Jingwei Hu

University of Washington Seattle | UW · Department of Applied Mathematics

Radiation transport problems are posed in a high-dimensional phase space, limiting the use of finely resolved numerical simulations. An emerging tool to efficiently reduce computational costs and memory footprint in such settings is dynamical low-rank approximation (DLRA). Despite its efficiency, numerical methods for DLRA need to be carefully cons...

Efficient and accurate numerical approximation of the full Boltzmann equation has been a longstanding challenging problem in kinetic theory. This is mainly due to the high dimensionality of the problem and the complicated collision operator. In this work, we propose a highly efficient adaptive low rank method for the Boltzmann equation, concerning...

In this work we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker–Planck equation and the Keller–Segel equations. The two proposed schemes are first-order accurate in time, explicitly solvable, and second-order and fourth-ord...

Efficient and accurate numerical approximation of the full Boltzmann equation has been a longstanding challenging problem in kinetic theory. This is mainly due to the high dimensionality of the problem and the complicated collision operator. In this work, we propose a highly efficient adaptive low rank method for the Boltzmann equation, concerning...

We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the particular case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-line...

In this work, we are concerned with a Fokker-Planck equation related to the nonlinear noisy leaky integrate-and-fire model for biological neural networks which are structured by the synaptic weights and equipped with the Hebbian learning rule. The equation contains a small parameter $\varepsilon$ separating the time scales of learning and reacting...

Learning time-dependent partial differential equations (PDEs) that govern evolutionary observations is one of the 4 core challenges for data-driven inference in many fields. In this work, we propose to capture the essential dynamics of numerically 5 challenging PDEs arising in multiscale modeling and simulation-kinetic equations. These equations ar...

We propose a dynamical low-rank method to reduce the computational complexity for solving the multi-scale multi-dimensional linear transport equation. The method is based on a macro-micro decomposition of the equation; the low-rank approximation is only used for the micro part of the solution. The time and spatial discretizations are done properly...

We consider general systems of ordinary differential equations with monotonic Gibbs entropy, and introduce an entropic scheme that simply imposes an entropy fix after every time step of any existing time integrator. It is proved that in the general case, our entropy fix has only infinitesimal influence on the numerical order of the original scheme,...

Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for its relative high accuracy and possibility of being accelerated by the fast Fourier transform. However, this me...

In this work, we introduce semi-implicit or implicit finite difference schemes for the continuity equation with a gradient flow structure. Examples of such equations include the linear Fokker-Planck equation and the Keller-Segel equations. The two proposed schemes are first order accurate in time, explicitly solvable, and second order and fourth or...

In this work we present multi-derivative implicit-explicit (IMEX) Runge--Kutta schemes. We derive their order conditions up to third order, and show that such methods can preserve positivity (and more generally strong stability) with a time-step restriction independent of the stiff term, under mild assumptions on the operators. We present sufficien...

Over the past decades, kinetic description of granular materials has received a lot of attention in mathematical community and applied fields such as physics and engineering. This article aims to review recent mathematical results in kinetic granular materials, especially for those which arose since the last review Villani (J Stat Phys 124(2):781–8...

In this work, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. While the...

It has recently been demonstrated that dynamical low-rank algorithms can provide robust and efficient approximation to a range of kinetic equations. This is true especially if the solution is close to some asymptotic limit where it is known that the solution is low-rank. A particularly interesting case is the fluid dynamic limit that is commonly ob...

We introduce a fast Fourier spectral method for the spatially homogeneous Boltzmann equation with non-cutoff collision kernels. Such kernels contain non-integrable singularity in the deviation angle which arise in a wide range of interaction potentials (e.g., the inverse power law potentials). Albeit more physical, the non-cutoff kernels bring a lo...

Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further ac...

We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains al...

We introduce a dynamical low-rank method to reduce the computational complexity for solving the multi-scale multi-dimensional linear transport equation. The method is based on a macro-micro decomposition of the equation. The proposed numerical method uses the low rank approximation only for the micro part of the solution. The time and spatial discr...

The Poisson–Nernst–Planck (PNP) equations is a macroscopic model widely used to describe the dynamics of ion transport in ion channels. In this paper, we introduce a semi-implicit finite difference scheme for the PNP equations in a bounded domain. A general boundary condition for the Poisson equation is considered. The fully discrete scheme is show...

We introduce a fast Fourier spectral method for the spatially homogeneous Boltzmann equation with non-cutoff collision kernels. Such kernels contain non-integrable singularity in the deviation angle which arise in a wide range of interaction potentials (e.g., the inverse power law potentials). Albeit more physical, the non-cutoff kernels bring a lo...

We propose a control variate multilevel Monte Carlo method for the kinetic BGK model of the Boltzmann equation subject to random inputs. The method combines a multilevel Monte Carlo technique with the computation of the optimal control variate multipliers derived from local or global variance minimization problems. Consistency and convergence analy...

We propose fully discrete, implicit-in-time finite-volume schemes for a general family of non-linear and non-local Fokker-Planck equations with a gradient-flow structure, usually known as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivity-preservation and energy-dissipation properties, essential for their practical...

Over the past decades, kinetic description of granular materials has received a lot of attention in mathematical community and applied fields such as physics and engineering. This article aims to review recent mathematical results in kinetic granular materials, especially for those which arose since the last review by Villani on the same subject. W...

We develop in this paper a Petrov-Galerkin spectral method for the inelastic Boltzmann equation in one dimension. Solutions to such equations typically exhibit heavy tails in the velocity space so that domain truncation or Fourier approximation would suffer from large truncation errors. Our method is based on the mapped Chebyshev functions on unbou...

Many hyperbolic and kinetic equations contain a non-stiff convection/transport part and a stiff relaxation/collision part (characterized by the relaxation or mean free time $\varepsilon$). To solve this type of problems, implicit-explicit (IMEX) Runge-Kutta or multistep methods have been widely used and their performance is understood well in the n...

In this work, we are concerned with the Fokker-Planck equations associated with the Nonlinear Noisy Leaky Integrate-and-Fire model for neuron networks. Due to the jump mechanism at the microscopic level, such Fokker-Planck equations are endowed with an unconventional structure: transporting the boundary flux to a specific interior point. While the...

We propose a novel deterministic particle method to numerically approximate the Landau equation for plasmas. Based on a new variational formulation in terms of gradient flows of the Landau equation, we regularize the collision operator to make sense of the particle solutions. These particle solutions solve a large coupled ODE system that retains al...

The Boltzmann equation, a six-dimensional integro-differential equation, governs the fluid flow behavior at molecular level for a wide range of physical phenomena, including shocks, turbulence, diffusion, and non-equilibrium chemistry which are beyond the reach of continuum fluid flow modelling based on the Navier-Stokes equations. Despite Boltzman...

When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow—an observation attributed to the thermostress convection effects at the microscale. The dynamics of the overall thermostress convection process is governed by the Boltzmann equation—an integrodifferen...

The Boltzmann equation may contain uncertainties in initial/boundary data or collision kernel. To study the impact of these uncertainties, a stochastic Galerkin (sG) method was proposed in [18] and studied in the kinetic regime. When the system is close to the fluid regime (the Knudsen number is small), the method would become prohibitively expensi...

When the molecules of a gaseous system are far apart, say in microscale gas flows where the surface to volume ratio is high and hence the surface forces dominant, the molecule-surface interactions lead to the formation of a local thermodynamically non-equilibrium region extending few mean free paths from the surface. The dynamics of such systems is...

When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow---an observation attributed to the thermo-stress convection effects at microscale. The dynamics of the overall thermo-stress convection process is governed by the Boltzmann equation---an integro-diffe...

We introduce a fast Fourier spectral method for the multi-species Boltzmann collision operator. The method retains the advantages of the single-species fast spectral method Gamba et al. (2017) including: (a) spectral accuracy, (b) reduced computational complexity compared to direct spectral method, (c) reduced memory requirement in the precomputati...

When the molecules of a gaseous system are far apart, say in microscale gas flows where the surface to volume ratio is high and hence the surface forces dominant, the molecule-surface interactions lead to the formation of a local thermodynamically non-equilibrium region extending few mean free paths from the surface. The dynamics of such systems is...

We introduce a fast Fourier spectral method for the multi-species Boltzmann collision operator. The method retains the riveting properties of the single-species fast spectral method (Gamba et al. SIAM J. Sci. Comput., 39 pp. B658--B674 2017) including: (a) spectral accuracy, (b) reduced computational complexity compared to direct spectral method, (...

In this paper, we propose a simple fast Fourier spectral method for the inelastic Boltzmann collision operator, with its application to one of the widely used models of granular gases, the inelastic Boltzmann equation with a heating source. Compared to the direct Fourier spectral method, our fast algorithm reduces the computational complexity from...

The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions, including compressible, turbulent, as well as flows involving further physics such as non-equilibrium internal energy exchange and chemical...

The Boltzmann equation, an integro-differential equation for the molecular distribution function in the physical and velocity phase space, governs the fluid flow behavior at a wide range of physical conditions, including compressible, turbulent, as well as flows involving further physics such as non-equilibrium internal energy exchange and chemical...

We introduce a second-order time discretization method for stiff kinetic equations. The method is asymptotic-preserving (AP) -- can capture the Euler limit without numerically resolving the small Knudsen number; and positivity-preserving -- can preserve the non-negativity of the solution which is a probability density function for arbitrary Knudsen...

We propose a generalized polynomial chaos-based stochastic Galerkin method (gPC-sG) for the Fokker–Planck–Landau (FPL) equation with random uncertainties. The method can handle uncertainties from initial or boundary data and the neutralizing background. By a gPC expansion and the Galerkin projection, we convert the FPL equation with uncertainty int...

Implicit–explicit (IMEX) Runge–Kutta (RK) schemes are popular high order time discretization methods for solving stiff kinetic equations. As opposed to the compressible Euler limit (leading order asymptotics of the Boltzmann equation as the Knudsen number ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...

We develop a family of second-order implicit-explicit (IMEX) schemes for the stiff BGK kinetic equation. The method is asymptotic-preserving (can capture the Euler limit without numerically resolving the small Knudsen number) as well as positivity-preserving --- a feature that is not possessed by any of the existing second or high order IMEX scheme...

We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensio...

Kinetic equations contain uncertainties in their collision kernels or scattering coefficients, initial or boundary data, forcing terms, geometry, etc. Quantifying the uncertainties in kinetic models have important engineering and industrial applications. In this article we survey recent efforts in the study of kinetic equations with random inputs,...

Hyperbolic and kinetic equations often possess small spatial and temporal scales that lead to various asymptotic limits. Numerical approximation of these equations is challenging due to the presence of stiff source, collision, forcing terms, or when different scales coexist. Asymptotic-preserving (AP) schemes are numerical methods that are efficien...

We propose a simple fast spectral method for the Boltzmann collision operator with general collision kernels. In contrast to the direct spectral method \cite{PR00, GT09} which requires $O(N^6)$ memory to store precomputed weights and has $O(N^6)$ numerical complexity, the new method has complexity $O(MN^4\log N)$, where $N$ is the number of discret...

Attenuation of seismic waves needs to be taken into account to improve the accuracy of seismic imaging. In viscoacoustic media, reverse time migration (RTM) can be performed with Q-compensation, which is also known as Q-RTM. Least-squares RTM (LSRTM) has also been shown to be able to compensate for attenuation through linearized inversion. However,...

We develop a stochastic Galerkin method for the Boltzmann equation with uncertainty. The method is based on the generalized polynomial chaos (gPC) approximation in the stochastic Galerkin framework, and can handle random inputs from collision kernel, initial data or boundary data. We show that a simple singular value decomposition of gPC related co...

We present a new asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions - a leading-order elastic collision together with a lower-order interparticle collision. When the mean free path is small, numerically solving this equation is prohibitively expensive due to the stiff collision terms. Furthermore, since...

We develop a class of stochastic numerical schemes for Hamilton-Jacobi equations with random inputs in initial data and/or the Hamiltonians. Since the gradient of the Hamilton- Jacobi equations gives a symmetric hyperbolic system, we utilize the generalized polynomial chaos (gPC) expansion with stochastic Galerkin procedure in random space and the...

We present a conservative spectral method for the fully nonlinear Boltzmann collision operator based on the weighted convolution structure in Fourier space developed by Gamba and Tharkabhushnanam. This method can simulate a broad class of collisions, including both elastic and inelastic collisions as well as angularly dependent cross sections in w...

The conventional velocity scan can be computationally expensive for large-scale seismic data sets, particularly when the presence of anisotropy requires multiparameter scanning. We introduce a fast algorithm for 3D azimuthally anisotropic velocity scan by generalizing the previously proposed 2D butterfly algorithm for hyperbolic Radon transforms. T...

We used a novel iterative estimation scheme for separa- tion of blended seismic data from simultaneous sources. The scheme is based on an augmented estimation problem that can be solved by iteratively constraining the deblended data using shaping regularization in the seislet domain. We for- mulated the forward modeling operator in the common- rece...

We evaluated a new spectral method and a new finite-difference (FD) method for seismic-wave extrapolation in time. Using staggered temporal and spatial grids, we derived a wave-extrapolation operator using a lowrank decomposition for a first-order system of wave equations and designed the corresponding FD scheme. The proposed methods extend previou...

This paper introduces a fast algorithm for the energy space boson Boltzmann collision operator. Compared to the direct O(N 3) calculation and the previous O(N 2 log N) method [8], the new algorithm runs in complexity O(N log 2 N), which is optimal up to a logarithmic factor (N is the number of gird points in energy space). The basic idea is to part...

We study the quasi-random choice method (QRCM) for the Liouville equation of geometrical optics with discontinuous local wave speed. This equation arises in the phase space computation of high frequency waves through interfaces, where waves undergo partial transmissions and reflections. The numerical challenges include interface, contact discontinu...

In this paper we develop high-order asymptotic-preserving methods for the spatially inhomogeneous quantum Boltzmann equation. We follow the work in Li and Pareschi, where asymptotic preserving exponential Runge-Kutta methods for the classical inhomogeneous Boltzmann equation were constructed. A major difficulty here is related to the non Gaussian s...

We introduce a new iterative estimation scheme for separation of blended seismic data from simultaneous sources. We first construct an augmented estimation problem, then use shaping regularization to constrain the model when iteratively solving the problem. Both model and data in the estimation problem correspond to different seismic sources. In th...

We design an asymptotic-preserving scheme for the semiconductor Boltzmann equation which leads to an energy-transport system for electron mass and internal energy as mean free path goes to zero. To overcome the stiffness induced by the convection terms, we adopt an even-odd decomposition to formulate the equation into a diffusive relaxation system....

Generalized Radon transforms, such as the hyperbolic Radon transform, cannot be implemented as efficiently in the frequency domain as convolutions, thus limiting their use in seismic data processing. We have devised a fast butterfly algorithm for the hyperbolic Radon transform. The basic idea is to reformulate the transform as an oscillatory integr...

We construct an efficient numerical scheme for the quantum Fokker-Planck-Landau (FPL) equation that works uniformly from kinetic to fluid regimes. Such a scheme in-evitably needs an implicit discretization of the nonlinear collision operator, which is difficult to invert. Inspired by work [9] we seek a linear operator to penalize the quantum FPL co...

We introduce a fast butterfly algorithm for the hyperbolic Radon transform commonly used in seismic data processing. For two-dimensional data, the algorithm runs in complexity O(N[superscript 2] logN), where N is representative of the number of points in either dimension of data space or model space. Using a series of examples, we show that the pro...

This paper introduces a fast spectral algorithm for the quantum Boltzmann collision operator. In the usual spectral framework, one of the terms in the operator cannot be evaluated efficiently. The new approach is based on the fundamental property of the exponential function which allows one to construct a new decomposition of the collision kernel t...

Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision terms. A class of asymptotic preserving schemes was introduced in [6] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own...

The kinetic flux vector splitting (KFVS) scheme, when used for quantum Euler equa-tions, as was done by Yang et al [22], requires the integration of the quantum Maxwellians (Bose-Einstein and Fermi-Dirac distributions), giving a numerical flux much more compli-cated than the classical counterpart. As a result, a nonlinear 2 by 2 system that connect...

Numerically solving the Boltzmann kinetic equations with the small Knudsen number is challenging due to the stiff nonlinear collision term. A class of asymptotic preserving schemes was introduced in [6] to handle this kind of problems. The idea is to penalize the stiff collision term by a BGK type operator. This method, however, encounters its own...