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Yingda Cheng

Publications and source records attributed to Yingda Cheng.

At least 19 recordsLinked to original sources
arXiv2026

Gregory Nested Picard Iteration Schemes for Open Quantum Systems Governed by the Lindblad Equation

Numerical simulation of quantum computing hardware and open quantum systems governed by the Lindblad equation is challenging due to the high dimensionality of the density matrix and the need to preserve fundamental physical properties. In our previous work, we developed an arbitrary-order, low-rank, completely positive and trace preserving (CPTP) method for the Lindblad equation with time-dependent Hamiltonians by nested Picard iteration (NPI). In this work, we develop Gregory NPI schemes, which are CPTP schemes constructed by Gregory-type quadrature on equispaced nodes. The methods, which are of order up to nine, substantially reduce the computational cost compared to our previously proposed NPI schemes with Gaussian quadrature rules, while retaining high-order accuracy and structure preservation. We analyze the stability of the resulting scheme for a physics-based test equation. Numerical experiments verify the convergence of the method and demonstrate the effectiveness of the low-rank approximation. We study the performance of a previously constructed CNOT gate for both closed and open quantum systems.

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arXiv2026

A fast scheme for the homogeneous Boltzmann equation based on lifting and tensor train approximation

We propose a fast deterministic scheme for the space-homogeneous Boltzmann equation that exploits the low-rank structure of the velocity distribution. This paper consists of two independent contributions. The first is a \emph{lifting-projection (LP) scheme}, inspired by the approach in the recent theoretical breakthroughs \cite{guillen2025landau, imbert2026monotonicity, guillen2025landau2} on the well-posedness of the Landau and Boltzmann equations. In particular, the approach lifts the nonlinear 3D Boltzmann equation to the 6D linear Kac master equation, advanced over a single time step, and projected back to its marginal in 3D. The second contribution is a \emph{low-rank tensor method} for evaluating the collision operator, in which the lifted solution is represented in tensor train (TT) format and computed via a TT cross approximation algorithm with interpolation, complemented by a TT-friendly conservation correction that enforces conservation of mass, momentum, and energy. When the solution is low-rank in velocity, the method scales linearly in $n$ when cubic interpolation is used (and quadratic in $n$ when spectral interpolation is used), where $n$ is the number of grid points in each velocity direction. Therefore, our methods offer significant computational savings over existing deterministic solvers in such cases. Numerical experiments on 2D and 3D benchmarks, including the BKW exact solution and anisotropic initial data, confirm the computational scaling, the expected order of accuracy and verify the effectiveness of the conservation correction.

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arXiv2026

Completely Positive and Trace Preserving Schemes with Tensor Train Compression for the Lindblad Equation

We propose a family of low-rank, completely positive and trace preserving schemes for the Lindblad equation, a common model for open quantum systems. Low-rank representation is employed at two levels: the density matrix is factorized into the product of tall-skinny matrices, and the columns of these matrices are further represented using the tensor train (TT) format, also know as matrix product states (MPS). This two-level low-rank format fits naturally into our existing Kraus is King scheme (arXiv:2409.08898v2 [math.NA]) for the Lindblad equation, whose underlying operations are arithmetic on the columns of the tall-skinny matrices. We show how these operations can be performed efficiently in the TT/MPS format, with particular emphasis on density matrix rank-truncation. We conclude with extensive numerical experiments demonstrating the convergence of this scheme and its efficiency in simulating systems with up to $10^{19}$ degrees of freedom using only modest compute resources.

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arXiv2025

Reduced Basis Methods for Parametric Steady-State Radiative Transfer Equation

The radiative transfer equation (RTE) is a fundamental mathematical model to describe physical phenomena involving the propagation of radiation and its interactions with the host medium. Deterministic methods can produce accurate solutions without any statistical noise, yet often at a price of expensive computational costs originating from the intrinsic high dimensionality of the model. With this work, we present the first systematic investigation of projection-based reduced order models (ROMs) following the reduced basis method (RBM) framework to simulate the parametric steady-state RTE with isotropic scattering and one energy group. Four ROMs are designed, with each defining a nested family of reduced surrogate solvers of different resolution/fidelity. They are based on either a Galerkin or least-squares Petrov-Galerkin projection and utilize either an $L_1$ or residual-based importance/error indicator. Two of the proposed ROMs are certified in the setting when the absorption cross section is positively bounded below uniformly. One technical focus and contribution lie in the proposed implementation strategies under the affine assumption of the parameter dependence of the model. These well-crafted broadly applicable strategies not only ensure the efficiency and accuracy of the offline training stage and the online prediction of reduced surrogate solvers, they also take into account the conditioning of the reduced systems as well as the stagnation-free residual evaluation for numerical robustness. Computational complexities are derived for both the offline training and online prediction stages of the proposed model order reduction strategies, and they are demonstrated numerically along with the accuracy and robustness of the reduced surrogate solvers. Numerically we observe four to six orders of magnitude speedup of our ROMs compared to full order models for some 2D2v examples.

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arXiv2025

Arbitrary High Order Low-rank Completely Positive and Trace Preserving (CPTP) Schemes for Lindblad Equations with Time-dependent Hamiltonian

In this paper, we develop a framework for designing arbitrary high order low-rank schemes for the Lindblad equation with time-dependent Hamiltonians. Our approach is based on nested Picard iterative integrators (NPI) and results in schemes in Kraus form that are completely positive and trace preserving (CPTP). The schemes are amenable to low rank formulations, making them suitable for problems where the matrix rank of the density matrix is small.

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arXiv2025

A new cross approximation for Tucker tensors and its application in Tucker-Anderson Acceleration

This paper proposes two new algorithms related to the Tucker tensor format. The first method is a new cross approximation for Tucker tensors, which we call Cross$^2$-DEIM. Cross$^2$-DEIM is an iterative method that uses a fiber sampling strategy, sampling $O(r)$ fibers in each mode, where $r$ denotes the target rank. The fibers are selected based on the discrete empirical interpolation method (DEIM). Cross$^2$-DEIM resemblances the Fiber Sampling Tucker Decomposition (FSTD)2 approximation, and has favorable computational scaling compared to existing methods in the literature. We demonstrate good performance of Cross$^2$-DEIM in terms of iteration count and intermediate memory. First we design a fast direct Poisson solver based on Cross$^2$-DEIM and the fast Fourier transform. This solver can be used as a stand alone or as a preconditioner for low-rank solvers for elliptic problems. The second method is a low-rank solver for nonlinear tensor equation in Tucker format by Anderson acceleration (AA), which we call Tucker-AA. Tucker-AA is an extension of low-rank AA (lrAA) proposed in our prior work for low-rank solution to nonlinear matrix equation. We apply Cross$^2$-DEIM with warm-start in Tucker-AA to deal with the nonlinearity in the equation. We apply low-rank operations in AA, and by an appropriate rank truncation strategy, we are able to control the intermediate rank growth. We demonstrated the performance for Tucker-AA for approximate solutions nonlinear PDEs in 3D.

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arXiv2025

lrAA: Low-Rank Anderson Acceleration

This paper proposes a new framework for computing low-rank solutions to nonlinear matrix equations arising from spatial discretization of nonlinear partial differential equations: low-rank Anderson acceleration (lrAA). lrAA is an adaptation of Anderson acceleration (AA), a well-known approach for solving nonlinear fixed point problems, to the low-rank format. In particular, lrAA carries out all linear and nonlinear operations in low-rank form with rank truncation using an adaptive truncation tolerance. We propose a simple scheduling strategy to update the truncation tolerance throughout the iteration according to a residual indicator. This controls the intermediate rank and iteration number effectively. To perform rank truncation for nonlinear functions, we propose a new cross approximation, which we call Cross-DEIM, with adaptive error control that is based on the discrete empirical interpolation method (DEIM). Cross-DEIM employs an iterative update between the approximate singular value decomposition (SVD) and cross approximation. It naturally incorporates a warm-start strategy for each lrAA iterate. We demonstrate the superior performance of lrAA applied to a range of linear and nonlinear problems, including those arising from finite difference discretizations of Laplace's equation, the Bratu problem, the elliptic Monge-Amp\'ere equation and the Allen-Cahn equation.

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arXiv2024

High-Order Implicit Low-Rank Method with Spectral Deferred Correction for Matrix Differential Equations

In this paper, we develop a low-rank method with high-order temporal accuracy using spectral deferred correction (SDC) to compute linear matrix differential equations. In [1], a low rank numerical method is proposed to correct the modeling error of the basis update and the Galerkin (BUG) method, which is a computational approach for DLRA. This method (merge-BUG/mBUG method) has been demonstrated to be first order convergent for general advection-diffusion problems. In this paper, we explore using SDC to elevate the convergence order of the mBUG method. In SDC, we start by computing a first-order solution by mBUG, and then perform successive updates by computing low-rank solutions to the Picard integral equation. Rather than a straightforward application of SDC with mBUG, we propose two aspects to improve computational efficiency. The first is to reduce the intermediate numerical rank by detailed analysis of dependence of truncation parameter on the correction levels. The second aspect is a careful choice of subspaces in the successive correction to avoid inverting large linear systems (from the K- and L-steps in BUG). We prove that the resulting scheme is high-order accurate for the Lipschitz continuous and bounded dynamical system. We consider numerical rank control in our framework by comparing two low-rank truncation strategies: the hard truncation strategy by truncated singular value decomposition and the soft truncation strategy by soft thresholding. We demonstrate numerically that soft thresholding offers better rank control in particular for higher-order schemes for weakly (or non-)dissipative problems.

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arXiv2024

Preconditioning Low Rank Generalized Minimal Residual Method (GMRES) for Implicit Discretizations of Matrix Differential Equations

This work proposes a new class of preconditioners for the low rank Generalized Minimal Residual Method (GMRES) for multiterm matrix equations arising from implicit timestepping of linear matrix differential equations. We are interested in computing low rank solutions to matrix equations, e.g. arising from spatial discretization of stiff partial differential equations (PDEs). The low rank GMRES method is a particular class of Krylov subspace method where the iteration is performed on the low rank factors of the solution. Such methods can exploit the low rank property of the solution to save on computational and storage cost. Of critical importance for the efficiency and applicability of the low rank GMRES method is the availability of an effective low rank preconditioner that operates directly on the low rank factors of the solution and that can limit the iteration count and the maximal Krylov rank. The preconditioner we propose here is based on the basis update and Galerkin (BUG) method, resulting from the dynamic low rank approximation. It is a nonlinear preconditioner for the low rank GMRES scheme that naturally operates on the low rank factors. Extensive numerical tests show that this new preconditioner is highly efficient in limiting iteration count and maximal Krylov rank. We show that the preconditioner performs well for general diffusion equations including highly challenging problems, e.g. high contrast, anisotropic equations. Further, it compares favorably with the state of the art exponential sum preconditioner. We also propose a hybrid BUG - exponential sum preconditioner based on alternating between the two preconditioners.

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arXiv2024

Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations

In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac-Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using a base implicit solve for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. We prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.

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arXiv2022

A micro-macro decomposed reduced basis method for the time-dependent radiative transfer equation

Kinetic transport equations are notoriously difficult to simulate because of their complex multiscale behaviors and the need to numerically resolve a high dimensional probability density function. Past literature has focused on building reduced order models (ROM) by analytical methods. In recent years, there is a surge of interest in developing ROM using data-driven or computational tools that offer more applicability and flexibility. This paper is a work towards that direction. Motivated by our previous work of designing ROM for the stationary radiative transfer equation in [30] by leveraging the low-rank structure of the solution manifold induced by the angular variable, we here further advance the methodology to the time-dependent model. Particularly, we take the celebrated reduced basis method (RBM) approach and propose a novel micro-macro decomposed reduced basis method (MMD-RBM). The MMD-RBM is constructed by exploiting, in a greedy fashion, the low-rank structures of both the micro- and macro-solution manifolds with respect to the angular and temporal variables. Our reduced order surrogate consists of: reduced bases for reduced order subspaces and a reduced quadrature rule in the angular space. The proposed MMD-RBM features several structure-preserving components: 1) an equilibrium-respecting strategy to construct reduced order subspaces which better utilize the structure of the decomposed system, and 2) a recipe for preserving positivity of the quadrature weights thus to maintain the stability of the underlying reduced solver. The resulting ROM can be used to achieve a fast online solve for the angular flux in angular directions outside the training set and for arbitrary order moment of the angular flux.

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arXiv2022

Adaptive sparse grid discontinuous Galerkin method: review and software implementation

This paper reviews the adaptive sparse grid discontinuous Galerkin (aSG-DG) method for computing high dimensional partial differential equations (PDEs) and its software implementation. The C\texttt{++} software package called AdaM-DG, implementing the aSG-DG method, is available on Github at \url{https://github.com/JuntaoHuang/adaptive-multiresolution-DG}. The package is capable of treating a large class of high dimensional linear and nonlinear PDEs. We review the essential components of the algorithm and the functionality of the software, including the multiwavelets used, assembling of bilinear operators, fast matrix-vector product for data with hierarchical structures. We further demonstrate the performance of the package by reporting numerical error and CPU cost for several benchmark test, including linear transport equations, wave equations and Hamilton-Jacobi equations.

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arXiv2022

Superconvergence and accuracy enhancement of discontinuous Galerkin solutions for Vlasov-Maxwell equations

This paper considers the discontinuous Galerkin (DG) methods for solving the Vlasov-Maxwell (VM) system, a fundamental model for collisionless magnetized plasma. The DG methods provide accurate numerical description with conservation and stability properties. However, to resolve the high dimensional probability distribution function, the computational cost is the main bottleneck even for modern-day supercomputers. This work studies the applicability of a post-processing technique to the DG solution to enhance its accuracy and resolution for the VM system. In particular, we prove the superconvergence of order $(2k+\frac{1}{2})$ in the negative order norm for the probability distribution function and the electromagnetic fields when piecewise polynomial degree $k$ is used. Numerical tests including Landau damping, two-stream instability and streaming Weibel instabilities are considered showing the performance of the post-processor.

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arXiv2021

Machine learning moment closure models for the radiative transfer equation III: enforcing hyperbolicity and physical characteristic speeds

This is the third paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work \cite{huang2021gradient}, we proposed an approach to learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional $P_N$ closure. However, while the ML moment closure has better accuracy, it is not able to guarantee hyperbolicity and has issues with long time stability. In our second paper \cite{huang2021hyperbolic}, we identified a symmetrizer which leads to conditions that enforce that the gradient based ML closure is symmetrizable hyperbolic and stable over long time. The limitation of this approach is that in practice the highest moment can only be related to four, or fewer, lower moments. In this paper, we propose a new method to enforce the hyperbolicity of the ML closure model. Motivated by the observation that the coefficient matrix of the closure system is a lower Hessenberg matrix, we relate its eigenvalues to the roots of an associated polynomial. We design two new neural network architectures based on this relation. The ML closure model resulting from the first neural network is weakly hyperbolic and guarantees the physical characteristic speeds, i.e., the eigenvalues are bounded by the speed of light. The second model is strictly hyperbolic and does not guarantee the boundedness of the eigenvalues. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, stability and generalizability of our hyperbolic ML closure model.

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arXiv2021

Machine learning moment closure models for the radiative transfer equation II: enforcing global hyperbolicity in gradient based closures

This is the second paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work \cite{huang2021gradient}, we proposed an approach to directly learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional $P_N$ closure. However, the ML moment closure model in \cite{huang2021gradient} is not able to guarantee hyperbolicity and long time stability. We propose in this paper a method to enforce the global hyperbolicity of the ML closure model. The main idea is to seek a symmetrizer (a symmetric positive definite matrix) for the closure system, and derive constraints such that the system is globally symmetrizable hyperbolic. It is shown that the new ML closure system inherits the dissipativeness of the RTE and preserves the correct diffusion limit as the Knunsden number goes to zero. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, long time stability and generalizability of our globally hyperbolic ML closure model.

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arXiv2021

Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure

In this paper, we take a data-driven approach and apply machine learning to the moment closure problem for radiative transfer equation in slab geometry. Instead of learning the unclosed high order moment, we propose to directly learn the gradient of the high order moment using neural networks. This new approach is consistent with the exact closure we derive for the free streaming limit and also provides a natural output normalization. A variety of benchmark tests, including the variable scattering problem, the Gaussian source problem with both periodic and reflecting boundaries, and the two-material problem, show both good accuracy and generalizability of our machine learning closure model.

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arXiv2021

A class of adaptive multiresolution ultra-weak discontinuous Galerkin methods for some nonlinear dispersive wave equations

In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) ultra-weak discontinuous Galerkin (UWDG) methods for solving some nonlinear dispersive wave equations including the Korteweg-de Vries (KdV) equation and its two dimensional generalization, the Zakharov-Kuznetsov (ZK) equation. The UWDG formulation, which relies on repeated integration by parts, was proposed for KdV equation in \cite{cheng2008discontinuous}. For the ZK equation which contains mixed derivative terms, we develop a new UWDG formulation. The $L^2$ stability and the optimal error estimate with a novel local projection are established for this new scheme on regular meshes. Adaptivity is achieved based on multiresolution and is particularly effective for capturing solitary wave structures. Various numerical examples are presented to demonstrate the accuracy and capability of our methods.

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