Kutta (SDIRK) and spectral deferred correction (SDC) frameworks to achieve high orders of temporal accuracy. We demonstrate with numerical results that the CH, and VCH equations maintain gradient stability in one and two spatial dimensions. We also explore time-adaptivity, so that meta-stable states and ripening events can be simulated both quickly and efficiently.

Benjamin Ong, Andrew Melfi, Andrew Christlieb

In this paper, we further develop a family of parallel time integrators known as Revisionist Integral Deferred Correction methods (RIDC) to allow for the semi-implicit solution of time dependent PDEs. Additionally, we show that our semi-implicit RIDC algorithm can harness the computational potential of multiple general purpose graphical processing units (GPUs) in a single node by utilizing existing CUBLAS libraries for matrix linear algebra routines in our implementation. In the numerical experiments, we show that our implementation computes a fourth order solution using four GPUs and four CPUs in approximately the same wall clock time as a first order solution computed using a single GPU and a single CPU.

Matthew Causley, Andrew Christlieb, Eric Wolf

Building upon recent results obtained in [7,8,9], we describe an efficient second order, A-stable scheme for solving the wave equation, based on the method of lines transpose (MOL$^T$), and the resulting semi-discrete (i.e. continuous in space) boundary value problem. In [7], A-stable schemes of high order were derived, and in [9] a high order, fast $\mathcal{O}(N)$ spatial solver was derived, which is matrix-free and is based on dimensional-splitting. In this work, are interested in building a wave solver, and our main concern is the development of boundary conditions. We demonstrate all desired boundary conditions for a wave solver, including outflow boundary conditions, in 1D and 2D. The scheme works in a logically Cartesian fashion, and the boundary points are embedded into the regular mesh, without incurring stability restrictions, so that boundary conditions are imposed without any reduction in the order of accuracy. We demonstrate how the embedded boundary approach works in the cases of Dirichlet and Neumann boundary conditions. Further, we develop outflow and periodic boundary conditions for the MOL$^T$ formulation. Our solver is designed to couple with particle codes, and so special attention is also paid to the implementation of point sources, and soft sources which can be used to launch waves into waveguides.

David Bortz, Andrew Christlieb

We propose a novel approach which employs random sampling to generate an accurate non-uniform mesh for numerically solving Partial Differential Equation Boundary Value Problems (PDE-BVP's). From a uniform probability distribution U over a 1D domain, we sample M discretizations of size N where M>>N. The statistical moments of the solutions to a given BVP on each of the M ultra-sparse meshes provide insight into identifying highly accurate non-uniform meshes. Essentially, we use the pointwise mean and variance of the coarse-grid solutions to construct a mapping Q(x) from uniformly to non-uniformly spaced mesh-points. The error convergence properties of the approximate solution to the PDE-BVP on the non-uniform mesh are superior to a uniform mesh for a certain class of BVP's. In particular, the method works well for BVP's with locally non-smooth solutions. We present a framework for studying the sampled sparse-mesh solutions and provide numerical evidence for the utility of this approach as applied to a set of example BVP's. We conclude with a discussion of how the near-perfect paralellizability of our approach suggests that these strategies have the potential for highly efficient utilization of massively parallel multi-core technologies such as General Purpose Graphics Processing Units (GPGPU's). We believe that the proposed algorithm is beyond embarrassingly parallel; implementing it on anything but a massively multi-core architecture would be scandalous.

Yue Zhao, Joshua W. Burby, Andrew Christlieb et al.

We introduce a data-driven approach to learn a generalized kinetic collision operator directly from molecular dynamics. Unlike the conventional (e.g., Landau) models, the present operator takes an anisotropic form that accounts for a second energy transfer arising from the collective interactions between the pair of collision particles and the environment. Numerical results show that preserving the broadly overlooked anisotropic nature of the collision energy transfer is crucial for predicting the plasma kinetics with non-negligible correlations, where the Landau model shows limitations.

Sami Merhi, Ruochuan Zhang, Mark A. Iwen et al.

In this paper we consider Sparse Fourier Transform (SFT) algorithms for approximately computing the best $s$-term approximation of the Discrete Fourier Transform (DFT) $\mathbf{\hat{f}} \in \mathbb{C}^N$ of any given input vector $\mathbf{f} \in \mathbb{C}^N$ in just $\left( s \log N\right)^{\mathcal{O}(1)}$-time using only a similarly small number of entries of $\mathbf{f}$. In particular, we present a deterministic SFT algorithm which is guaranteed to always recover a near best $s$-term approximation of the DFT of any given input vector $\mathbf{f} \in \mathbb{C}^N$ in $\mathcal{O} \left( s^2 \log ^{\frac{11}{2}} (N) \right)$-time. Unlike previous deterministic results of this kind, our deterministic result holds for both arbitrary vectors $\mathbf{f} \in \mathbb{C}^N$ and vector lengths $N$. In addition to these deterministic SFT results, we also develop several new publicly available randomized SFT implementations for approximately computing $\mathbf{\hat{f}}$ from $\mathbf{f}$ using the same general techniques. The best of these new implementations is shown to outperform existing discrete sparse Fourier transform methods with respect to both runtime and noise robustness for large vector lengths $N$.

Benjamin Ong, Andrew Christlieb, Bryan Quaife

In this paper, a new two-parameter family of regularized kernels is introduced, suitable for applying high-order time stepping to N-body systems. These high-order kernels are derived by truncating a Taylor expansion of the non-regularized kernel about $(r^2+ε^2)$, generating a sequence of increasingly more accurate kernels. This paper proves the validity of this two-parameter family of regularized kernels, constructs error estimates, and illustrates the benefits of using high-order kernels through numerical experiments.