arXiv subjects

Jack Xin

Publications and source records attributed to Jack Xin.

At least 109 records · Page 6Linked to original sources

arXiv2015

Computational Modeling of Spectral Data Fitting with Nonlinear Distortions

Substances such as chemical compounds are invisible to human eyes, they are usually captured by sensing equipments with their spectral fingerprints. Though spectra of pure chemicals can be identified by visual inspection, the spectra of their mixtures take a variety of complicated forms. Given the knowledge of spectral references of the constituent chemicals, the task of data fitting is to retrieve their weights, and this usually can be obtained by solving a least squares problem. Complications occur if the basis functions (reference spectra) may not be used directly to best fit the data. In fact, random distortions (spectral variability) such as shifting, compression, and expansion have been observed in some source spectra when the underlying substances are mixed. In this paper, we formulate mathematical model for such nonlinear effects and build them into data fitting algorithms. If minimal knowledge of the distortions is available, a deterministic approach termed {\it augmented least squares} is developed and it fits the spectral references along with their derivatives to the mixtures. If the distribution of the distortions is known a prior, we consider to solve the problem with maximum likelihood estimators which incorporate the shifts into the variance matrix. The proposed methods are substantiated with numerical examples including data from Raman spectroscopy (RS), nuclear magnetic resonance (NMR), and differential optical absorption spectroscopy (DOAS) and show satisfactory results.

math.NA↗

arXiv2014

PhaseLiftOff: an Accurate and Stable Phase Retrieval Method Based on Difference of Trace and Frobenius Norms

Phase retrieval aims to recover a signal $x \in \mathbb{C}^{n}$ from its amplitude measurements $| |^2$, $i=1,2,...,m$, where $a_i$'s are over-complete basis vectors, with $m$ at least $3n -2$ to ensure a unique solution up to a constant phase factor. The quadratic measurement becomes linear in terms of the rank-one matrix $X = x x^*$. Phase retrieval is then a rank-one minimization problem subject to linear constraint for which a convex relaxation based on trace-norm minimization (PhaseLift) has been extensively studied recently. At $m=O(n)$, PhaseLift recovers with high probability the rank-one solution. In this paper, we present a precise proxy of rank-one condition via the difference of trace and Frobenius norms which we call PhaseLiftOff. The associated least squares minimization with this penalty as regularization is equivalent to the rank-one least squares problem under a mild condition on the measurement noise. Stable recovery error estimates are valid at $m=O(n)$ with high probability. Computation of PhaseLiftOff minimization is carried out by a convergent difference of convex functions algorithm. In our numerical example, $a_i$'s are Gaussian distributed. Numerical results show that PhaseLiftOff outperforms PhaseLift and its nonconvex variant (log-determinant regularization), and successfully recovers signals near the theoretical lower limit on the number of measurements without the noise.

math.OC↗

arXiv2014

A Computational Study of Residual KPP Front Speeds in Time-Periodic Cellular Flows in the Small Diffusion Limit

The minimal speeds ($c^*$) of the Kolmogorov-Petrovsky-Piskunov (KPP) fronts at small diffusion ($ε\ll 1$) in a class of time-periodic cellular flows with chaotic streamlines is investigated in this paper. The variational principle of $c^*$ reduces the computation to that of a principal eigenvalue problem on a periodic domain of a linear advection-diffusion operator with space-time periodic coefficients and small diffusion. To solve the advection dominated time-dependent eigenvalue problem efficiently over large time, a combination of finite element and spectral methods, as well as the associated fast solvers, are utilized to accelerate computation. In contrast to the scaling $c^*=\mathcal{O}(ε^{1/4})$ in steady cellular flows, a new relation $c^* = \mathcal{O}(1)$ as $ε\ll 1$ is revealed in the time-periodic cellular flows due to the presence of chaotic streamlines. Residual propagation speed emerges from the Lagrangian chaos which is quantified as a sub-diffusion process.

nlin.CD↗

arXiv2013

A Method for Finding Structured Sparse Solutions to Non-negative Least Squares Problems with Applications

Demixing problems in many areas such as hyperspectral imaging and differential optical absorption spectroscopy (DOAS) often require finding sparse nonnegative linear combinations of dictionary elements that match observed data. We show how aspects of these problems, such as misalignment of DOAS references and uncertainty in hyperspectral endmembers, can be modeled by expanding the dictionary with grouped elements and imposing a structured sparsity assumption that the combinations within each group should be sparse or even 1-sparse. If the dictionary is highly coherent, it is difficult to obtain good solutions using convex or greedy methods, such as non-negative least squares (NNLS) or orthogonal matching pursuit. We use penalties related to the Hoyer measure, which is the ratio of the $l_1$ and $l_2$ norms, as sparsity penalties to be added to the objective in NNLS-type models. For solving the resulting nonconvex models, we propose a scaled gradient projection algorithm that requires solving a sequence of strongly convex quadratic programs. We discuss its close connections to convex splitting methods and difference of convex programming. We also present promising numerical results for example DOAS analysis and hyperspectral demixing problems.

stat.ML↗

arXiv2012

Turbulent Flame Speeds of G-equation Models in Unsteady Cellular Flows

We perform a computationl study of front speeds of G-equation models in time dependent cellular flows. The G-equations arise in premixed turbulent combustion, and are Hamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strain G equations are also non-convex with degenerate diffusion. The computation is based on monotone finite difference discretization and weighted essentially nonoscillatory (WENO) methods. We found that the large time front speeds lock into the frequency of time periodic cellular flows in curvature-strain G-equations similar to what occurs in the basic inviscid G-equation. However, such frequency locking phenomenon disappears in viscous G-equation, and in the inviscid G-equation if time periodic oscillation of the cellular flow is replaced by time stochastic oscillation.

nlin.PS↗

arXiv2012

A Numerical Study of Turbulent Flame Speeds of Curvature and Strain G-equations in Cellular Flows

We study front speeds of curvature and strain G-equations arising in turbulent combustion. These G-equations are Hamilton-Jacobi type level set partial differential equations (PDEs) with non-coercive Hamiltonians and degenerate nonlinear second order diffusion. The Hamiltonian of strain G-equation is also non-convex. Numerical computation is performed based on monotone discretization and weighted essentially nonoscillatory (WENO) approximation of transformed G-equations on a fixed periodic domain. The advection field in the computation is a two dimensional Hamiltonian flow consisting of a periodic array of counter-rotating vortices, or cellular flows. Depending on whether the evolution is predominantly in the hyperbolic or parabolic regimes, suitable explicit and semi-implicit time stepping methods are chosen. The turbulent flame speeds are computed as the linear growth rates of large time solutions. A new nonlinear parabolic PDE is proposed for the reinitialization of level set functions to prevent piling up of multiple bundles of level sets on the periodic domain. We found that the turbulent flame speed $s_T$ of the curvature G-equation is enhanced as the intensity $A$ of cellular flows increases, at a rate between those of the inviscid and viscous G-equations. The $s_T$ of the strain G-equation increases in small $A$, decreases in larger $A$, then drops down to zero at a large enough but finite value $A_{*}$. The flame front ceases to propagate at this critical intensity $A_*$, and is quenched by the cellular flow.

math.NA↗

arXiv2011

Analysis and Comparison of Large Time Front Speeds in Turbulent Combustion Models

Predicting turbulent flame speed (the large time front speed) is a fundamental problem in turbulent combustion theory. Several models have been proposed to study the turbulent flame speed, such as the G-equations, the F-equations (Majda-Souganidis model) and reaction-diffusion-advection (RDA) equations. In the first part of this paper, we show that flow induced strain reduces front speeds of G-equations in periodic compressible and shear flows. The F-equations arise in asymptotic analysis of reaction-diffusion-advection equations and are quadratically nonlinear analogues of the G-equations. In the second part of the paper, we compare asymptotic growth rates of the turbulent flame speeds from the G-equations, the F-equations and the RDA equations in the large amplitude ($A$) regime of spatially periodic flows. The F and G equations share the same asymptotic front speed growth rate; in particular, the same sublinear growth law $A\over \log(A)$ holds in cellular flows. Moreover, in two space dimensions, if one of these three models (G-equation, F-equation and the RDA equation) predicts the bending effect (sublinear growth in the large flow), so will the other two. The nonoccurrence of speed bending is characterized by the existence of periodic orbits on the torus and the property of their rotation vectors in the advective flow fields. The cat's eye flow is discussed as a typical example of directional dependence of the front speed bending. The large time front speeds of the viscous F-equation have the same growth rate as those of the inviscid F and G-equations in two dimensional periodic incompressible flows.

math.AP↗

arXiv2011

A convex model for non-negative matrix factorization and dimensionality reduction on physical space

A collaborative convex framework for factoring a data matrix $X$ into a non-negative product $AS$, with a sparse coefficient matrix $S$, is proposed. We restrict the columns of the dictionary matrix $A$ to coincide with certain columns of the data matrix $X$, thereby guaranteeing a physically meaningful dictionary and dimensionality reduction. We use $l_{1,\infty}$ regularization to select the dictionary from the data and show this leads to an exact convex relaxation of $l_0$ in the case of distinct noise free data. We also show how to relax the restriction-to-$X$ constraint by initializing an alternating minimization approach with the solution of the convex model, obtaining a dictionary close to but not necessarily in $X$. We focus on applications of the proposed framework to hyperspectral endmember and abundances identification and also show an application to blind source separation of NMR data.

stat.ML↗

arXiv2010

Asymptotics for turbulent flame speeds of the viscous G-equation enhanced by cellular and shear flows

G-equations are well-known front propagation models in turbulent combustion and describe the front motion law in the form of local normal velocity equal to a constant (laminar speed) plus the normal projection of fluid velocity. In level set formulation, G-equations are Hamilton-Jacobi equations with convex ($L^1$ type) but non-coercive Hamiltonians. Viscous G-equations arise from either numerical approximations or regularizations by small diffusion. The nonlinear eigenvalue $\bar H$ from the cell problem of the viscous G-equation can be viewed as an approximation of the inviscid turbulent flame speed $s_T$. An important problem in turbulent combustion theory is to study properties of $s_T$, in particular how $s_T$ depends on the flow amplitude $A$. In this paper, we will study the behavior of $\bar H=\bar H(A,d)$ as $A\to +\infty$ at any fixed diffusion constant $d > 0$. For the cellular flow, we show that $$ \bar H(A,d)\leq O(\sqrt {\mathrm {log}A}) \quad \text{for all $d>0$}. $$ Compared with the inviscid G-equation ($d=0$), the diffusion dramatically slows down the front propagation. For the shear flow, the limit \nit $\lim_{A\to +\infty}{\bar H(A,d)\over A} = λ(d) >0$ where $λ(d)$ is strictly decreasing in $d$, and has zero derivative at $d=0$. The linear growth law is also valid for $s_T$ of the curvature dependent G-equation in shear flows.

math.AP↗

arXiv2007

A Dynamic Algorithm for Blind Separation of Convolutive Sound Mixtures

We study an efficient dynamic blind source separation algorithm of convolutive sound mixtures based on updating statistical information in the frequency domain, andminimizing the support of time domain demixing filters by a weighted least square method. The permutation and scaling indeterminacies of separation, and concatenations of signals in adjacent time frames are resolved with optimization of $l^1 \times l^\infty$ norm on cross-correlation coefficients at multiple time lags. The algorithm is a direct method without iterations, and is adaptive to the environment. Computations on recorded and synthetic mixtures of speech and music signals show excellent performance.

math.ST↗

arXiv2006

A Many to One Discrete Auditory Transform

A many to one discrete auditory transform is presented to map a sound signal to a perceptually meaningful spectrum on the scale of human auditory filter band widths (critical bands). A generalized inverse is constructed in closed analytical form, preserving the band energy and band signal to noise ratio of the input sound signal. The forward and inverse transforms can be implemented in real time. Experiments on speech and music segments show that the inversion gives a perceptually equivalent though mathematically different sound from the input.

math.NA↗

arXiv2005

Variational Principle of KPP Front Speeds in Temporally Random Shear Flows

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows inside an infinite cylinder, under suitable assumptions of the shear field. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. The variational principle then allows us to bound and compute the front speeds. We show the linear and quadratic laws of speed enhancement as well as a resonance-like dependence of front speed on the temporal shear correlation length. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows.

math.AP↗

arXiv2005

Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds

We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We apply a dynamic argument to justify that the constructed KPP traveling fronts propagate at minimal speeds, and derive the speed variational formula. The dynamic method avoids the degeneracy in traveling front equations, and utilizes the parabolic maximum principle of the governing reaction-diffusion-advection equation. The dynamic method does not rely on existence of traveling fronts.

math.AP↗

arXiv2005

An Orthogonal Discrete Auditory Transform

An orthogonal discrete auditory transform (ODAT) from sound signal to spectrum is constructed by combining the auditory spreading matrix of Schroeder et al and the time one map of a discrete nonlocal Schrödinger equation. Thanks to the dispersive smoothing property of the Schrödinger evolution, ODAT spectrum is smoother than that of the discrete Fourier transform (DFT) consistent with human audition. ODAT and DFT are compared in signal denoising tests with spectral thresholding method. The signals are noisy speech segments. ODAT outperforms DFT in signal to noise ratio (SNR) when the noise level is relatively high.

math.NA↗

arXiv2005

Min-Max Variational Principle and Front Speeds in Random Shear Flows

Speed ensemble of bistable (combustion) fronts in mean zero stationary Gaussian shear flows inside two and three dimensional channels is studied with a min-max variational principle. In the small root mean square regime of shear flows, a new class of multi-scale test functions are found to yield speed asymptotics. The quadratic speed enhancement law holds with probability arbitrarily close to one under the almost sure continuity (dimension two) and mean square Hölder regularity (dimension three) of the shear flows. Remarks are made on the conditions for the linear growth of front speed expectation in the large root mean square regime.

math.AP↗

arXiv2005

A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears

Variational principle for Kolmogorov-Petrovsky-Piskunov (KPP) minimal front speeds provides an efficient tool for statistical speed analysis, as well as a fast and accurate method for speed computation. A variational principle based analysis is carried out on the ensemble of KPP speeds through spatially stationary random shear flows inside infinite channel domains. In the regime of small root mean square (rms) shear amplitude, the enhancement of the ensemble averaged KPP front speeds is proved to obey the quadratic law under certain shear moment conditions. Similarly, in the large rms amplitude regime, the enhancement follows the linear law. In particular, both laws hold for the Ornstein-Uhlenbeck process in case of two dimensional channels. An asymptotic ensemble averaged speed formula is derived in the small rms regime and is explicit in case of the Ornstein-Uhlenbeck process of the shear. Variational principle based computation agrees with these analytical findings, and allows further study on the speed enhancement distributions as well as the dependence of enhancement on the shear covariance. Direct simulations in the small rms regime suggest quadratic speed enhancement law for non-KPP nonlinearities.

math.AP↗

arXiv2004

An Invertible Discrete Auditory Transform

A discrete auditory transform (DAT) from sound signal to spectrum is presented and shown to be invertible in closed form. The transform preserves energy, and its spectrum is smoother than that of the discrete Fourier transform (DFT) consistent with human audition. DAT and DFT are compared in signal denoising tests with spectral thresholding method. The signals are noisy speech segments. It is found that DAT can gain 5 to 7 decibel (dB) in signal to noise ratio (SNR) over DFT except when the noise level is relatively low.

math.NA↗

arXiv2004

Existence of KPP Type Fronts in Space-Time Periodic Shear Flows and a Study of Minimal Speeds Based on Variational Principle

We prove the existence of reaction-diffusion traveling fronts in mean zero space-time periodic shear flows for nonnegative reactions including the classical KPP (Kolmogorov-Petrovsky-Piskunov) nonlinearity. For the KPP nonlinearity, the minimal front speed is characterized by a variational principle involving the principal eigenvalue of a space-time periodic parabolic operator. Analysis of the variational principle shows that adding a mean-zero space time periodic shear flow to an existing mean zero space-periodic shear flow leads to speed enhancement. Computation of KPP minimal speeds is performed based on the variational principle and a spectrally accurate discretization of the principal eigenvalue problem. It shows that the enhancement is monotone decreasing in temporal shear frequency, and that the total enhancement from pure reaction-diffusion obeys quadratic and linear laws at small and large shear amplitudes.

math.AP↗