Ming-jun Lai

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Papers by Ming-jun Lai

The Bijectivity of the Map Based on Harmonic Gbc Functions

Social Science Research Network, 2022

A Bivariate Spline based Collocation Method for Numerical Solution to Optimal Transport Problem

arXiv (Cornell University), Oct 25, 2023

In this paper, we study a spline collocation method for a numerical solution to the optimal trans... more In this paper, we study a spline collocation method for a numerical solution to the optimal transport problem We mainly solve the Monge-Ampére equation with the second boundary condition numerically by proposing a center matching algorithm. We prove a pointwise convergence of our iterative algorithm under the assumption the boundedness of spline iterates. We use the Monge-Ampére equation with Dirichlet boundary condition and some known solutions to the Monge-Ampére equation with second boundary condition to demonstrate the effectiveness of our algorithm. Then we use our method to solve some real-life problems. One application problem is to use the optimal transportation for the conversion of fisheye view images into standard rectangular images.

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Trivariate Spline Collocation Methods for Numerical Solution to 3D Monge-Ampère Equation

Journal of Scientific Computing, Apr 6, 2023

We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D ... more We use trivariate spline functions for the numerical solution of the Dirichlet problem of the 3D elliptic Monge-Ampére equation. Mainly we use the spline collocation method introduced in [SIAM J. Numerical Analysis, 2405-2434,2022] to numerically solve iterative Poisson equations and use an averaged algorithm to ensure the convergence of the iterations. We shall also establish the rate of convergence under a sufficient condition and provide some numerical evidence to show the numerical rates. Then we present many computational results to demonstrate that this approach works very well. In particular, we tested many known convex solutions as well as nonconvex solutions over convex and nonconvex domains and compared them with several existing numerical methods to show the efficiency and effectiveness of our approach.

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Sparse Solutions of Underdetermined Linear Systems

SpringerReference, Feb 8, 2012

We first explain the research problem of finding the sparse solution of underdetermined linear sy... more We first explain the research problem of finding the sparse solution of underdetermined linear systems with some applications. Then we explain three different approaches how to solve the sparse solution: the ℓ 1 approach, the orthogonal greedy approach, and the ℓ q approach with 0 < q ≤ 1. We mainly survey recent results and present some new or simplified proofs. In particular, we give a good reason why the orthogonal greedy algorithm converges and why it can be used to find the sparse solution. About the restricted isometry property (RIP) of matrices, we provide an elementary proof to a known result that the probability that the random matrix with iid Gaussian variables possesses the PIP is strictly positive.

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A Bivariate Spline Method for Second Order Elliptic Equations in Non-Divergence Form

arXiv (Cornell University), Oct 18, 2016

A bivariate spline method is developed to numerically solve second order elliptic partial differe... more A bivariate spline method is developed to numerically solve second order elliptic partial differential equations (PDE) in non-divergence form. The existence, uniqueness, stability as well as approximation properties of the discretized solution will be established by using the well-known Ladyzhenskaya-Babuska-Brezzi (LBB) condition. Bivariate splines, discontinuous splines with smoothness constraints are used to implement the method. A plenty of computational results based on splines of various degrees are presented to demonstrate the effectiveness and efficiency of our method.

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A Polygonal Spline Method for General Second-Order Elliptic Equations and Its Applications

Springer proceedings in mathematics & statistics, 2017

We explain how to use polygonal splines to numerically solve second-order elliptic partial differ... more We explain how to use polygonal splines to numerically solve second-order elliptic partial differential equations. The convergence of the polygonal spline method will be studied. Also, we will use this approach to numerically study the solution of some mixed parabolic and hyperbolic partial differential equations. Comparison with standard bivariate spline method will be given to demonstrate that our polygonal splines have some better numerical performance.

<title>Bivariate box spline wavelets in Sobolev spaces</title&gt

Proceedings of SPIE, Oct 19, 1998

We use bivariate boxspline functions to construct nonseparable wavelets in Sobolev spaces.

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Construction of multivariate compactly supported prewavelets in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math> space and pre-Riesz bases in S...

Journal of Approximation Theory, Oct 1, 2006

We give a new constructive method for finding compactly supported prewavelets in L 2 spaces in th... more We give a new constructive method for finding compactly supported prewavelets in L 2 spaces in the multivariate setting. This method works for any dimensional space. When this method is generalized to the Sobolev space setting, it produces a pre-Riesz basis for H s (R d) which can be useful for applications.

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<title>Examples of bivariate nonseparable continuous compactly supported orthonormal wavelets</title&gt

Proceedings of SPIE, Oct 30, 1997

We give many examples of bivariate nonseparable compactly supported orthonormal wavelets which ar... more

Graph-based Semi-supervised Local Clustering with Few Labeled Nodes

Local clustering aims at extracting a local structure inside a graph without the necessity of kno... more Local clustering aims at extracting a local structure inside a graph without the necessity of knowing the entire graph structure. As the local structure is usually small in size compared to the entire graph, one can think of it as a compressive sensing problem where the indices of target cluster can be thought as a sparse solution to a linear system. In this paper, we apply this idea based on two pioneering works under the same framework and propose a new semi-supervised local clustering approach using only few labeled nodes. Our approach improves the existing works by making the initial cut to be the entire graph and hence overcomes a major limitation of the existing works, which is the low quality of initial cut. Extensive experimental results on various datasets demonstrate the effectiveness of our approach.

On Locality of Harmonic Generalized Barycentric Coordinates and Their Application to Solution of the Poisson Equation

arXiv (Cornell University), Jul 23, 2022

We first extend the construction of generalized barycentric coordinates (GBC) based on the vertic... more We first extend the construction of generalized barycentric coordinates (GBC) based on the vertices on the boundary of a polygon Ω to a new kind of GBCs based on vertices inside the Ω of interest. For clarity, the standard GBCs are called boundary GBCs while the new GBCs are called interior GBCs. Then we present an analysis on these two kinds of harmonic GBCs to show that each GBC function whose value is 1 at a vertex (boundary or interior vertex of Ω) decays to zero away from its supporting vertex exponentially fast except for a trivial example. Based on the exponential decay property, we explain how to approximate the harmonic GBC functions locally. That is, due to the locality of these two kinds of GBCs, one can approximate each of these GBC functions by its local versions which is supported over a sub-domain of Ω. The local version of these GBC function will help reduce the computational time for shape deformation in graphical design. Next, with these two kinds of GBC functions at hand, we can use them to approximate the solution of the Dirichlet problem of the Poisson equation. This may provide a more efficient way to solve the Poisson equation by using a computer which has graphical processing unit(GPU) with thousands or more processes than the standard methods using a computer with one or few CPU kernels.

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Construction of C1 polygonal splines over quadrilateral partitions

Computer Aided Geometric Design, 2022

In this paper we construct smooth bivariate spline functions over a polygonal partition, e.g. a c... more In this paper we construct smooth bivariate spline functions over a polygonal partition, e.g. a convex quadrilateral partition by using vertex spline techniques. Vertex splines, introduced in [10] are smooth piecewise polynomial functions supported over a collection of triangles sharing a vertex. In this paper, we extend the concept of vertex splines to the partition of polygons and describe how to construct C 1 vertex polygonal splines over a collection of quadrilaterals. We begin with our construction of C 1 vertex splines over a collection of parallelograms, although they may not be axisorientated. Then the construction is generalized to the setting of general quadrilaterals. We will use various monomials of Wachspress GBC functions of degrees 5 and 7 to explain how to construct C 1 vertex splines together with additional special splines called edge and face splines. With these splines at hand, we construct quasi-interpolatory formulas, whose approximation properties will be shown. Numerical interpolation and approximation results will be presented. Finally, three applications of these splines are explained: the first one is to form smooth locally supported GBC functions, the second one is to construct smooth suitcase corners, and the third one to construct C 1 surfaces over quadrilateral partitions with extraordinary points(EP). Several examples will be demonstrated to show the convenience of using these splines.

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On the Computation of Battle-Lemarie's Wavelets

Mathematics of Computation, Oct 1, 1994

We propose a matrix approach to the computation of Battle-Lemarié's wavelets. The Fourier transfo... more We propose a matrix approach to the computation of Battle-Lemarié's wavelets. The Fourier transform of the scaling function is the product of the inverse F(x) of a square root of a positive trigonometric polynomial and the Fourier transform of a B-spline of order m. The polynomial is the symbol of a bi-infinite matrix B associated with a B-spline of order 2m. We approximate this bi-infinite matrix B2m by its finite section As , a square matrix of finite order. We use As to compute an approximation \s of x whose discrete Fourier transform is F(x). We show that xs converges pointwise to x exponentially fast. This gives a feasible method to compute the scaling function for any given tolerance. Similarly, this method can be used to compute the wavelets.

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Multivariate Splines and Their Applications

Springer eBooks, 2009

Semi-supervised Local Cluster Extraction by Compressive Sensing

arXiv (Cornell University), Nov 20, 2022

Local clustering problem aims at extracting a small local structure inside a graph without the ne... more Local clustering problem aims at extracting a small local structure inside a graph without the necessity of knowing the entire graph structure. As the local structure is usually small in size compared to the entire graph, one can think of it as a compressive sensing problem where the indices of target cluster can be thought as a sparse solution to a linear system. In this paper, we propose a new semi-supervised local cluster extraction approach by applying the idea of compressive sensing based on two pioneering works under the same framework. Our approves improves the existing works by making the initial cut to be the entire graph and hence overcomes a major limitation of existing works, which is the low quality of initial cut. Extensive experimental results on multiple benchmark datasets demonstrate the effectiveness of our approach.

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C2 Macro-element Spaces

Cambridge University Press eBooks, May 5, 2010

On DC based Methods for Phase Retrieval

arXiv (Cornell University), Oct 21, 2018

In this paper, we develop a new computational approach which is based on minimizing the differenc... more In this paper, we develop a new computational approach which is based on minimizing the difference of two convex functionals (DC) to solve a broader class of phase retrieval problems. The approach splits a standard nonlinear least squares minimizing function associated with the phase retrieval problem into the difference of two convex functions and then solves a sequence of convex minimization sub-problems. For each subproblem, the Nesterov's accelerated gradient descent algorithm or the Barzilai-Borwein (BB) algorithm is used. In the setting of sparse phase retrieval, a standard ℓ 1 norm term is added into the minimization mentioned above. The subproblem is approximated by a proximal gradient method which is solved by the shrinkage-threshold technique directly without iterations. In addition, a modified Attouch-Peypouquet technique is used to accelerate the iterative computation. These lead to more effective algorithms than the Wirtinger flow (WF) algorithm and the Gauss-Newton (GN) algorithm and etc.. A convergence analysis of both DC based algorithms shows that the iterative solutions is convergent linearly to a critical point and will be closer to a global minimizer than the given initial starting point. Our study is a deterministic analysis while the study for the Wirtinger flow (WF) algorithm and its variants, the Gauss-Newton (GN) algorithm, the trust region algorithm is based on the probability analysis. In particular, the DC based algorithms are able to retrieve solutions using a number m of measurements which is about twice of the number n of entries in the solution with high frequency of successes. When m ≈ n, the ℓ 1 DC based algorithm is able to retrieve sparse signals. Finally, the paper discusses the nonexistence of the solution to the exact recovery of the phase retrieval problem for arbitrary given measurement values. In addition, for a given set of measurement values, if there is a solution, an estimate of the upper bound of the number of distinct solutions is also given.

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Efficient spatial modelling using the SPDE approach with bivariate splines

arXiv (Cornell University), Mar 12, 2015

Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associ... more Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation of piecewise polynomial representations of various degrees. It leads to GMRFs that can be inferred efficiently and can be easily extended to non-stationary fields. The solutions approximated with higher order bivariate splines converge faster, hence the computational cost can be alleviated. Numerical simulations using both real and simulated data also demonstrate that our framework increases the flexibility and efficiency.

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The Probabilistic Estimates on the Largest and Smallest q-Singular Values of Pre-Gaussian Random Matrices

We study the q-singular values of random matrices with pre-Gaussian entries defined in terms of t... more We study the q-singular values of random matrices with pre-Gaussian entries defined in terms of the q-quasinorm with 0 < q ≤ 1. In this paper, we mainly consider the decay of the lower and upper tail probabilities of the largest q-singular value s (q) 1 , when the number of rows of the matrices becomes very large. Based on the results in probabilistic estimates on the largest q-singular value, we also give probabilistic estimates on the smallest q-singular value for pre-Gaussian random matrices.

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Sharp Sufficient Conditions for Stable Recovery of Block Sparse Signals by Block Orthogonal Matching Pursuit

arXiv (Cornell University), May 10, 2016

In this paper, we use the block orthogonal matching pursuit (BOMP) algorithm to recover block spa... more In this paper, we use the block orthogonal matching pursuit (BOMP) algorithm to recover block sparse signals x from measurements y = Ax + v, where v is an ℓ 2-bounded noise vector (i.e., v 2 ≤ ǫ for some constant ǫ). We investigate some sufficient conditions based on the block restricted isometry property (block-RIP) for exact (when v = 0) and stable (when v 0) recovery of block sparse signals x. First, on the one hand, we show that if A satisfies the block-RIP with δ K+1 < 1/ √ K + 1, then every block K-sparse signal x can be exactly or stably recovered by BOMP in K iterations. On the other hand, we show that, for any K ≥ 1 and 1/ √ K + 1 ≤ δ < 1, there exists a matrix A satisfying the block-RIP with δ K+1 = δ and a block K-sparse signal x such that BOMP may fail to recover x in K iterations. Then, we study some sufficient conditions for recovering block α-strongly-decaying K-sparse signals. We show that if A satisfies the block-RIP with δ K+1 < √ 2/2, then every α-strongly-decaying block K-sparse signal can be exactly or stably recovered by BOMP in K iterations under some conditions on α. Our newly found sufficient condition on the block-RIP of A is less restrictive than that for ℓ 1 minimization for this special class of sparse signals. Furthermore, for any K ≥ 1, α > 1 and √ 2/2 ≤ δ < 1, the recovery of x may fail in K iterations for a sensing matrix A which satisfies the block-RIP with δ K+1 = δ. Finally, we study some sufficient conditions for partial recovery of block sparse signals. Specifically, if A satisfies the block-RIP with δ K+1 < √ 2/2, then BOMP is guaranteed to recover some blocks of x if these blocks satisfy a sufficient condition. We further show that this condition is also sharp.

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