arXiv subjects

Shingyu Leung

Publications and source records attributed to Shingyu Leung.

At least 19 recordsLinked to original sources

arXiv2026

A Moment-Based Eulerian Method for Variance-Based Finite-Time Lyapunov Exponent Computation in Stochastic Flows

Variance-based finite-time Lyapunov exponents (vFTLEs) provide a stochastic analogue of deterministic FTLE by measuring the covariance of stochastic arrival locations. Existing PDF-based formulations compute this covariance by solving a Fokker--Planck equation for each initial point, which becomes expensive when the diagnostic is required on a dense grid. In this work, we develop a moment-based Eulerian approximation to vFTLE in the small-noise regime. Starting from a stochastic trajectory expansion about the deterministic flow, we derive a closed covariance equation for the leading stochastic displacement. By embedding this trajectory-wise covariance dynamics into physical space, we obtain an Eulerian transport--reaction equation for a symmetric covariance tensor field. The covariance associated with each initial point is recovered by evaluating this tensor field at the deterministic arrival location, and a moment-based vFTLE is then defined from its largest eigenvalue. The proposed method replaces a family of Fokker--Planck solves by the evolution of a single covariance tensor field, requiring only $d(d+1)/2$ scalar fields in $d$ dimensions. It also retains directional information through the eigenvectors of the covariance tensor, allowing the dominant directions of stochastic spreading to be visualized. We establish the leading-order consistency of the method with PDF-based vFTLE in the small-noise limit, clarify its relation to scalar stochastic sensitivity, and show how the same covariance equation connects process-noise spreading with deterministic deformation. In particular, deterministic FTLE is recovered, up to an additive constant, from an isotropic initial covariance when no process noise is present, while continuous process noise produces a time-integrated deformation covariance.

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arXiv2026

Geodesic Interpolation on the Grassmann Manifold: GLERP and Recursive GIDER Interpolants

This article develops a geodesic interpolation framework for data on the Grassmann manifold. The motivation is that many matrix-valued data sets represent subspaces rather than fixed bases: if the columns of two matrices differ only by a right orthogonal transformation, then they describe the same point on $Gr(r,m)$. Interpolation should therefore be invariant under this basis ambiguity. We first introduce $GLERP$, a Grassmann analogue of spherical linear interpolation defined by the Grassmann exponential and logarithm maps. The method follows the constant-speed geodesic joining two nearby subspaces and is second-order accurate for smooth Grassmann-valued curves under the usual normal-neighborhood condition. We then define $GIDER_n$, a recursive higher-order construction obtained by replacing each affine interpolation step in Neville's algorithm by $GLERP$. The resulting interpolant matches $n+1$ subspace data exactly, is basis-invariant, and is locally of order $n+1$ for sufficiently smooth curves. We compare the construction with tangent-space interpolation and projection-matrix interpolation, discuss intrinsic and extrinsic error measures, and present numerical tests. The results confirm the expected convergence orders and show that $GIDER_n$and tangent-space interpolation are nearly indistinguishable in a smooth local regime, while projection-matrix interpolation provides a useful extrinsic baseline. We also outline how the recursive construction can be used as the basis of a Grassmann ENO procedure.

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arXiv2026

Analytic First Derivatives of SIDER Interpolation

Spherical interpolation is required in numerical and geometric applications in which the unknowns are constrained to remain on the unit sphere. Spherical Interpolation of orDER $n$ (SIDER-$n$) was introduced as the high-order reconstruction component of spherical essentially non-oscillatory interpolation, where the reconstruction is built entirely from spherical linear interpolation (SLERP) operations and therefore preserves the spherical constraint exactly. This paper develops analytic first-derivative formulas for SIDER curves of arbitrary order. The central observation is that the recursive definition of SIDER can be differentiated by direct chain-rule propagation through its binary tree of SLERP operations. After deriving the total derivative of SLERP with moving endpoints, we obtain compact recursions for the derivative of SIDER-$n$, including simplified formulas at interpolation nodes and practical formulas at middle points between consecutive sampling locations. The latter are relevant when a reconstruction is evaluated halfway between data samples, as occurs in several high-order reconstruction-based numerical algorithms. The base case SIDER2 is treated explicitly, and SIDER3 and SIDER4 are used to illustrate the recursive mechanism. We also prove that the derivative is tangent to the sphere at every reconstructed point, including both sampling points and middle points. The resulting formulas extend the original SIDER/SENO framework by supplying differential information for sphere-valued reconstructions, with potential use in high-order finite-volume, ENO/WENO, SENO-type, and related methods for conservation laws and evolution problems.

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arXiv2026

Local Consistency and Higher-Order Structure of Spherical Interpolation

Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation construction for data on the unit sphere $\mathbb{S}^2$, built from repeated spherical linear interpolation (SLERP). This paper gives a local consistency analysis of SIDER for smooth spherical curves sampled at equally spaced parameter values. The analysis is carried out in geodesic normal coordinates, which allows the SIDER recursion to be compared with classical Neville interpolation while retaining the curvature-dependent corrections introduced by SLERP. We first derive local expansions of SLERP and show that SIDER2 has third-order accuracy; its leading error has the same shifted nodal structure as Euclidean quadratic interpolation. We then prove that the adjacent SIDER2 errors entering SIDER3 have a common leading coefficient, so that the SIDER3 recurrence cancels the cubic term and yields fourth-order accuracy. Carrying the expansion one order further gives the corresponding coefficient compatibility for SIDER3 and proves fifth-order accuracy of SIDER4. Finally, we introduce a degree-filtered formal expansion framework for the general SIDER recursion. This framework proves that, for each fixed $n$, SIDER-$n$ preserves the required polynomial degree structure in the normalized stencil variable. Together with the interpolation conditions at the $n+1$ nodes, this yields the local consistency estimate $d_{\mathbb{S}^2}\bigl(\gamma(\theta h),P_i^{[n]}(\theta;h)\bigr)=O(h^{n+1})$ under the stated smoothness and small-stencil assumptions.

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arXiv2026

A Quaternion--BCH Framework for the Local Accuracy of SIDER Interpolation

Spherical Interpolation of orDER $n$ (SIDER-$n$) is a recursive high-order interpolation method for data on the unit sphere $\mathbb{S}^2$, built from repeated spherical linear interpolation (SLERP). This paper develops a quaternion--Lie algebra framework for proving the local consistency of SIDER for smooth spherical curves sampled at equally spaced parameter values. Points on $\mathbb{S}$ are represented as pure unit quaternions, and interpolation errors are measured in fixed-base quaternion logarithmic coordinates. In this setting, each SLERP operation admits an exact Baker--Campbell--Hausdorff (BCH) representation, which converts the geometric interpolation problem into an algebraic problem involving filtered Lie-polynomial expansions. The BCH expansion shows that SLERP is affine to leading order, has no quadratic correction, and has a first nonlinear correction that is cubic and commutator-valued. Using this structure, we prove that SIDER2 has a third-order divided-error form with the same leading nodal factor as ordinary quadratic interpolation. We then show that the recursive SIDER step raises the order by one: the affine part gives the Neville-type finite-difference cancellation, while the nonlinear BCH remainder preserves the sharp filtered degree structure after the nodal factor is removed. Consequently, for every fixed $n\geq2$, $d_{\mathbb{S}^2}\bigl(\gamma(\theta h),P_i^{[n]}(\theta;h)\bigr) = O(h^{n+1}) $under the stated smoothness and small-stencil assumptions. The proof also identifies the shift-invariance of the leading divided-error coefficient as the algebraic compatibility condition underlying the SIDER recurrence.

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arXiv2026

A Semi-Lagrangian Spherical Essentially Non-Oscillatory (SENO) Scheme for Advection Equations of S2-valued Functions

We develop a numerical scheme for solving the advection equation of $\mathbb{S}^2$-valued functions of real variables, which models the time-evolution of a $\mathbb{S}^2$-valued mapping on the real line by a known velocity field. The idea is to extend the semi-Lagrangian method for the linear scalar advection equation. We first construct the backward flow map between two adjacent time levels and then interpolate the discrete ordered data of $\mathbb{S}^2$. To handle $\mathbb{S}^2$-functions which have kinks or sharp discontinuity in their components, we incorporate the \textit{Spherical Essentially Non-Oscillatory} (SENO) interpolation method, which effectively reduces the spurious oscillations in high-order reconstructions. We will show multiple examples to demonstrate the accuracy and effectiveness of the proposed algorithm for the partial differential equation of $\mathbb{S}^2$-functions.

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arXiv2025

A multilayer level-set method for eikonal-based traveltime tomography

We present a novel multilayer level-set method (MLSM) for eikonal-based first-arrival traveltime tomography. Unlike classical level-set approaches that rely solely on the zero-level set, the MLSM represents multiple phases through a sequence of $i_n$-level sets ($n = 0, 1, 2, \cdots$). Near each $i_n$-level set, the function is designed to behave like a local signed-distance function, enabling a single level-set formulation to capture arbitrarily many interfaces and subregions. Within this Eulerian framework, first-arrival traveltimes are computed as viscosity solutions of the eikonal equation, and Fr\'{e}chet derivatives of the misfit are obtained via the adjoint state method. To stabilize the inversion, we incorporate several regularization strategies, including multilayer reinitialization, arc-length penalization, and Sobolev smoothing of model parameters. In addition, we introduce an illumination-based error measure to assess reconstruction quality. Numerical experiments demonstrate that the proposed MLSM efficiently recovers complex discontinuous slowness models with multiple phases and interfaces.

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arXiv2025

Fast Operator-Splitting Methods for Nonlinear Elliptic Equations

Nonlinear elliptic problems arise in many fields, including plasma physics, astrophysics, and optimal transport. In this article, we propose a novel operator-splitting/finite element method for solving such problems. We begin by introducing an auxiliary function in a new way for a semilinear elliptic partial differential equation, leading to the development of a convergent operator-splitting/finite element scheme for this equation. The algorithm is then extended to fully nonlinear elliptic equations of the Monge-Amp\`ere type, including the Dirichlet Monge-Amp\`ere equation and Pucci's equation. This is achieved by reformulating the fully nonlinear equations into forms analogous to the semilinear case, enabling the application of the proposed splitting algorithm. In our implementation, a mixed finite element method is used to approximate both the solution and its Hessian matrix. Numerical experiments show that the proposed method outperforms existing approaches in efficiency and accuracy, and can be readily applied to problems defined on domains with curved boundaries.

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arXiv2025

The Closest Point Heat Method for Solving Eikonal Equations on Implicit Surfaces

We introduce the Closest Point Heat Method (CPHM), a novel approach for solving the surface Eikonal equation on general smooth surfaces. Building on the strengths of the classical heat method, such as simplicity of implementation and computational efficiency, CPHM integrates closest point techniques to reduce dependence on surface meshes. This embedding framework naturally extends the heat method to implicit surfaces while preserving both its efficiency and intrinsic geometric properties. Numerical experiments on benchmark geometries confirm the accuracy and convergence of the proposed method and demonstrate its effectiveness on complex shapes.

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arXiv2025

A Spherical Crank-Nicolson Integrator Based on the Exponential Map and the Spherical Linear Interpolation

We propose implicit integrators for solving stiff differential equations on unit spheres. Our approach extends the standard backward Euler and Crank-Nicolson methods in Cartesian space by incorporating the geometric constraint inherent to the unit sphere without additional projection steps to enforce the unit length constraint on the solution. We construct these algorithms using the exponential map and spherical linear interpolation (SLERP) formula on the unit sphere. Specifically, we introduce a spherical backward Euler method, a projected backward Euler method, and a second-order symplectic spherical Crank-Nicolson method. While all methods require solving a system of nonlinear equations to advance the solution to the next time step, these nonlinear systems can be efficiently solved using Newton's iterations. We will present several numerical examples to demonstrate the effectiveness and convergence of these numerical schemes. These examples will illustrate the advantages of our proposed methods in accurately capturing the dynamics of stiff systems on unit spheres.

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arXiv2024

SLERP-TVDRK (STVDRK) Methods for Ordinary Differential Equations on Spheres

We mimic the conventional explicit Total Variation Diminishing Runge-Kutta (TVDRK) schemes and propose a class of numerical integrators to solve differential equations on a unit sphere. Our approach utilizes the exponential map inherent to the sphere and employs spherical linear interpolation (SLERP). These modified schemes, named SLERP-TVDRK methods or STVDRK, offer improved accuracy compared to typical projective RK methods. Furthermore, they eliminate the need for any projection and provide a straightforward implementation. While we have successfully constructed STVDRK schemes only up to third-order accuracy, we explain the challenges in deriving STVDRK-r for r \ge 4. To showcase the effectiveness of our approach, we will demonstrate its application in solving the eikonal equation on the unit sphere and simulating p-harmonic flows using our proposed method.

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arXiv2024

Solving Partial Differential Equations on Evolving Surfaces via the Constrained Least-Squares and Grid-Based Particle Method

We present a framework for solving partial different equations on evolving surfaces. Based on the grid-based particle method (GBPM) [18], the method can naturally resample the surface even under large deformation from the motion law. We introduce a new component in the local reconstruction step of the algorithm and demonstrate numerically that the modification can improve computational accuracy when a large curvature region is developed during evolution. The method also incorporates a recently developed constrained least-squares ghost sample points (CLS-GSP) formulation, which can lead to a better-conditioned discretized matrix for computing some surface differential operators. The proposed framework can incorporate many methods and link various approaches to the same problem. Several numerical experiments are carried out to show the accuracy and effectiveness of the proposed method.

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arXiv2024

A Constrained Least-Squares Ghost Sample Points (CLS-GSP) Method for Differential Operators on Point Clouds

We introduce a novel meshless method called the Constrained Least-Squares Ghost Sample Points (CLS-GSP) method for solving partial differential equations on irregular domains or manifolds represented by randomly generated sample points. Our approach involves two key innovations. Firstly, we locally reconstruct the underlying function using a linear combination of radial basis functions centered at a set of carefully chosen \textit{ghost sample points} that are independent of the point cloud samples. Secondly, unlike conventional least-squares methods, which minimize the sum of squared differences from all sample points, we regularize the local reconstruction by imposing a hard constraint to ensure that the least-squares approximation precisely passes through the center. This simple yet effective constraint significantly enhances the diagonal dominance and conditioning of the resulting differential matrix. We provide analytical proofs demonstrating that our method consistently estimates the exact Laplacian. Additionally, we present various numerical examples showcasing the effectiveness of our proposed approach in solving the Laplace/Poisson equation and related eigenvalue problems.

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arXiv2024

Local Trajectory Variation Exponent (LTVE) for Visualizing Dynamical Systems

The identification and visualization of Lagrangian structures in flows plays a crucial role in the study of dynamic systems and fluid dynamics. The Finite Time Lyapunov Exponent (FTLE) has been widely used for this purpose; however, it only approximates the flow by considering the positions of particles at the initial and final times, ignoring the actual trajectory of the particle. To overcome this limitation, we propose a novel quantity that extends and generalizes the FTLE by incorporating trajectory metrics as a measure of similarity between trajectories. Our proposed method utilizes trajectory metrics to quantify the distance between trajectories, providing a more robust and accurate measure of the LCS. By incorporating trajectory metrics, we can capture the actual path of the particle and account for its behavior over time, resulting in a more comprehensive analysis of the flow. Our approach extends the traditional FTLE approach to include trajectory metrics as a means of capturing the complexity of the flow.

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arXiv2024

Hadamard integrators for wave equations in time and frequency domain: Eulerian formulations via butterfly algorithms

Starting from the Kirchhoff-Huygens representation and Duhamel's principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagator. Second, using the Fourier transform in time, we derive the corresponding Eulerian short-time propagator in frequency domain; on top of this propagator, we further develop a time-frequency-time (TFT) method for the Cauchy problem of time-domain wave equations. Third, we further propose the time-frequency-time-frequency (TFTF) method for the corresponding point-source Helmholtz equation, which provides Green's functions of the Helmholtz equation for all angular frequencies within a given frequency band. Fourth, to implement TFT and TFTF methods efficiently, we introduce butterfly algorithms to compress oscillatory integral kernels at different frequencies. As a result, the proposed methods can construct wave field beyond caustics implicitly and advance spatially overturning waves in time naturally with quasi-optimal computational complexity and memory usage. Furthermore, once constructed the Hadamard integrators can be employed to solve both time-domain wave equations with various initial conditions and frequency-domain wave equations with different point sources. Numerical examples for two-dimensional wave equations illustrate the accuracy and efficiency of the proposed methods.

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arXiv2023

Operator Splitting/Finite Element Methods for the Minkowski Problem

The classical Minkowski problem for convex bodies has deeply influenced the development of differential geometry. During the past several decades, abundant mathematical theories have been developed for studying the solutions of the Minkowski problem, however, the numerical solution of this problem has been largely left behind, with only few methods available to achieve that goal. In this article, focusing on the two-dimensional Minkowski problem with Dirichlet boundary conditions, we introduce two solution methods, both based on operator-splitting. One of these two methods deals directly with the Dirichlet condition, while the other method uses an approximation of this Dirichlet condition. This relaxation of the Dirichlet condition makes this second method better suited than the first one to treat those situations where the Minkowski and the Dirichlet condition are not compatible. Both methods are generalizations of the solution method for the canonical Monge-Amp\`{e}re equation discussed by Glowinski et al. (Journal of Scientific Computing, 79(1), 1-47, 2019); as such they take advantage of a divergence formulation of the Minkowski problem, well-suited to a mixed finite element approximation, and to the the time-discretization via an operator-splitting scheme, of an associated initial value problem. Our methodology can be easily implemented on convex domains of rather general shape (with curved boundaries, possibly). The numerical experiments we performed validate both methods and show that if one uses continuous piecewise affine finite element approximations of the smooth solution of the Minkowski problem and of its three second order derivatives, these two methods provide nearly second order accuracy for the $L^2$ and $L^{\infty}$ error. One can extend easily the methods discussed in this article, to address the solution of three-dimensional Minkowski problem.

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arXiv2023

A Simple Embedding Method for Scalar Hyperbolic Conservation Laws on Implicit Surfaces

We have developed a new embedding method for solving scalar hyperbolic conservation laws on surfaces. The approach represents the interface implicitly by a signed distance function following the typical level set method and some embedding methods. Instead of solving the equation explicitly on the surface, we introduce a modified partial differential equation in a small neighborhood of the interface. This embedding equation is developed based on a push-forward operator that can extend any tangential flux vectors from the surface to a neighboring level surface. This operator is easy to compute and involves only the level set function and the corresponding Hessian. The resulting solution is constant in the normal direction of the interface. To demonstrate the accuracy and effectiveness of our method, we provide some two- and three-dimensional examples.

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arXiv2022

Within-Cluster Variability Exponent for Identifying Coherent Structures in Dynamical Systems

We propose a clustering-based approach for identifying coherent flow structures in continuous dynamical systems. We first treat a particle trajectory over a finite time interval as a high-dimensional data point and then cluster these data from different initial locations into groups. The method then uses the normalized standard deviation or mean absolute deviation to quantify the deformation. Unlike the usual finite-time Lyapunov exponent (FTLE), the proposed algorithm considers the complete traveling history of the particles. We also suggest two extensions of the method. To improve the computational efficiency, we develop an adaptive approach that constructs different subsamples of the whole particle trajectory based on a finite time interval. To start the computation in parallel to the flow trajectory data collection, we also develop an on-the-fly approach to improve the solution as we continue to provide more measurements for the algorithm. The method can efficiently compute the WCVE over a different time interval by modifying the available data points.

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