arXiv subjects

Jiuhua Hu

Publications and source records attributed to Jiuhua Hu.

10 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.

math.NA↗

arXiv2026

Power laws in the sea ice floe size distribution: a stochastic theory

Sea ice is a complex system, and observations have shown that ice segments (i.e., floes) have a wide range of sizes, with a floe size distribution that follows a power law. However, a theory for the power law and its exponent have remained elusive. Here, floe-resolving numerical simulations are investigated with a discrete element model, in order to gain further information by gathering statistics of fracture and welding events. Then, based on the insights from the floe-resolving simulations, a stochastic fragmentation-coagulation theory is proposed. Exact solutions are found with a power law. The power-law exponent can take a variety of values, and it depends on the fracture and welding rates. Such behavior is reminiscent of seasonal changes in the power-law exponent, which have been reported in past analyses of observational data.

physics.ao-ph↗

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.

math.NA↗

arXiv2024

LEMDA: A Lagrangian-Eulerian Multiscale Data Assimilation Framework

Lagrangian trajectories are widely used as observations for recovering the underlying flow field via Lagrangian data assimilation (DA). However, the strong nonlinearity in the observational process and the high dimensionality of the problems often cause challenges in applying standard Lagrangian DA. In this paper, a Lagrangian-Eulerian multiscale DA (LEMDA) framework is developed. It starts with exploiting the Boltzmann kinetic description of the particle dynamics to derive a set of continuum equations, which characterize the statistical quantities of particle motions at fixed grids and serve as Eulerian observations. Despite the nonlinearity in the continuum equations and the processes of Lagrangian observations, the time evolutions of the posterior distribution from LEMDA can be written down using closed analytic formulae. This offers an exact and efficient way of carrying out DA, which avoids using ensemble approximations and the associated tunings. The analytically solvable properties also facilitate the derivation of an effective reduced-order Lagrangian DA scheme that further enhances computational efficiency. The Lagrangian DA within the framework has advantages when a moderate number of particles is used, while the Eulerian DA can effectively save computational costs when the number of particle observations becomes large. The Eulerian DA is also valuable when particles collide, such as using sea ice floe trajectories as observations. LEMDA naturally applies to multiscale turbulent flow fields, where the Eulerian DA recovers the large-scale structures, and the Lagrangian DA efficiently resolves the small-scale features in each grid cell via parallel computing. Numerical experiments demonstrate the skilful results of LEMDA and its two components.

physics.flu-dyn↗

arXiv2021

Partially Explicit Time Discretization for Time Fractional Diffusion Equation

Time fractional PDEs have been used in many applications for modeling and simulations. Many of these applications are multiscale and contain high contrast variations in the media properties. It requires very small time step size to perform detailed computations. On the other hand, in the presence of small spatial grids, very small time step size is required for explicit methods. Explicit methods have many advantages as we discuss in the paper. In this paper, we propose a partial explicit method for time fractional PDEs. The approach solves the forward problem on a coarse computational grid, which is much larger than spatial heterogeneities, and requires only a few degrees of freedom to be treated implicitly. Via the construction of appropriate spaces and careful stability analysis, we can show that the time step can be chosen not to depend on the contrast or scale as the coarse mesh size. Thus, one can use larger time step size in an explicit approach. We present stability theory for our proposed method and our numerical results confirm the stability findings and demonstrate the performance of the approach.

math.NA↗

arXiv2021

Space-time Non-local multi-continua upscaling for parabolic equations with moving channelized media

In this paper, we consider a parabolic problem with time-dependent heterogeneous coefficients. Many applied problems have coupled space and time heterogeneities. Their homogenization or upscaling requires cell problems that are formulated in space-time representative volumes for problems with scale separation. In problems without scale separation, local problems include multiple macroscopic variables and oversampled local problems, where these macroscopic parameters are computed. These approaches, called Non-local multi-continua, are proposed for problems with complex spatial heterogeneities in a number of previous papers. In this paper, we extend this approach for space-time heterogeneities, by identifying macroscopic parameters in space-time regions. Our proposed method space-time Non-local multi-continua (space-time NLMC) is an efficient numerical solver to deal with time-dependent heterogeneous coefficients. It provides a flexible and systematic way to construct multiscale basis functions to approximate the solution. These multiscale basis functions are constructed by solving a local energy minimization problems in the oversampled space-time regions such that these multiscale basis functions decay exponentially outside the oversampled domain. Unlike the classical time-stepping methods combined with full-discretization technique, our space-time NLMC efficiently constructs the multiscale basis functions in a space-time domain and can provide a computational savings compared to space-only approaches as we discuss in the paper. We present two numerical experiments, which show that the proposed approach can provide a good accuracy.

math.NA↗

arXiv2020

Convergence of the CEM-GMsFEM for Stokes flows in heterogeneous perforated domains

In this paper, we consider the incompressible Stokes flow problem in a perforated domain and employ the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) to solve this problem. The proposed method provides a flexible and systematical approach to construct crucial divergence-free multiscale basis functions for approximating the displacement field. These basis functions are constructed by solving a class of local energy minimization problems over the eigenspaces that contain local information on the heterogeneities. These multiscale basis functions are shown to have the property of exponential decay outside the corresponding local oversampling regions. By adapting the technique of oversampling, the spectral convergence of the method with error bounds related to the coarse mesh size is proved.

math.NA↗

arXiv2020

Wavelet-based Edge Multiscale Parareal Algorithm for Parabolic Equations with Heterogeneous Coefficients

We propose in this paper the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm to efficiently solve parabolic equations with heterogeneous coefficients. This algorithm combines the advantages of multiscale methods that can deal with heterogeneity in the spacial domain effectively, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. We derive the convergence rate of this algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step and the fine-scale time step. Several numerical tests are presented to demonstrate the performance of our algorithm, which verify our theoretical results perfectly.

math.NA↗

arXiv2019

Homogenization of time-fractional diffusion equations with periodic coefficients

We consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data $a(x)\in L^{2}(D)$ in a bounded domain $D\subset \mathbb{R}^d$ with a sufficiently smooth boundary. We analyze the homogenized solution under the assumption that the diffusion coefficient $κ^ε(x)$ is smooth and periodic with the period $ε>0$ being sufficiently small. We derive that its first order approximation has a convergence rate of $\mathcal{O}(ε^{1/2})$ when the dimension $d\leq 2$ and $\mathcal{O}(ε^{1/6})$ when $d=3$. Several numerical tests are presented to show the performance of the first order approximation.

math.NA↗

arXiv2016

Total domination polynomials of graphs

Given a graph $G$, a total dominating set $D_t$ is a vertex set that every vertex of $G$ is adjacent to some vertices of $D_t$ and let $d_t(G,i)$ be the number of all total dominating sets with size $i$. The total domination polynomial, defined as $D_t(G,x)=\sum\limits_{i=1}^{| V(G)|} d_t(G,i)x^i$, recently has been one of the considerable extended research in the field of domination theory. In this paper, we obtain the vertex-reduction and edge-reduction formulas of total domination polynomials. As consequences, we give the total domination polynomials for paths and cycles. Additionally, we determine the sharp upper bounds of total domination polynomials for trees and characterize the corresponding graphs attaining such bounds. Finally, we use the reduction-formulas to investigate the relations between vertex sets and total domination polynomials in $G$.

math.CO↗