arXiv subjects

Qin Li

Publications and source records attributed to Qin Li.

At least 217 records · Page 12Linked to original sources

arXiv2018

Online learning in optical tomography: a stochastic approach

We study the inverse problem of radiative transfer equation (RTE) using stochastic gradient descent method (SGD) in this paper. Mathematically, optical tomography amounts to recovering the optical parameters in RTE using the incoming-outgoing pair of light intensity. We formulate it as a PDE-constraint optimization problem, where the mismatch of computed and measured outgoing data is minimized with same initial data and RTE constraint. The memory and computation cost it requires, however, is typically prohibitive, especially in high dimensional space. Smart iterative solvers that only use partial information in each step is called for thereafter. Stochastic gradient descent method is an online learning algorithm that randomly selects data for minimizing the mismatch. It requires minimum memory and computation, and advances fast, therefore perfectly serves the purpose. In this paper we formulate the problem, in both nonlinear and its linearized setting, apply SGD algorithm and analyze the convergence performance.

math.OC↗

arXiv2018

Explicit Finite Element Error Estimates for Nonhomogeneous Neumann problems

The paper develops an explicit a priori error estimate for finite element solution to nonhomogeneous Neumann problems. For this purpose, the hypercircle equation over finite element spaces is constructed and the explicit upper bound of the constant in the trace theorem is given. Numerical examples are shown in the final section, which implies the proposed error {estimate} has the convergence rate as $0.5$.

math.NA↗

arXiv2018

Interpretation of wake instability at slip line in rotating detonation

In studies on instabilities of flowfield in rotating detonation, one of the most common concerns is the instability at the slip line originating from the conjunction of the detonation wave and oblique shock. Using Euler equations associated with 7-species-and-8-reaction finite-rate chemical reaction model of hydrogen/air mixtures, further studies are performed to simulate the 2-D rotating detonation, and the flow mechanism of instability at the slip line is investigated in depth. The results show that the distinct wake profile exists at the slip line, which is different from the typical mixing layer. Analysis indicates that the generation of wake is caused by the transition shock between the detonation wave and oblique shock. Because of the wake profile, the vorticity distribution therein appears in a double-layer layout, and different evolution exist in different vorticity layers. Based on the velocity profile across the slip line, the analysis by the linear stability theory is made, and two unstable modes which have different shape profiles and phase velocities are found. Discrete Fourier transformation is utilized to analyze the numerical results, and similar shape profiles are obtained. A general coincidence in velocity of vortex movement is also attained between the theoretical predictions and simulations. Investigations show that the wake instability is responsible for the unstable mechanism, and corresponding unstable structures differs from the canonical ones in typical mixing layers.

physics.flu-dyn↗

arXiv2017

Extending Counter-Streaming Motion from an Active Region Filament to Sunspot Light Bridge

We analyze the high-resolution observations from the 1.6m telescope at Big Bear Solar Observatory that cover an active region filament. Counter-streaming motions are clearly observed in the filament. The northern end of the counter-streaming motions extends to a light bridge, forming a spectacular circulation pattern around a sunspot, with clockwise motion in the blue wing and counterclockwise motion in the red wing as observed in Halpha off-bands. The apparent speed of the flow is around 10 to 60 km/s in the filament, decreasing to 5 to 20 km/s in the light bridge. The most intriguing results are the magnetic structure and the counter-streaming motions in the light bridge. Similar to those in the filament, magnetic fields show a dominant transverse component in the light bridge. However, the filament is located between opposite magnetic polarities, while the light bridge is between strong fields of the same polarity. We analyze the power of oscillations with the image sequences of constructed Dopplergrams, and find that the filament's counter-streaming motion is due to physical mass motion along fibrils, while the light bridge's counter-streaming motion is due to oscillation in the direction along the line-of-sight. The oscillation power peaks around 4 minutes. However, the section of the light bridge next to the filament also contains a component of the extension of the filament in combination with the oscillation, indicating that some strands of the filament are extended to and rooted in that part of the light bridge.

astro-ph.SR↗

arXiv2017

Stability of Stationary Inverse Transport Equation in Diffusion Scaling

We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales characterized by the magnitude of a dimensionless parameter---the Knudsen number ($K_n$). In the diffusive scaling ($K_n \ll 1$), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well-posedness of inverse transport equation, which raises the question of how uniqueness being lost as $K_n \rightarrow 0$. We tackle this problem by examining the stability of inverse problem with varying $K_n$. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of $K_n^p~ (p = 1\text{ or} ~2)$, and as a result lead to ill-posedness in the zero limit of $K_n$. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings.

math.NA↗

arXiv2017

Stability of inverse transport equation in diffusion scaling and Fokker-Planck limit

We consider the inverse problem of reconstructing the scattering and absorption coefficients using boundary measurements for a time dependent radiative transfer equation (RTE). As the measurement is mostly polluted by errors, both experimental and computational, an important question is to quantify how the error is amplified in the process of reconstruction. In the forward setting, the solution to the RTE behaves differently in different regimes, and the stability of the inverse problem vary accordingly. In particular, we consider two scalings in this paper. The first one concerns with a diffusive scaling whose macroscopic limit is a diffusion equation. In this case, we showed, following the similar approach as in [Chen, Li and Wang, arXiv:1703.00097], that the stability degrades when the limit is taken. The second one considers a highly forward peaked scattering, wherein the scattering operator is approximated by a Fokker-Planck operator as a limit. In this case, we showed that a fully recover of the scattering coefficient is less possible in the limit, whereas obtaining a rescaled version of the scattering coefficient becomes more practice friendly.

math.NA↗

arXiv2017

A new numerical approach to inverse transport equation with error analysis

The inverse radiative transfer problem finds broad applications in medical imaging, atmospheric science, astronomy, and many other areas. This problem intends to recover the optical properties, denoted as absorption and scattering coefficient of the media, through the source-measurement pairs. A typical computational approach is to form the inverse problem as a PDE-constraint optimization, with the minimizer being the to-be-recovered coefficients. The method is tested to be efficient in practice, but lacks analytical justification: there is no guarantee of the existence or uniqueness of the minimizer, and the error is hard to quantify. In this paper, we provide a different algorithm by levering the ideas from singular decomposition analysis. Our approach is to decompose the measurements into three components, two out of which encode the information of the two coefficients respectively. We then split the optimization problem into two subproblems and use those two components to recover the absorption and scattering coefficients separately. In this regard, we prove the well-posedness of the new optimization, and the error could be quantified with better precision. In the end, we incorporate the diffusive scaling and show that the error is harder to control in the diffusive limit.

math.NA↗

arXiv2017

High-resolution Observation of Downflows at One End of a Pre-eruption Filament

Studying the dynamics of filaments at pre-eruption phase can shed light on the precursor of eruptive events. Such studies in high-resolution (in the order of 0.1") are highly desirable yet very rare so far. In this work, we present a detailed observation of a pre-eruption evolution of a filament obtained by the 1.6 m New Solar Telescope (NST) at Big Bear Solar Observatory (BBSO). One end of the filament is anchored at the sunspot in NOAA active region (AR) 11515, which is well observed by NST H$α$ off-bands four hours before till one hour after the filament eruption. A M1.6 flare is associated with the eruption. We observed persistent downflowing materials along the H$α$ multi-threaded component of the loop towards the AR end during the pre-eruption phase. We traced the trajectories of plasma blobs along the H$α$ threads and obtained the plane-of-sky velocity of 45 km s$^{-1}$ on average. We further estimated the real velocities of the downflows and the altitude of the filament by matching the observed H$α$ threads with magnetic field lines extrapolated from a nonlinear force-free field (NLFFF) model. Observation of chromospheric brightenings (BZs) at the footpoints of the falling plasma blobs is also presented in the paper. The lower limit of the kinetic energy per second of the downflows through the BZs is found to be $\sim$ 10$^{21}$ erg. Larger FOV observations from BBSO full disk H$α$ images show that the AR end of the filament started ascending four hours before the flare. We attribute the observed downflows at the AR end of the filament to the draining effect of the filament rising prior to its eruption. During the slow-rise phase, the downflows continuously drained away $\sim$ 10$^{15}$g mass from the filament over a few hours, which is believed to be essential for the instability at last, and could be an important precursor of eruptive events.

astro-ph.SR↗

arXiv2017

Batalin-Vilkovisky quantization and the algebraic index

Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov's deformation quantization of a symplectic manifold X and the BV quantization of a one-dimensional sigma model with target X. This model is a quantum field theory of AKSZ type and is quantized rigorously using Costello's homotopic theory of effective renormalization. We show that Fedosov's Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration produces a natural trace map on the deformation quantized algebra. This formulation allows us to exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem via semi-classical analysis, i.e., one-loop Feynman diagram computations.

math.QA↗

arXiv2017

BV quantization of the Rozansky-Witten model

We investigate the perturbative aspects of Rozansky-Witten's 3d $σ$-model using Costello's approach to the Batalin-Vilkovisky (BV) formalism. We show that the BV quantization (in Costello's sense) of the model, which produces a perturbative quantum field theory, can be obtained via the configuration space method of regularization due to Kontsevich and Axelrod-Singer. We also study the factorization algebra structure for quantum observables following Costello-Gwilliam. In particular, we show that the cohomology of local quantum observables on a genus $g$ handle body is given by $H^*(X,(\wedge^*T_X)^{\otimes g})$ (where $X$ is the target manifold), and prove that the partition function reproduces the Rozansky-Witten invariants.

math.QA↗

arXiv2017

An asymptotic preserving method for transport equations with oscillatory scattering coefficients

We design a numerical scheme for transport equations with oscillatory periodic scattering coefficients. The scheme is asymptotic preserving in the diffusion limit as Knudsen number goes to zero. It also captures the homogenization limit as the length scale of the scattering coefficient goes to zero. The proposed method is based on the construction of multiscale finite element basis and a Galerkin projection based on the even-odd decomposition. The method is analyzed in the asymptotic regime, as well as validated numerically.

math.NA↗

arXiv2017

Uniform regularity for linear kinetic equations with random input based on hypocoercivity

In this paper we study the effect of randomness in kinetic equations that preserve mass. Our focus is in proving the analyticity of the solution with respect to the randomness, which naturally leads to the convergence of numerical methods. The analysis is carried out in a general setting, with the regularity result not depending on the specific form of the collision term, the probability distribution of the random variables, or the regime the system is in, and thereby termed "uniform". Applications include the linear Boltzmann equation, BGK model, Carlemann model, among many others; and the results hold true in kinetic, parabolic and high field regimes. The proof relies on the explicit expression of the high order derivatives of the solution in the random space, and the convergence in time is mainly based on hypocoercivity, which, despite the popularity in PDE analysis of kinetic theory, has rarely been used for numerical algorithms.

math.NA↗

arXiv2017

Further investigations on the interface instability between fresh injections and burnt products in 2-D rotating detonation

Instabilities in rotating detonation are concerned because of their potential influence on the stability of operation. Previous studies on instability of 2-D rotating detonation mainly cared about the one of the contact discontinuity originated from the conjunction of the detonation and oblique shock. Hishida et al. first found the rippled structure existed in the interface between fresh injections and burnt product from the previous cycle (Shock Waves 19, 2009), and a mechanism of Kelvin-Helmholtz instability was suggested as well. Similar structures were observed as well in simulations by current authors, where a fifth-order WENO-type scheme with improved resolution and 7-species-and-8-reaction chemical model were used. In order to achieve a deep understanding on the flow mechanism, more careful simulations are carried out by using three grids with increasing resolution. The results show that besides the previously-mentioned Kelvin-Helmholtz instability, there are two other mechanisms which take effect in the interface instability, i.e., the effect of baroclinic torque and Rayleigh-Taylor instability. Occurrence conditions for two instabilities are checked and testified. Especially, the spike- and bubble-like structures are observed at the interface, which show appearances different from canonical structures by Kelvin-Helmholtz instability.

physics.flu-dyn↗

arXiv2017

Triple-pulse X-ray spot size measurement of the Dragon-II linear induction accelerator

The Dragon-II linear induction accelerator is able to provide triple-pulse electron beams with an adjustable pulse spacing at the minimum of 300 ns. As a main determinant of the image quality, the X-ray spot size is usually quoted as an evaluation of the resolving power. The pinhole imaging method is applied to measure the two-dimensional spatial distribution of the triple-pulse X-ray source, by which the azimuthal asymmetry of the source can be analyzed. The spot size of each pulse is characterized by both the full-width at half-maximum and the LANL definition. The weighted RMS radius at different sum ratios of photon intensity is also calculated, which directly takes into account the contribution of every single photon within the boundary of the spot.

physics.acc-ph↗

arXiv2016

Exploring the locally low dimensional structure in solving random elliptic PDEs

We propose a stochastic multiscale finite element method (StoMsFEM) to solve random elliptic partial differential equations with a high stochastic dimension. The key idea is to simultaneously upscale the stochastic solutions in the physical space for all random samples and explore the low stochastic dimensions of the stochastic solution within each local patch. We propose two effective methods to achieve this simultaneous local upscaling. The first method is a high order interpolation method in the stochastic space that explores the high regularity of the local upscaled quantities with respect to the random variables. The second method is a reduced-order method that explores the low rank property of the multiscale basis functions within each coarse grid patch. Our complexity analysis shows that compared with the standard FEM on a fine grid, the StoMsFEM can achieve computational saving in the order of $(H/h)^{d}/(\log(H/h))^k$, where $H/h$ is the ratio between the coarse and the fine gird sizes, $d$ is the physical dimension and $k$ is the local stochastic dimension. Several numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed methods. In the high contrast example, we observe a factor of 2000 speed-up.

math.NA↗

arXiv2016

A sparse decomposition of low rank symmetric positive semi-definite matrices

Suppose that $A \in \mathbb{R}^{N \times N}$ is symmetric positive semidefinite with rank $K \le N$. Our goal is to decompose $A$ into $K$ rank-one matrices $\sum_{k=1}^K g_k g_k^T$ where the modes $\{g_{k}\}_{k=1}^K$ are required to be as sparse as possible. In contrast to eigen decomposition, these sparse modes are not required to be orthogonal. Such a problem arises in random field parametrization where $A$ is the covariance function and is intractable to solve in general. In this paper, we partition the indices from 1 to $N$ into several patches and propose to quantify the sparseness of a vector by the number of patches on which it is nonzero, which is called patch-wise sparseness. Our aim is to find the decomposition which minimizes the total patch-wise sparseness of the decomposed modes. We propose a domain-decomposition type method, called intrinsic sparse mode decomposition (ISMD), which follows the "local-modes-construction + patching-up" procedure. The key step in the ISMD is to construct local pieces of the intrinsic sparse modes by a joint diagonalization problem. Thereafter a pivoted Cholesky decomposition is utilized to glue these local pieces together. Optimal sparse decomposition, consistency with different domain decomposition and robustness to small perturbation are proved under the so called regular-sparse assumption (see Definition 1.2). We provide simulation results to show the efficiency and robustness of the ISMD. We also compare the ISMD to other existing methods, e.g., eigen decomposition, pivoted Cholesky decomposition and convex relaxation of sparse principal component analysis [25] and [40].

math.NA↗

arXiv2016

Validity and regularization of classical half-space equations

Recent result [Wu and Guo, Comm. Math. Phys., 2015] has shown that over the 2D unit disk, the classical half-space equation (CHS) for the neutron transport does not capture the correct boundary layer behaviour as long believed. In this paper we develop a regularization technique for CHS to any arbitrary order and use its first-order regularization to show that in the case of the 2D unit disk, although CHS misrepresents the boundary layer behaviour, it does give the correct boundary condition for the interior macroscopic (Laplace) equation. Therefore CHS is still a valid equation to recover the correct boundary condition for the interior Laplace equation over the 2D unit disk.

math.AP↗

arXiv2016

A convergent method for linear half-space kinetic equations

We give a unified proof for the well-posedness of a class of linear half-space equations with general incoming data and construct a Galerkin method to numerically resolve this type of equations in a systematic way. Our main strategy in both analysis and numerics includes three steps: adding damping terms to the original half-space equation, using an inf-sup argument and even-odd decomposition to establish the well-posedness of the damped equation, and then recovering solutions to the original half-space equation. The proposed numerical methods for the damped equation is shown to be quasi-optimal and the numerical error of approximations to the original equation is controlled by that of the damped equation. This efficient solution to the half-space problem is useful for kinetic-fluid coupling simulations.

math.AP↗