Xingye Yue

Chenghua Duan, Chun Liu, Cheng Wang et al.

We study numerical methods for porous media equation (PME). There are two important characteristics: the finite speed propagation of the free boundary and the potential waiting time, which make the problem not easy to handle. Based on different dissipative energy laws, we develop two numerical schemes by an energetic variational approach. Firstly, based on $f \log f$ as the total energy form of the dissipative law, we obtain the trajectory equation, and then construct a fully discrete scheme. It is proved that the scheme is uniquely solvable on an admissible convex set by taking the advantage of the singularity of the total energy. Next, based on $\frac{1}{2 f}$ as the total energy form of the dissipation law, we construct a linear numerical scheme for the corresponding trajectory equation. Both schemes preserve the corresponding discrete dissipation law. Meanwhile, under some smoothness assumption, it is proved, by a higher order expansion technique, that both schemes are second-order convergent in space and first-order convergent in time. Each scheme yields a good approximation for the solution and the free boundary. No oscillation is observed for the numerical solution around the free boundary. Furthermore, the waiting time problem could be naturally treated, which has been a well-known difficult issue for all the existence methods. Due to its linear nature, the second scheme is more efficient.

Chenghua Duan, Chun Liu, Cheng Wang et al.

In this paper, we focus on numerical solutions for random genetic drift problem, which is governed by a degenerated convection-dominated parabolic equation. Due to the fixation phenomenon of genes, Dirac delta singularities will develop at boundary points as time evolves. Based on an energetic variational approach (EnVarA), a balance between the maximal dissipation principle (MDP) and least action principle (LAP), we obtain the trajectory equation. In turn, a numerical scheme is proposed using a convex splitting technique, with the unique solvability (on a convex set) and the energy decay property (in time) justified at a theoretical level. Numerical examples are presented for cases of pure drift and drift with semi-selection. The remarkable advantage of this method is its ability to catch the Dirac delta singularity close to machine precision over any equidistant grid.

Minxin Chen, Chun Liu, Shixin Xu et al.

In this paper, we focus on numerical methods for the genetic drift problems, which is governed by a degenerated convection-dominated parabolic equation. Due to the degeneration and convection, Dirac singularities will always be developed at boundary points as time evolves. In order to find a \emph{complete solution} which should keep the conservation of total probability and expectation, three different schemes based on finite volume methods are used to solve the equation numerically: one is a upwind scheme, the other two are different central schemes. We observed that all the methods are stable and can keep the total probability, but have totally different long-time behaviors concerning with the conservation of expectation. We prove that any extra infinitesimal diffusion leads to a same artificial steady state. So upwind scheme does not work due to its intrinsic numerical viscosity. We find one of the central schemes introduces a numerical viscosity term too, which is beyond the common understanding in the convection-diffusion community. Careful analysis is presented to prove that the other central scheme does work. Our study shows that the numerical methods should be carefully chosen and any method with intrinsic numerical viscosity must be avoided.

Xiaotao Zheng, Xingye Yue, Zhihong Xia et al.

Semilinear parabolic partial differential equations (PDEs) are fundamental to modeling complex dynamical systems across scientific domains. The Deep Backward Stochastic Differential Equation (BSDE) method is a promising approach for high-dimensional PDEs; however, existing convergence results apply only to globally Lipschitz generators, excluding important cases such as Allen--Cahn and Hamilton--Jacobi--Bellman (HJB) equations. This paper presents both a theoretical and a computational advance for Deep BSDE methods. Theoretically, we establish the convergence theory for non--Lipschitz generators--covering Allen--Cahn equations with cubic nonlinearity and HJB equations with quadratic gradient growth--based on a bounded double--well lemma and a truncated-BSDE analysis within the Bouchard--Touzi--Zhang theory. Computationally, we instantiate the framework with XNet, a shallow architecture with $\mathcal O(L)$ parameters that preserves strong approximation while substantially reducing optimization and computational cost. Numerical experiments on 100--dimensional PDEs corroborate the predicted convergence behavior and demonstrate significant efficiency gains over standard feedforward implementations.

Ziting Pei, Xingye Yue, Xiaotao Zheng

G-expectation, as a sublinear expectation, provides a powerful framework for modeling uncertainty in financial markets. Motivated by the need for robust valuation under model uncertainty, this work develops a unified risk-neutral valuation approach within the G-expectation environment, yielding a nonlinear generalization of the Black-Scholes model, termed the G-Black-Scholes equation. To enhance computational efficiency and reduce numerical cost, we introduce a logarithmic transformation of the asset price, which yields an alternative nonlinear PDE. Based on this transformed formulation, we design both explicit and implicit finite difference schemes that are rigorously demonstrated to be consistent, stable, monotone, and convergent to the viscosity solution. Numerical examples confirm that the proposed schemes achieve high accuracy, while the logarithmic transformation relaxes the stability constraints of explicit schemes and improves computational efficiency.

Ziting Pei, Shige Peng, Xingye Yue et al.

A $G$-normal random variable $X\sim \mathcal{N}(0,[\underlineσ^2,\overlineσ^2])$ does not admit a unique probability law due to volatility uncertainty. For a given test function $ϕ$, the $G$-expectation admits the stochastic control representation$$\mathbb{E}[ϕ(X)] = \sup_{σ\in[\underlineσ,\overlineσ]} {E}\!\left[ϕ(X_T^σ)\mid X_0^σ=0\right] ={E}\!\left[ϕ(X_T^\ast)\mid X_0^\ast=0\right].$$ This formulation interprets the nonlinear expectation as a linear expectation under the law induced by the optimally controlled diffusion $X^\ast$, namely, the terminal law of $X_T^\ast$. This observation motivates the notion of a \emph{responsive distribution}, a measurement-dependent probability density $f_ϕ$ such that, for a given test function $ϕ$, $$\mathbb{E}[ϕ(X)] = \int_{\mathbb{R}} ϕ(x)\,f_ϕ(x)\,dx.$$ Based on this viewpoint, we propose a coupled backward--forward trinomial tree framework for computing the $G$-expectation and constructing the corresponding responsive distribution. The backward trinomial tree discretizes the associated stochastic optimal control problem and yields approximations of the value function (i.e., the $G$-expectation) and the optimal feedback control, while the forward trinomial tree propagates the induced transition probabilities and produces a discrete approximation of the responsive distribution. We establish rigorous convergence results for both components of the method. Numerical results not only validate the theoretical convergence of the coupled schemes but also provide a powerful, practical sampling tool to visualize the complex responsive distributions under various measurements.

Haoran Xu, Kunyang Li, Xingye Yue

We propose a novel non-compact, positivity-preserving scheme for linear non-divergence form elliptic equations. Based on the Feynman--Kac formula, the solution is represented as a conditional expectation associated with a diffusion process.Instead of using compact Markov chain approximations, we construct a wide-stencil scheme by approximating the expectation with carefully designed transition probabilities, ensuring both consistency and positivity preservation. The method is effective for anisotropic diffusion problems with mixed derivatives, where classical schemes typically fail unless the covariance matrix is diagonally dominant. A key feature of the proposed framework is its robust treatment of boundary conditions. For Dirichlet boundaries, we introduce a quadtree-based non-uniform stopping-time strategy, achieving $O(h)$ accuracy. For Neumann boundaries, a discrete specular reflection mechanism is employed, yielding $O(h^{1/2})$ convergence. Periodic boundaries are handled through modular wrapping, also achieving $O(h)$ accuracy. The resulting schemes are unconditionally stable and positivity-preserving due to their probabilistic structure. Numerical experiments confirm the theoretical convergence rates under all boundary conditions considered.

Xishun Wang, Zhouwang Yang, Xingye Yue et al.

Subspace segmentation assumes that data comes from the union of different subspaces and the purpose of segmentation is to partition the data into the corresponding subspace. Low-rank representation (LRR) is a classic spectral-type method for solving subspace segmentation problems, that is, one first obtains an affinity matrix by solving a LRR model and then performs spectral clustering for segmentation. This paper proposes a group norm regularized factorization model (GNRFM) inspired by the LRR model for subspace segmentation and then designs an Accelerated Augmented Lagrangian Method (AALM) algorithm to solve this model. Specifically, we adopt group norm regularization to make the columns of the factor matrix sparse, thereby achieving a purpose of low rank, which means no Singular Value Decompositions (SVD) are required and the computational complexity of each step is greatly reduced. We obtain affinity matrices by using different LRR models and then performing cluster testing on different sets of synthetic noisy data and real data, respectively. Compared with traditional models and algorithms, the proposed method is faster and more robust to noise, so the final clustering results are better. Moreover, the numerical results show that our algorithm converges fast and only requires approximately ten iterations.