arXiv subjects

Yunzhang Li

Publications and source records attributed to Yunzhang Li.

11 recordsLinked to original sources

arXiv2026

A Fully Discrete Local Discontinuous Galerkin Method for Quasilinear Stochastic Convection-Diffusion-Type Equations

In this paper, we develop and analyze a fully discrete local discontinuous Galerkin (LDG) method with IMEX-Euler time discretization for a class of multi-dimensional quasilinear stochastic convection-diffusion-type equations driven by multiplicative $\mathcal Q$-Wiener noise. The leading diffusion matrix may depend on the solution as well as the spatial and temporal variables, while the lower-order drift and noise coefficients may depend on both the solution and its gradient. Under a suitable stochastic parabolicity condition, we establish unconditional high-moment stability estimates for the fully discrete scheme in the quasilinear setting. In the semilinear setting, where the leading diffusion matrix is independent of the solution but may vary in space and time, we further prove optimal high-moment strong error estimates of order $\mathcal O(h^{r+1})$ in space and $\mathcal O(k^{1/2})$ in time. A pathwise error estimate is then derived by combining the high-moment error bound with a discrete Kolmogorov argument. Numerical experiments are presented to illustrate the stability and convergence properties of the proposed method.

math.NA↗

arXiv2026

A Geometric Inverse Source Problem for Stochastic Parabolic Equations

This paper addresses the geometric inverse problem of simultaneously recovering two unknown deterministic source supports in a stochastic parabolic equation, where one source appears in the drift term and the other in the diffusion term. We establish that partial boundary flux measurements alone uniquely determine both supports. Moreover, we prove the existence of minimizers for a perimeter-regularized objective functional. For smooth interfaces, we derive a Hadamard-type boundary representation of the shape derivative. To the best of our knowledge, this is the first such formula for the simultaneous recovery of a drift-source support and a diffusion-source support in an SPDE setting. The derivative exhibits a genuinely stochastic two-channel structure: the adjoint state governs the sensitivity of the drift-source interface, while the martingale component controls the sensitivity of the diffusion-source interface. Based on this formula, we develop a shape-gradient reconstruction method. Numerical experiments demonstrate its effectiveness and its capacity to distinguish between the two source channels under both full and partial boundary observations.

math.OC↗

arXiv2026

A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations

We propose a deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations. Building on the DBDP method of Huré, Pham, and Warin~\cite{HCPHWX20}, the proposed method reformulates the local backward losses through conditional expectations and trains the resulting regression problems sequentially in time. This conditional-expectation formulation replaces pathwise Brownian fluctuations in the Euler residual by their averaged effect and therefore provides an intrinsic variance-reduction mechanism before loss evaluation. In practice, the conditional expectations are approximated by local multi-path Monte Carlo averages, which leads to smoother training targets and improved numerical stability. Numerical experiments show that DBR performs competitively on standard high-dimensional benchmarks and is more stable than DBDP1 on the challenging unbounded benchmark considered in Example~2. Under an idealized population-loss minimization setting, we provide an error analysis and establish a half-order convergence result under suitable approximation and integrability assumptions. We also discuss an extension to variational inequalities.

math.NA↗

arXiv2026

Fully Discrete High-Order DG Schemes for Waves: Dispersion and Observability

This paper investigates the spectral structure, numerical dispersion, and observability of fully discrete approximations of the one-dimensional wave equation by $P^k$ (local) discontinuous Galerkin methods. Characterizing the coupled space-time numerical dispersion reveals a trapping mechanism that forces the group velocities of both physical and spurious modes to vanish at selected frequencies. We then establish an exponential blow-up of order $\exp(h^{-(1-\varepsilon)})$ for the observability constant under this trapping mechanism. To overcome this divergence for arbitrary $k$, we propose a spectral filtering strategy to restore uniform observability. Theoretical analysis and numerical experiments indicate that higher-order methods may facilitate this recovery by preserving a larger genuine physical frequency band, thereby reducing filtering cost and observation time.

math.OC↗

arXiv2025

Non-Markovian dynamics: the memory-dependent probability density evolution equations

This paper aims to investigate the non-Markovian dynamics. The governing equations are derived for the probability density functions (PDFs) of non-Markovian stochastic responses to Langevin equation excited by combined fractional Gaussian noise (FGN) and Gaussian white noise (GWN). The main difficulty here is that the Langevin equation excited by FGN cannot be augmented by a filter excited by GWN, leading to the inapplicability of Itô stochastic calculus theory. Thus, in the present work, based on the fractional Wick Itô Skorohod integral and rough path theory, a new non-Markovian probability density evolution method is established to derive theoretically the memory-dependent probability density evolution equation (PDEEs) for the PDFs of non-Markovian stochastic responses to Langevin equation excited by combined FGN and GWN, which is a breakthrough to stochastic dynamics. Then, we extend an efficient algorithm, the local discontinuous Galerkin method, to numerically solve the memory-dependent PDEEs. Remarkably, this proposed method attains a higher accuracy compared to the prevalent methods such as finite difference, path integral (PI) and Monte Carlo methods, and boasts a broader applicability than the PI method, which fails to solve the memory-dependent PDEEs. Finally, several numerical examples are illustrated to verify the proposed scheme.

math.PR↗

arXiv2024

Averaging principle for semilinear slow-fast rough partial differential equations

In this paper, we investigate the averaging principle for a class of semilinear slow-fast partial differential equations driven by finite-dimensional rough multiplicative noise. Specifically, the slow component is driven by a general random $γ$-Hölder rough path for some $γ\in (1/3,1/2)$, while the fast component is driven by a Brownian rough path. Using controlled rough path theory and the classical Khasminskii's time discretization scheme, we demonstrate that the slow component converges strongly to the solution of the corresponding averaged equation under the Hölder topology.

math.PR↗

arXiv2024

Local discontinuous Galerkin method for nonlinear BSPDEs of Neumann boundary conditions with deep backward dynamic programming time-marching

This paper aims to present a local discontinuous Galerkin (LDG) method for solving backward stochastic partial differential equations (BSPDEs) with Neumann boundary conditions. We establish the $L^2$-stability and optimal error estimates of the proposed numerical scheme. Two numerical examples are provided to demonstrate the performance of the LDG method, where we incorporate a deep learning algorithm to address the challenge of the curse of dimensionality in backward stochastic differential equations (BSDEs). The results show the effectiveness and accuracy of the LDG method in tackling BSPDEs with Neumann boundary conditions.

math.NA↗

arXiv2024

Fractional Backward Stochastic Partial Differential Equations with Applications to Stochastic Optimal Control of Partially Observed Systems driven by Lévy Processes

In this paper, we study the Cauchy problem for backward stochastic partial differential equations (BSPDEs) involving fractional Laplacian operator. Firstly, by employing the martingale representation theorem and the fractional heat kernel, we construct an explicit form of the solution for fractional BSPDEs with space invariant coefficients, thereby demonstrating the existence and uniqueness of strong solution. Then utilizing the freezing coefficients method as well as the continuation method, we establish Hölder estimates and well-posedness for general fractional BSPDEs with coefficients dependent on space-time variables. As an application, we use the fractional adjoint BSPDEs to investigate stochastic optimal control of the partially observed systems driven by $α$-stable Lévy processes.

math.PR↗

arXiv2024

Particle approximation for a conditional McKean--Vlasov stochastic differential equation

In this paper, we construct a type of interacting particle systems to approximate a class of stochastic different equations whose coefficients depend on the conditional probability distributions of the processes given partial observations. After proving the well-posedness and regularity of the particle systems, we establish a quantitative convergence result for the empirical measures of the particle systems in the Wasserstein space, as the number of particles increases. Moreover, we discuss an Euler--Maruyama scheme of the particle system and validate its strong convergence. A numerical experiment is conducted to illustrate our results.

math.PR↗

arXiv2024

Room-temperature intrinsic ferromagnetism of two-dimensional Na2Cl crystals originated by s- and p-orbitals

Ferromagnetism, as one of the most valuable properties of materials, has attracted sustained and widespread interest in basic and applied research from ancient compasses to modern electronic devices. Traditionally, intrinsic ferromagnetism has been attributed to the permanent magnetic moment induced by partially filled d- or f-orbitals. However, the development of ferromagnetic materials has been limited by this electronic structure convention. Thus, the identification of additional materials that are not constrained by this conventional rule but also exhibit intrinsic ferromagnetism is highly expected and may impact all the fields based on ferromagnetism. Here, we report the direct observation of room-temperature intrinsic ferromagnetism in two-dimensional (2D) Na2Cl crystals, in which there are only partially filled s- and p-orbitals rather than d- or f-orbitals, using the superconducting quantum interference device (SQUID) and magnetic force microscope (MFM). These Na2Cl crystals formed in reduced graphene oxide (rGO) membranes have an unconventional stoichiometric structure leading to unique electron and spin distributions. And the structure of these 2D Na2Cl crystals, including the Na and Cl sites, is characterized in situ for the first time and directly observed by cryo-electron microscopy (cryo-EM) based on the observed difference in contrast between Na stacked with Cl and single Na. These findings break the conventional rule of intrinsic ferromagnetism and provide new insights into the design of novel magnetic and electronic devices and transistors with a size down to the atomic scale.

cond-mat.mtrl-sci↗

arXiv2023

Discrete-time Approximation of Stochastic Optimal Control with Partial Observation

We consider a class of stochastic optimal control problems with partial observation, and study their approximation by discrete-time control problems. We establish a convergence result by using weak convergence technique of Kushner and Dupuis [Numerical Methods for Stochastic Control Problems in Continuous Time (2001), Springer-Verlag, New York], together with the notion of relaxed control rule introduced by El Karoui, Huu Nguyen and Jeanblanc-Picqué [SIAM J. Control Optim., 26 (1988) 1025-1061]. In particular, with a well chosen discrete-time control system, we obtain a first implementable numerical algorithm (with convergence) for the partially observed control problem. Moreover, our discrete-time approximation result would open the door to study convergence of more general numerical approximation methods, such as machine learning based methods. Finally, we illustrate our convergence result by the numerical experiments on a partially observed control problem in a linear quadratic setting.

math.OC↗