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We propose a high precision algorithm for solving the Gelfand-Levitan-Marchenko equation. The algorithm is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson's type. To approximate integrals, we use the high-precision one-sided and two-sided Gregory quadrature formulas. Also we use the Woodbury formula to construct a computational algorithm. This makes it possible to use the almost Toeplitz structure of the matrices for the fast calculations.

Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high-dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. By varying the diffusion coefficient, there are in fact infinitely many overdamped Langevin dynamics which are reversible with respect to the target probability measure at hand. This suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. We analytically study this problem here, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some appropriate homogenized limit. Numerical results, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient.

When applying the classical multistep schemes for solving differential equations, one often faces the dilemma that smaller time steps are needed with higher-order schemes, making it impractical to use high-order schemes for stiff problems. We construct in this paper a new class of BDF and implicit-explicit (IMEX) schemes for parabolic type equations based on the Taylor expansions at time $t^{n+\beta}$ with $\beta > 1$ being a tunable parameter. These new schemes, with a suitable $\beta$, allow larger time steps at higher-order for stiff problems than that is allowed with a usual higher-order scheme. For parabolic type equations, we identify an explicit uniform multiplier for the new second- to fourth-order schemes, and conduct rigorously stability and error analysis by using the energy argument. We also present ample numerical examples to validate our findings.

A critical issue in approximating solutions of ordinary differential equations using neural networks is the exact satisfaction of the boundary or initial conditions. For this purpose, neural forms have been introduced, i.e., functional expressions that depend on neural networks which, by design, satisfy the prescribed conditions exactly. Expanding upon prior progress, the present work contributes in three distinct aspects. First, it presents a novel formalism for crafting optimized neural forms. Second, it outlines a method for establishing an upper bound on the absolute deviation from the exact solution. Third, it introduces a technique for converting problems with Neumann or Robin conditions into equivalent problems with parametric Dirichlet conditions. The proposed optimized neural forms were numerically tested on a set of diverse problems, encompassing first-order and second-order ordinary differential equations, as well as first-order systems. Stiff and delay differential equations were also considered. The obtained solutions were compared against solutions obtained via Runge-Kutta methods and exact solutions wherever available. The reported results and analysis verify that in addition to the exact satisfaction of the boundary or initial conditions, optimized neural forms provide closed-form solutions of superior interpolation capability and controllable overall accuracy.

We study the stability and sensitivity of an absorbing layer for the Boltzmann equation by examining the Bhatnagar-Gross-Krook (BGK) approximation and using the perfectly matched layer (PML) technique. To ensure stability, we discard some parameters in the model and calculate the total sensitivity indices of the remaining parameters using the ANOVA expansion of multivariate functions. We conduct extensive numerical experiments on two test cases to study stability and compute the total sensitivity indices, which allow us to identify the essential parameters of the model.

Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise evaluations, which in turn result in redundant function terms or erroneous equations. This study proposes a framework to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. The framework operates through two alternating update processes: discovering and embedding. The discovering phase employs symbolic representation and a novel reinforcement learning (RL)-guided hybrid PDE generator to efficiently produce diverse open-form PDEs with tree structures. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. PDEs with higher rewards are utilized to iteratively optimize the generator via the RL strategy and the best-performing PDE is selected by a parameter-free stability metric. The embedding phase integrates the initially identified PDE from the discovering process as a physical constraint into the predictive model for robust training. The traversal of PDE trees automates the construction of the computational graph and the embedding process without human intervention. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data and outperform other physics-informed neural network-based discovery methods. This work opens new potential for exploring real-world systems with limited understanding.

We analyze a bilinear optimal control problem for the Stokes--Brinkman equations: the control variable enters the state equations as a coefficient. In two- and three-dimensional Lipschitz domains, we perform a complete continuous analysis that includes the existence of solutions and first- and second-order optimality conditions. We also develop two finite element methods that differ fundamentally in whether the admissible control set is discretized or not. For each of the proposed methods, we perform a convergence analysis and derive a priori error estimates; the latter under the assumption that the domain is convex. Finally, assuming that the domain is Lipschitz, we develop an a posteriori error estimator for each discretization scheme and obtain a global reliability bound.

We propose a local discontinuous Galerkin (LDG) method for fractional Korteweg-de Vries equation involving the fractional Laplacian with exponent $\alpha\in (1,2)$ in one and two space dimensions. By decomposing the fractional Laplacian into a first order derivative and a fractional integral, we prove $L^2$-stability of the semi-discrete LDG scheme incorporating suitable interface and boundary fluxes. We analyze the error estimate by considering linear convection term and utilizing the estimate, we derive the error estimate for general nonlinear flux and demonstrate an order of convergence $\mathcal{O}(h^{k+1/2})$. Moreover, the stability and error analysis have been extended to multiple space dimensional case. Additionally, we devise a fully discrete LDG scheme using the four-stage fourth-order Runge-Kutta method. We prove that the scheme is strongly stable under an appropriate time step constraint by establishing a \emph{three-step strong stability} estimate. Numerical illustrations are shown to demonstrate the efficiency of the scheme by obtaining an optimal order of convergence.

As one kind important phase field equations, Cahn-Hilliard equations contain spatial high order derivatives, strong nonlinearities, and even singularities. When using the physics informed neural network (PINN) to simulate the long time evolution, it is necessary to decompose the time domain to capture the transition of solutions in different time. Moreover, the baseline PINN can't maintain the mass conservation property for the equations. We propose a mass-preserving spatio-temporal adaptive PINN. This method adaptively dividing the time domain according to the rate of energy decrease, and solves the Cahn-Hilliard equation in each time step using an independent neural network. To improve the prediction accuracy, spatial adaptive sampling is employed in the subdomain to select points with large residual value and add them to the training samples. Additionally, a mass constraint is added to the loss function to compensate the mass degradation problem of the PINN method in solving the Cahn-Hilliard equations. The mass-preserving spatio-temporal adaptive PINN is employed to solve a series of numerical examples. These include the Cahn-Hilliard equations with different bulk potentials, the three dimensional Cahn-Hilliard equation with singularities, and the set of Cahn-Hilliard equations. The numerical results demonstrate the effectiveness of the proposed algorithm.

We introduce a nonconforming hybrid finite element method for the two-dimensional vector Laplacian, based on a primal variational principle for which conforming methods are known to be inconsistent. Consistency is ensured using penalty terms similar to those used to stabilize hybridizable discontinuous Galerkin (HDG) methods, with a carefully chosen penalty parameter due to Brenner, Li, and Sung [Math. Comp., 76 (2007), pp. 573-595]. Our method accommodates elements of arbitrarily high order and, like HDG methods, it may be implemented efficiently using static condensation. The lowest-order case recovers the $P_1$-nonconforming method of Brenner, Cui, Li, and Sung [Numer. Math., 109 (2008), pp. 509-533], and we show that higher-order convergence is achieved under appropriate regularity assumptions. The analysis makes novel use of a family of weighted Sobolev spaces, due to Kondrat'ev, for domains admitting corner singularities.