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An adaptive mesh refinement method for numerically solving optimal control problems is developed using Legendre-Gauss-Radau direct collocation. In regions of the solution where the desired accuracy tolerance has not been met, the mesh is refined by either increasing the degree of the approximating polynomial in a mesh interval or dividing a mesh interval into subintervals. In regions of the solution where the desired accuracy tolerance has been met, the mesh size may be reduced by either merging adjacent mesh intervals or decreasing the degree of the approximating polynomial in a mesh interval. Coupled with the mesh refinement method described in this paper is a newly developed relative error estimate that is based on the differences between solutions obtained from the collocation method and those obtained by solving initial-value and terminal-value problems in each mesh interval using an interpolated control obtained from the collocation method. Because the error estimate is based on explicit simulation, the solution obtained via collocation is in close agreement with the solution obtained via explicit simulation using the control on the final mesh, which ensures that the control is an accurate approximation of the true optimal control. The method is demonstrated on three examples from the open literature, and the results obtained show an improvement in final mesh size when compared against previously developed mesh refinement methods. 

Abstract BackgroundDiagnostic pathology depends on complex, structured reasoning to interpret clinical, histologic, and molecular data. Replicating this cognitive process algorithmically remains a significant challenge. As large language models (LLMs) gain traction in medicine, it is critical to determine whether they have clinical utility by providing reasoning in highly specialized domains such as pathology. MethodsWe evaluated the performance of four reasoning LLMs (OpenAI o1, OpenAI o3-mini, Gemini 2.0 Flash Thinking Experimental, and DeepSeek-R1 671B) on 15 board-style open-ended pathology questions. Responses were independently reviewed by 11 pathologists using a structured framework that assessed language quality (accuracy, relevance, coherence, depth, and conciseness) and seven diagnostic reasoning strategies. Scores were normalized and aggregated for analysis. We also evaluated inter-observer agreement to assess scoring consistency. Model comparisons were conducted using one-way ANOVA and Tukey’s Honestly Significant Difference (HSD) test. ResultsGemini and DeepSeek significantly outperformed OpenAI o1 and OpenAI o3-mini in overall reasoning quality (p < 0.05), particularly in analytical depth and coherence. While all models achieved comparable accuracy, only Gemini and DeepSeek consistently applied expert-like reasoning strategies, including algorithmic, inductive, and Bayesian approaches. Performance varied by reasoning type: models performed best in algorithmic and deductive reasoning and poorest in heuristic and pattern recognition. Inter-observer agreement was highest for Gemini (p < 0.05), indicating greater consistency and interpretability. Models with more in-depth reasoning (Gemini and DeepSeek) were generally less concise. ConclusionAdvanced LLMs such as Gemini and DeepSeek can approximate aspects of expert-level diagnostic reasoning in pathology, particularly in algorithmic and structured approaches. However, limitations persist in contextual reasoning, heuristic decision-making, and consistency across questions. Addressing these gaps, along with trade-offs between depth and conciseness, will be essential for the safe and effective integration of AI tools into clinical pathology workflows. 

We present an integrated procedure for the synthesis of infinite-layer nickelates using molecular-beam epitaxy with gas-phase reduction by atomic hydrogen. We first discuss challenges in the growth and characterization of perovskite NdNiO3/SrTiO3, arising from post growth crack formation in stoichiometric films. We then detail a procedure for fully reducing NdNiO3 films to the infinite-layer phase, NdNiO2, using atomic hydrogen; the resulting films display excellent structural quality, smooth surfaces, and lower residual resistivities than films reduced by other methods. We utilize the in situ nature of this technique to investigate the role that SrTiO3 capping layers play in the reduction process, illustrating their importance in preventing the formation of secondary phases at the exposed nickelate surface. A comparative bulk- and surface-sensitive study indicates that the formation of a polycrystalline crust on the film surface serves to limit the reduction process. 

The trajectory optimization of a reusable launch vehicle entry is studied. The objective is to maximize the crossrange during atmospheric entry subject to a constraint on the stagnation point heating rate. The problem is solved by partitioning the domain of the independent variable into multiple subdomains such that each subdomain consists of a segment where the heating rate constraint is either active or inactive. Additional necessary conditions for optimality are enforced in segments where the heating rate constraint is active. A multiple-domain Legendre-Gauss-Radau direct collocation method is then used to solve the partitioned problem. Key features of the stagnation point heating rate constraint are presented, and it is observed that the aforementioned approach is capable of solving the problem under consideration more accurately than traditional direct collocation methods. 

A Monte Carlo analysis of a contingency optimal guidance strategy is conducted. The guidance strategy is applied to a Mars Entry problem in which it is assumed that the surface level atmospheric density is a random variable. First, a nominal guidance strategy is employed such that the optimal control problem is re-solved at constant guidance cycles. When the trajectory lies within a particular distance from a path constraint boundary, the nominal guidance strategy is replaced with a contingency guidance strategy, where the contingency guidance strategy attempts to prevent a violation in the the relevant path constraint. The contingency guidance strategy utilizes the reference optimal control problem formulation, but modifies the objective functional to maximize the margin between the path constraint limit and path constraint function value. The ability of the contingency guidance strat- egy to prevent violations in the path constraints is assessed via a Monte Carlo simulation. 

A structure detection method is developed for solving state-variable inequality path con- strained optimal control problems. The method obtains estimates of activation and deactiva- tion times of active state-variable inequality path constraints (SVICs), and subsequently al- lows for the times to be included as decision variables in the optimization process. Once the identification step is completed, the method partitions the problem into a multiple-domain formulation consisting of constrained and unconstrained domains. Within each domain, Legendre-Gauss-Radau (LGR) orthogonal direct collocation is used to transcribe the infinite- dimensional optimal control problem into a finite-dimensional nonlinear programming (NLP) problem. Within constrained domains, the corresponding time derivative of the active SVICs that are explicit in the control are enforced as equality path constraints, and at the beginning of the constrained domains, the necessary tangency conditions are enforced. The accuracy of the proposed method is demonstrated on a well-known optimal control problem where the analytical solution contains a state constrained arc. 

A Monte Carlo analysis of a contingency optimal guidance strategy is conducted. The guidance strategy is applied to a Mars Entry problem in which it is assumed that the surface level atmospheric density is a random variable. First, a nominal guidance strategy is employed such that the optimal control problem is re-solved at constant guidance cycles. When the trajectory lies within a particular distance from a path constraint boundary, the nominal guidance strategy is replaced with a contingency guidance strategy, where the contingency guidance strategy attempts to prevent a violation in the the relevant path constraint. The contingency guidance strategy utilizes the reference optimal control problem formulation, but modifies the objective functional to maximize the margin between the path constraint limit and path constraint function value. The ability of the contingency guidance strategy to prevent violations in the path constraints is assessed via a Monte Carlo simulation.