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Zhichao Peng

Publications and source records attributed to Zhichao Peng.

At least 19 recordsLinked to original sources
arXiv2026

Highly Efficient Rank-Adaptive Sweep-based SI-DSA for the Radiative Transfer Equation via Mild Space Augmentation

Low-rank methods have emerged as a promising strategy for reducing the memory footprint and computational cost of discrete-ordinates discretizations of the radiative transfer equation (RTE). However, most existing rank-adaptive approaches rely on rank-proportional space augmentation, which can negate efficiency gains when the effective solution rank becomes moderately large. To overcome this limitation, we develop a rank-adaptive sweep-based source iteration with diffusion synthetic acceleration (SI-DSA) for the first-order steady-state RTE. The core of our method is a sweep-based low-rank SI with an inner-loop iteration that performs efficient rank adaptation via mild space augmentation. In each inner iteration, the spatial basis is augmented with a small, rank-independent number of basis vectors without truncation, while a single truncation is performed only after the inner loop converges. Efficient rank adaptation is achieved through residual-based greedy angular subsampling strategy together with incremental updates of projection operators, enabling non-intrusive reuse of existing transport-sweep implementations. In the outer iteration, a DSA preconditioner is applied to accelerate convergence. Numerical experiments show that the proposed solver achieves accuracy and iteration counts comparable to those of full-rank SI-DSA while substantially reducing memory usage and runtime, even for challenging multiscale problems in which the effective rank reaches 30-45% of the full rank.

math.NA↗
arXiv2026

Numerical Study of Eigenvector Deflation to Accelerate the WaveHoltz Method

We present a numerical study of eigenvector deflation as a means of accelerating the WaveHoltz method for solving the Helmholtz equation. For energy-conserving (Dirichlet or Neumann) boundary conditions the WaveHoltz fixed-point iteration converges slowly at high frequency, requiring approximately $\mathcal{O}(ω^{2d})$ iterations in $d$ dimensions. We show that deflating the eigenvectors whose eigenvalues lie nearest the driving frequency substantially reduces iteration counts, and we examine two ways of incorporating the eigenvectors: direct eigenvector deflation (DEVD), in which the forcing and iterate are projected against the deflation set, and augmented-Krylov eigenvector deflation (AUKED) using deflated conjugate gradient (DCG), augmented GMRES (AGMRES), and augmented (recycled) BICGSTAB (ABICGSTAB). The required eigenpairs can be computed efficiently with the EigenWave approach, and we demonstrate, in two dimensions, that when the number of deflation vectors grows quadratically with $ω$ the asymptotic convergence rate remains essentially constant. Because the eigenvectors on structured grids are naturally represented as matrices, we further apply SVD-based compression to reduce their storage. Numerical experiments on single curvilinear grids discretized with summation-by-parts operators, and on overset grids illustrate the robustness and efficiency of the approach, with the deflated solver breaking even against the undeflated solver after as few as two right-hand sides, when accounting for the cost of precomputing the eigenvectors.

math.NA↗
arXiv2026

Incremental Tensor-Train Compression from Streaming TT-Formatted Data: Applications to Reduced-Order Modeling

High-dimensional tensor data streams arise naturally in scientific and engineering applications, such as simulations of kinetic equations and quantum systems, where samples become available sequentially and are often already represented in compressed low-rank tensor formats. Existing streaming tensor-train (TT) algorithms typically construct or update representations from dense tensor data or randomized sketches. However, when high-dimensional data are generated directly in TT or related low-rank formats, reconstructing dense tensors solely for the purpose of compression is unnecessary and computationally prohibitive. We develop a deterministic incremental TT compression algorithm that operates directly on streaming TT-formatted data. Given a new TT tensor, the proposed method updates an accumulated TT representation through core-wise projection, residual orthogonalization, and adaptive enrichment, retaining only the complementary information that cannot be represented within a prescribed tolerance. By operating entirely at the level of TT cores, the algorithm avoids reconstructing either the incoming tensor or the accumulated full tensor. We establish approximation error bounds for the proposed incremental approach. Moreover, we show that the accumulated TT representation corresponds to a compressed analogue of standard proper orthogonal decomposition for full-order snapshot data, enabling reduced-order models to be constructed directly from streaming low-rank solution data through operations on TT cores, without first reconstructing full snapshots. Numerical experiments on parametric radiative transfer equations demonstrate that the proposed method achieves comparable reconstruction accuracy with substantially reduced wall time and yields efficient and accurate ROMs directly from compressed low-rank data.

math.NA↗
arXiv2026

On-the-Fly ROM-Based Acceleration of SI-DSA for Implicit Time Marching of the Radiative Transfer Equation

In implicit time marching of the radiative transfer equation (RTE), the resulting linear systems are commonly solved using source iteration with diffusion synthetic acceleration (SI-DSA). Despite its widespread success, the performance of the DSA preconditioner may deteriorate when the RTE cannot be well approximated by its diffusion limit. Moreover, classical SI-DSA does not exploit low-rank structures of the solution manifold across time steps when the solution evolves smoothly. To address these limitations, we develop an on-the-fly reduced-order-model (ROM)-based acceleration for SI-DSA in implicit time marching of the RTE. Instead of relying on a diffusion approximation, the proposed approach constructs ROMs directly from the underlying kinetic formulation while exploiting low-rank structures in the temporal evolution of the solution. The method is fully offline-free and constructs ROMs to enhance both initial guesses and preconditioners on the fly during time marching. To handle streaming solution data, we design efficient and memory-lean ROM construction and adaptive update strategies based on dynamical mode decomposition, incremental low-rank singular value decomposition, and error indicators. Numerical experiments demonstrate that the proposed method consistently accelerates implicit time marching, delivering $1.4\times$ to $2.0\times$ speedup over classical SI-DSA while incurring only marginal overhead for ROM construction and updates.

math.NA↗
arXiv2026

Enhancing Future Prediction of Linear and Nonlinear Reduced-Order Models for Transport-Dominated Problems Using Lagrangian Data

Designing effective reduced-order models (ROMs) for parametrized transport-dominated problems remains challenging because of the well-known Kolmogorov barrier. Autoencoder-based nonlinear ROMs have been developed to improve the compression ability for such systems. However, despite their stronger compression ability, autoencoder-based ROMs constructed in the Eulerian frame may fail to accurately predict future solutions, due to the poor coherence between historical and future solutions in the Eulerian frame. In contrast, we show that representing transport-dominated dynamics in the Lagrangian frame can lead to a significantly faster decay of the Kolmogorov n-width and improve coherence between historical and future solutions. Building on these insights, we develop two non-intrusive ROMs leveraging Lagrangian data: a Lagrangian autoencoder-based ROM and a Lagrangian parametric dynamic mode decomposition. Numerical experiments demonstrate that these Lagrangian ROMs achieve more accurate and stable future predictions than their Eulerian counterparts.

math.NA↗
arXiv2026

Enhancing Logical Expressiveness in Graph Neural Networks via Path-Neighbor Aggregation

Graph neural networks (GNNs) can effectively model structural information of graphs, making them widely used in knowledge graph (KG) reasoning. However, existing studies on the expressive power of GNNs mainly focuses on simple single-relation graphs, and there is still insufficient discussion on the power of GNN to express logical rules in KGs. How to enhance the logical expressive power of GNNs is still a key issue. Motivated by this, we propose Path-Neighbor enhanced GNN (PN-GNN), a method to enhance the logical expressive power of GNN by aggregating node-neighbor embeddings on the reasoning path. First, we analyze the logical expressive power of existing GNN-based methods and point out the shortcomings of the expressive power of these methods. Then, we theoretically investigate the logical expressive power of PN-GNN, showing that it not only has strictly stronger expressive power than C-GNN but also that its $(k+1)$-hop logical expressiveness is strictly superior to that of $k$-hop. Finally, we evaluate the logical expressive power of PN-GNN on six synthetic datasets and two real-world datasets. Both theoretical analysis and extensive experiments confirm that PN-GNN enhances the expressive power of logical rules without compromising generalization, as evidenced by its competitive performance in KG reasoning tasks.

cs.AI↗
arXiv2025

Synthetic Acceleration Preconditioners for Parametric Radiative Transfer Equations based on Trajectory-Aware Reduced Order Models

The parametric radiative transfer equation (RTE) arises in multi-query applications, such as design optimization, inverse problems, and uncertainty quantification, which require solving the RTE multiple times for various parameters. Classical synthetic acceleration (SA) preconditioners are designed based on low-order approximations of a kinetic correction equation, e.g., its diffusion limit in diffusion synthetic acceleration (DSA). Despite their widespread success, these methods rely on empirical physical assumptions and do not leverage low-rank structures across parameters of the parametric problem. To address these limitations, our previous work introduced a reduced-order model (ROM) enhanced preconditioner called ROMSAD, which exploits low-rank structures across parameters and the original kinetic description of the correction equation. While ROMSAD improves overall efficiency compared with DSA, its efficiency reduces after the first iteration, because the construction of the underlying ROM ignores the preconditioner-dependence of the residual trajectory, leading to a mismatch between the offline and online residual trajectories. To overcome this issue, we introduce a trajectory-aware framework that iteratively constructs ROMs to eliminate the mismatch between offline and online residual trajectories. Numerical tests demonstrate superior efficiency over DSA, and substantial gains in both efficiency and robustness over ROMSAD. For a parametric lattice problem, trajectory-aware ROM preconditioners achieve rapid convergence within only $2$-$3$ iterations online.

math.NA↗
arXiv2025

An Inexact Low-Rank Source Iteration for Steady-State Radiative Transfer Equation with Diffusion Synthetic Acceleration

We propose an inexact low-rank source iteration with diffusion synthetic acceleration (SI-DSA) for solving the multidimensional steady-state radiative transfer equation (RTE) in the second-order formulation. The angular flux is represented in either a low-rank matrix or hierarchical Tucker tensor (HTT) format, enabling substantial reductions in computational resources. Each SI step is solved using a preconditioned low-rank conjugate gradient (CG) method with a diffusion preconditioner. To further improve efficiency, we introduce an adaptive inexact strategy that dynamically relaxes the inner CG tolerance during early SI iterations. The method exploits the tensor-product structure of the discretized operators to perform all matrix-vector operations in low-rank form. Numerical experiments on 2D2V benchmark problems, including diffusion-dominated, transport-dominated, and multiscale problems, demonstrate that the proposed approach achieves errors on the order of $10^{-4}$ to $10^{-5}$ relative to full-rank reference solutions, while reducing the degrees of freedom by up to two orders of magnitude. In the diffusion-dominated case, the low-rank solver achieves speedups exceeding $90\times$ over its full-rank counterpart and remains competitive in solving challenging transport-dominated and multiscale problems while providing substantial storage savings. To our knowledge, this work provides the first low-rank SI-DSA framework for multidimensional steady-state RTE.

math.NA↗
arXiv2025

Superconvergence Extraction of Upwind Discontinuous Galerkin Method Solving the Radiative Transfer Equation

We theoretically analyze the superconvergence of the upwind discontinuous Galerkin (DG) method for both the steady-state and time-dependent radiative transfer equation (RTE), and apply the Smooth-Increasing Accuracy-Conserving (SIAC) filters to enhance the accuracy order. Direct application of SIAC filters on low-dimensional macroscopic moments, often the quantities of practical interest, can effectively improve the approximation accuracy with marginal computational overhead. Using piecewise $k$-th order polynomials for the approximation and assuming constant cross sections, we prove $(2k+2)$-th order superconvergence for the steady-state problem at Radau points on each element and $(2k+1/2)$-th order superconvergence for the global $L^2$ and negative-order Sobolev norms for the time-dependent problem. Numerical experiments confirm the efficacy of the filtering, demonstrating post-filter convergence orders of $2k+2$ for steady-state and $2k+1$ for time-dependent problems. More significantly, the SIAC filter delivers substantial gains in computational efficiency. For a time-dependent problem, we observed an approximately $2.22 \times$ accuracy improvement and a $19.94 \times$ reduction in computational time. For the steady-state problems, the filter achieved a $4$--$9 \times$ acceleration without any loss of accuracy.

math.NA↗
arXiv2025

Adaptive and hybrid reduced order models to mitigate Kolmogorov barrier in a multiscale kinetic transport equation

In this work, we develop reduced order models (ROMs) to predict solutions to a multiscale kinetic transport equation with a diffusion limit under the parametric setting. When the underlying scattering effect is not sufficiently strong, the system governed by this equation exhibits transport-dominated behavior. Suffering from the Kolmogorov barrier for transport-dominant problems, classical linear ROMs may become inefficient in this regime. To address this issue, we first develop a piecewise linear ROM by introducing a novel goal-oriented adaptive time partitioning strategy. To avoid local over-refinement or under-refinement, we propose an adaptive coarsening and refinement strategy that remains robust with various initial empirical partitions. Additionally, for problems where a local linear approximation is not sufficiently efficient, we further develop a hybrid ROM, which combines autoencoder-based nonlinear ROMs and piecewise linear ROMs. Compared to previous autoencoder-based ROMs, this hybridized method reduces the offline autoencoder's training cost by only applying it to time intervals that are adaptively identified as the most challenging. Numerical experiments demonstrate that our proposed approaches successfully predict full-order solutions at unseen parameter values with both efficiency and accuracy. To the best of our knowledge, this is the first attempt to address the Kolmogorov barrier for multiscale kinetic transport problems with the coexistence of both transport- and diffusion-dominant behaviors.

math.NA↗
arXiv2025

A $P$-Adaptive Hermite Method for Nonlinear Dispersive Maxwell's Equations

In this work, we introduce a novel Hermite method to handle Maxwell's equations for nonlinear dispersive media. The proposed method achieves high-order accuracy and is free of any nonlinear algebraic solver, requiring solving instead small local linear systems for which the dimension is independent of the order. The implementation of order adaptive algorithms is straightforward in this setting, making the resulting p-adaptive Hermite method appealing for the simulations of soliton-like wave propagation.

math.NA↗
arXiv2025

A Flexible GMRES Solver with Reduced Order Model Enhanced Synthetic Acceleration Preconditioenr for Parametric Radiative Transfer Equation

Parametric radiative transfer equation (RTE) occurs in multi-query applications such as uncertainty quantification, inverse problems, and sensitivity analysis, which require solving RTE multiple times for a range of parameters. Consequently, efficient iterative solvers are highly desired. Classical Synthetic Acceleration (SA) preconditioners for RTE build on low order approximations to an ideal kinetic correction equation such as its diffusion limit in Diffusion Synthetic Acceleration (DSA). Their performance depends on the effectiveness of the underlying low order approximation. In addition, they do not leverage low rank structures with respect to the parameters of the parametric problem. To address these issues, we proposed a ROM-enhanced SA strategy, called ROMSAD, under the Source Iteration framework in Peng (2024). In this paper, we further extend the ROMSAD preconditioner to flexible general minimal residual method (FGMRES). The main new advancement is twofold. First, after identifying the ideal kinetic correction equation within the FGMRES framework, we reformulate it into an equivalent form, allowing us to develop an iterative procedure to construct a ROM for this ideal correction equation without directly solving it. Second, we introduce a greedy algorithm to build the underlying ROM for the ROMSAD preconditioner more efficiently. Our numerical examples demonstrate that FGMRES with the ROMSAD preconditioner (FGMRES-ROMSAD) is more efficient than GMRES with the right DSA preconditioner. Furthermore, when the underlying ROM in ROMSAD is not highly accurate, FGMRES-ROMSAD exhibits greater robustness compared to Source Iteration accelerated by ROMSAD.

math.NA↗
arXiv2024

AAROC: Reduced Over-Collocation Method with Adaptive Time Partitioning and Adaptive Enrichment for Parametric Time-Dependent Equations

Nonlinear and nonaffine terms in parametric partial differential equations can potentially lead to a computational cost of a reduced order model (ROM) that is comparable to the cost of the original full order model (FOM). To address this, the Reduced Residual Reduced Over-Collocation method (R2-ROC) is developed as a hyper-reduction method within the framework of the reduced basis method in the collocation setting. R2-ROC greedily selects two sets of reduced collocation points based on the (generalized) empirical interpolation method for both solution snapshots and residuals, thereby avoiding the computational inefficiency. The vanilla R2-ROC method can face instability when applied to parametric fluid dynamic problems. To address this, an adaptive enrichment strategy has been proposed to stabilize the ROC method. However, this strategy can involve in an excessive number of reduced collocation points, thereby negatively impacting online efficiency. To ensure both efficiency and accuracy, we propose an adaptive time partitioning and adaptive enrichment strategy-based ROC method (AAROC). The adaptive time partitioning dynamically captures the low-rank structure, necessitating fewer reduced collocation points being sampled in each time segment. Numerical experiments on the parametric viscous Burgers' equation and lid-driven cavity problems demonstrate the efficiency, enhanced stability, and accuracy of the proposed AAROC method.

math.NA↗
arXiv2024

Reduced Order Model Enhanced Source Iteration with Synthetic Acceleration for Parametric Radiative Transfer Equation

Applications such as uncertainty quantification and optical tomography, require solving the radiative transfer equation (RTE) many times for various parameters. Efficient solvers for RTE are highly desired. Source Iteration with Synthetic Acceleration (SISA) is a popular and successful iterative solver for RTE. Synthetic Acceleration (SA) acts as a preconditioning step to accelerate the convergence of Source Iteration (SI). After each source iteration, classical SA strategies introduce a correction to the macroscopic particle density by solving a low order approximation to a kinetic correction equation. For example, Diffusion Synthetic Acceleration (DSA) uses the diffusion limit. However, these strategies may become less effective when the underlying low order approximations are not accurate enough. Furthermore, they do not exploit low rank structures concerning the parameters of parametric problems. To address these issues, we propose enhancing SISA with data-driven ROMs for the parametric problem and the corresponding kinetic correction equation. First, the ROM for the parametric problem can be utilized to obtain an improved initial guess. Second, the ROM for the kinetic correction equation can be utilized to design a novel SA strategy called ROMSAD. In the early stage, ROMSAD adopts a ROM based approximation, which builds on the kinetic description of the correction equation and leverages low rank structures concerning the parameters. This ROM-based approximation has greater efficiency than DSA in the early stage of SI. In the later stage, ROMSAD automatically switches to DSA to leverage its robustness. Additionally, we propose an approach to construct the ROM for the kinetic correction equation without directly solving it. In a series of of numerical tests, we compare the performance of the proposed methods with SI-DSA and DSA preconditioned GMRES solver.

math.NA↗
arXiv2023

Deterministic and Bayesian Characterization of Quantum Computing Devices

Motivated by the noisy and fluctuating behavior of current quantum computing devices, this paper presents a data-driven characterization approach for estimating transition frequencies and decay times in a Lindbladian dynamical model of a superconducting quantum device. The data includes parity events in the transition frequency between the first and second excited states. A simple but effective mathematical model, based upon averaging solutions of two Lindbladian models, is demonstrated to accurately capture the experimental observations. A deterministic point estimate of the device parameters is first performed to minimize the misfit between data and Lindbladian simulations. These estimates are used to make an informed choice of prior distributions for the subsequent Bayesian inference. An additive Gaussian noise model is developed for the likelihood function, which includes two hyper-parameters to capture the noise structure of the data. The outcome of the Bayesian inference are posterior probability distributions of the transition frequencies, which for example can be utilized to design risk neutral optimal control pulses. The applicability of our approach is demonstrated on experimental data from the Quantum Device and Integration Testbed (QuDIT) at Lawrence Livermore National Laboratory, using a tantalum-based superconducting transmon device.

quant-ph↗
arXiv2023

Mathematical approaches for characterization, control, calibration and validation of a quantum computing device

Quantum computing has received significant amounts of interest from many different research communities over the last few years. Although there are many introductory texts that focus on the algorithmic parts of quantum computing, there is a dearth of publications that describe the modeling, calibration and operation of current quantum computing devices. One aim of this report is to fill that void by providing a case study that walks through the entire procedure from the characterization and optimal control of a qudit device at Lawrence Livermore National Laboratory (LLNL) to the validation of the results. A goal of the report is to provide an introduction for students and researchers, especially computational mathematicians, who are interested in but new to quantum computing. Both experimental and mathematical aspects of this procedure are discussed. We present a description of the LLNL QuDIT testbed, the mathematical models that are used to describe it, and the numerical methods that are used to to design optimal controls. We also present experimental and computational methods that can be used to characterize a quantum device. Finally, an experimental validation of an optimized control pulse is presented, which relies on the accuracy of the characterization and the optimal control methodologies.

quant-ph↗
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.

math.NA↗
arXiv2022

A learning-based projection method for model order reduction of transport problems

The Kolmogorov $n$-width of the solution manifolds of transport-dominated problems can decay slowly. As a result, it can be challenging to design efficient and accurate reduced order models (ROMs) for such problems. To address this issue, we propose a new learning-based projection method to construct nonlinear adaptive ROMs for transport problems. The construction follows the offline-online decomposition. In the offline stage, we train a neural network to construct adaptive reduced basis dependent on time and model parameters. In the online stage, we project the solution to the learned reduced manifold. Inheriting the merits from both deep learning and the projection method, the proposed method is more efficient than the conventional linear projection-based methods, and may reduce the generalization error of a solely learning-based ROM. Unlike some learning-based projection methods, the proposed method does not need to take derivatives of the neural network in the online stage.

math.NA↗