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1. Episodic Bayesian Optimal Control with Unknown Randomness Distributions

   https://doi.org/10.1287/opre.2023.0446  

   Shapiro, Alexander; Zhou, Enlu; Lin, Yifan; Wang, Yuhao (May 2026, Operations Research)

   Bayesian Learning for Data-Driven Stochastic Optimal Control Stochastic optimal control (SOC) provides a principled approach to dynamic decision making under uncertainty. It models the transition of the system state with a dynamic equation driven by randomness, with the assumption that the distribution of randomness is known. However, in practical problems, modeling randomness frequently depends on data or observations, which introduces uncertainty regarding the distribution of randomness. To address the data-driven SOC problem, in “Episodic Bayesian Optimal Control with Unknown Randomness Distributions,” Shapiro, Zhou, Lin, and Wang propose a new approach that incorporates Bayesian learning with optimal control in an episodic manner. They show the convergence and convergence rate results for their approach. They also develop an efficient computational method for a class of problems that have convex cost functions and linear state dynamics. 

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2. Optimal Control of the Viscous Wave Equation via the Pontryagin Maximum Principle

   https://doi.org/10.1002/num.70103  

   Borzì, A; Roy, S (May 2026, Numerical Methods for Partial Differential Equations)

   ABSTRACT A tracking‐type optimal control problem governed by the viscous wave equation with a distributed‐source control and ‐ control costs is investigated. For this class of PDE‐constrained linear‐convex problems, a Pontryagin maximum principle (PMP) in the PDE setting is derived, and it is shown that the pointwise maximization condition is also sufficient for optimality. Based on the PMP, a sequential quadratic Hamiltonian (SQH) method is implemented, and a sufficient decrease property is established by introducing an adaptive penalization parameter. Convergence of the SQH method is discussed for both interval‐valued and discrete‐valued control sets. For the state and adjoint equations, a second‐order finite‐difference discretization is analysed. Numerical experiments validate both the approximation properties of the discretization and the effectiveness of the PMP‐based optimization framework. 

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3. DESIGN OF EXPERIMENTS VIA MULTI-FIDELITY SURROGATES AND STATISTICAL SENSITIVITY MEASURES

   https://doi.org/10.1615/JMachLearnModelComput.2024055261  

   Gillcrist, David J; Alemazkoor, Negin; Chen, Yanlai; Tootkaboni, Mazdak (January 2024, Journal of Machine Learning for Modeling and Computing)

   Parameter estimation and optimal experimental design problems have been widely studied across scienceand engineering. The two are inextricably linked, with optimally designed experiments leading to better-estimated parameters. This link becomes even more crucial when available experiments produce minimal data due to practical constraints of limited experimental budgets. This work presents a novel framework that allows for the identification of optimal experimental arrangement, from a finite set of possibilities, for precise parameter estimation. The proposed framework relies on two pillars. First, we use multi-fidelity modeling to create reliable surrogates that relate unknown parameters to a measurable quantity of interest for a multitude of available choices defined through a set of candidate control vectors. Secondly, we quantify the estimation potential of an arrangement from the set of control vectors through the examination of statistical sensitivity measures calculated from the constructed surrogates. The measures of sensitivity are defined using analysis of variance as well as directional statistics. Two numerical examples are provided, where we demonstrate how the correlation between the estimation potential and the frequency of precise parameter estimation can inform the choice of optimal arrangement. 

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4. Uniform error estimate of an asymptotic preserving scheme for the Lévy-Fokker-Planck equation

   https://doi.org/10.1090/mcom/3975  

   Sun, Weiran; Wang, Li (May 2024, Mathematics of Computation)

   We establish a uniform-in-scaling error estimate for the asymptotic preserving (AP) scheme proposed by Xu and Wang [Commun. Math. Sci. 21 (2023), pp. 1–23] for the Lévy-Fokker-Planck (LFP) equation. The main difficulties stem not only from the interplay between the scaling and numerical parameters but also the slow decay of the tail of the equilibrium state. We tackle these problems by separating the parameter domain according to the relative size of the scaling parameter $\mathrm{\epsilon <\#comment/>}$ : in the regime where $\mathrm{\epsilon <\#comment/>}$ is large, we design a weighted norm to mitigate the issue caused by the fat tail, while in the regime where $\mathrm{\epsilon <\#comment/>}$ is small, we prove a strong convergence of LFP towards its fractional diffusion limit with an explicit convergence rate. This method extends the traditional AP estimates to cases where uniform bounds are unavailable. Our result applies to any dimension and to the whole span of the fractional power. 

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5. Matrix-Free Stabilized BDF Schemes for Semilinear Parabolic Equations with Unconditional Maximum Bound Principle Preservation and Energy Stability

   Dai, Haishen; Lei, Huan; Zheng, Bin (March 2026, ArXivorg)

   We develop a family of stabilized backward differentiation formula (sBDF) schemes of orders one through four for semilinear parabolic equations. The proposed methods are designed to achieve three properties that are rarely available simultaneously in high-order time discretizations: unconditional preservation of the maximum bound principle (MBP), unconditional discrete energy stability, and practical matrix-free implementation. The construction integrates carefully designed stabilization terms, fixed-point iterations, and a pointwise cut-off strategy. The nonlinear algebraic systems arising from the implicit sBDF discretizations are solved by fixed-point iteration, resulting in fully matrix-free algorithms. This makes the approach particularly attractive for practical computations on general domains and under mixed boundary conditions, where FFT-based exponential time differencing methods are often unavailable or inefficient. We further present a unified analysis for the fully implemented schemes, explicitly incorporating the interplay among time discretization, nonlinear iteration, and cutoff. Unconditional contractivity of the fixed-point iterations and error estimates are established. For the Allen–Cahn equation, we additionally prove an unconditional discrete energy dissipation law. Numerical experiments confirm the theoretical convergence rates and demonstrate the robustness and efficiency of the proposed methods, particularly relative to ETD-based approaches for problems with mixed boundary conditions. 

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