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Yilu Liu

Publications and source records attributed to Yilu Liu.

At least 19 recordsLinked to original sources
arXiv2026

The Sharp $C^{1,1}$ Target Threshold for Minimizing Constraint Maps

Let $K\subset\R^m$ be the closure of a bounded domain whose boundary is a compact embedded hypersurface, and let \( \overline M=\R^m\setminus\operatorname{int}K \) be the allowed target. We prove that if $\partial K$ is of class $C^{1,1}$, then every $\overline M$-valued local minimizer of the Dirichlet energy $E(u;B):=\int_B|Du|^2\dd x$ is locally $W^{2,\infty}$, and hence locally $C^{1,1}$, on its continuity set. For every $0<\alpha<1$, we also construct a convex body with $C^{1,\alpha}$ boundary and an everywhere continuous global minimizer that is not $C^{1,1}$. Thus $C^{1,1}$ is the sharp target regularity threshold in the H\"older scale.

math.AP↗
arXiv2026

The Higher-Dimensional Nitsche Conjecture: Sharp Bounds and Rigidity

Let $n\ge3$ and let $h:\A(r,1)\to\A(R,1)\subset\mathbb R^n$ be an onto homeomorphism with harmonic coordinate functions. We prove the sharp Nitsche bound \[ R\le R_{n,+}(r):=\frac{nr}{n-1+r^n}, \] and, when $h$ interchanges the two ends, the strictly stronger sharp bound \[ R\le R_{n,-}(r):=\frac{nr^{n-1}}{1+(n-1)r^n}. \] Both critical cases are rigid: equality forces, up to an orthogonal transformation, the corresponding end-preserving or end-reversing radial harmonic homeomorphism. No continuous extension to the closed annulus, boundary homeomorphism, boundary Jacobian, or sign condition on the Jacobian is assumed. The proof converts the nonzero degree of each interior direction map into a probability coupling and establishes a sharp contraction principle for vector measures under positive zonal kernels, using the strict concavity of spherical-cap barycenters. At either critical value, a second-order endpoint defect forces equality for a limiting transfer kernel, whose equality classification yields an orthogonal coupling graph. The remaining trace is locked by a Dirichlet-to-Neumann spectral gap in the end-preserving case and by endpoint H\"older regularity and uniform convergence of the direction maps in the end-reversing case.

math.AP↗
arXiv2026

Global Minimality and Rigidity of the Constraint Map Vortex

We consider the minimization problem \[ \min\left\{\int_{B_1}|Du|^2:\ u\in W^{1,2}(B_1;\mathbb R^n),\quad u=x\ \text{on }\partial B_1,\quad |u|\ge a\right\}, \quad 0<a<1. \] Figalli, Guerra, Kim, and Shahgholian proved that the canonical radial vortex is the unique global minimizer for $n\ge7$, and asked whether the same holds in dimensions \(3\le n \le6\). We answer this question affirmatively, thereby completing the global minimality and rigidity of the constraint map vortex in every dimension \(n\ge3\).

math.AP↗
arXiv2026

A Model-threshold Dimension Bound and Sharp Critical Ends for Smooth Singular Sets of Constant Positive $\sigma_k$-curvature Metrics

Let $k\in\mathbb N$ satisfy $1 0$ must obey \[ p\leq p_k(n), \] where $p_k$ is the model threshold determined by $\Hh^{p+1}\times\Sn^{n-p-1}$. When $k=2$ and $n=m^2$, we construct a smooth complete equality example on $\Sn^n\setminus\Sn^{(m^2-m-2)/2}$. We also prove that the strict inequality $p<p_k(n)$ holds under a finite positive linear-contact hypothesis.

math.DG↗
arXiv2026

Time Distribution Principle Using Measured Traveling Waves in Power Grid

Accurate time synchronization is essential for distributed systems. Conventional methods, such as satellite-based synchronization and communication-based approaches, face challenges including signal vulnerability and dependence on communication delay symmetry. This paper proposes a novel time synchronization principle based on traveling wave (TW) measurements in power grids. By leveraging the inherent symmetry of forward and backward TW propagation, the proposed method achieves high-precision time distribution without relying on external time references. The methodology is validated through electromagnetic transient simulations on a modified IEEE 14-bus system. The results demonstrate that under normal conditions, the proposed approach achieves microsecond-level synchronization accuracy. These findings suggest that the TW-based time synchronization principle is a potential alternative to traditional methods, offering improved security and reduced dependence on communication quality.

eess.SY↗
arXiv2026

Nonconvex Sublevel Sets For The Planar Translating Mean Curvature Equation

Translating solitons arise as models for type~II singularities of mean-convex mean curvature flow. We construct a smooth bounded uniformly convex domain \(\Om\Subset\R^2\) such that the zero-Dirichlet solution of the planar translating mean curvature equation has a nonconvex sublevel set. The construction is based on a corrected near-critical grim-reaper profile and explicit barriers on a long convex channel.

math.AP↗
arXiv2026

Keeping Data Centers Online in Weak Grids: PLL-Free VM-DPC With Adaptive Reactive-Power Support for Centralized UPS Systems

Data center power systems are increasingly exposed to weak-grid conditions due to the rapid growth of converter-dominated networks and highly dynamic artificial intelligence (AI) workloads. In centralized uninterruptible power supply (UPS) architectures, the front-end rectifier continuously processes the incoming facility power, making its dynamic performance critical for ensuring stable operation and reliable power delivery to information technology (IT) equipment. Under weak-grid conditions, conventional phase-locked loop (PLL)-based proportional-integral (PI) rectifier controllers may exhibit instability due to strong interactions between converter control dynamics and grid impedance. This paper investigates the stability of centralized UPS data center systems operating under weak-grid conditions using a detailed switching-level model developed in MATLAB/Simulink and validated in real time using an OPAL-RT platform. To enhance weak-grid stability and improve converter-grid interaction, a voltage-modulated direct power control (VM-DPC) strategy with adaptive reactive power support is applied to the front-end rectifier. The proposed approach directly regulates active and reactive power without PLL synchronization while dynamically supporting the point of common coupling (PCC) voltage during rapid IT load variations. Results demonstrate that conventional PI-based rectifier control becomes unstable under SCR<=2 conditions, leading to dc-link oscillations and degradation of downstream power delivery. In contrast, the proposed VM-DPC strategy restores stable operation, improves system damping, and maintains reliable power transfer to highly dynamic IT loads under weak-grid operation.

eess.SY↗
arXiv2026

Critical Inertia Estimation for the Three U.S. Interconnections

The rapid integration of inverter-based resources (IBRs) is reducing system inertia across U.S. power grids, raising concerns about frequency stability following large contingencies. This paper presents a simulation-based assessment of critical inertia, defined as the minimum system inertia required to prevent first-stage under-frequency load shedding (UFLS) after the largest credible contingency, across the three major U.S. interconnections: Eastern Interconnection (EI), WECC, and ERCOT. Reduced-inertia scenarios are created by progressively replacing synchronous generators with IBRs, and dynamic simulations are performed using full-scale PSS/E and PowerWorld models. The results show that ERCOT reaches critical inertia at approximately 58 percent IBR penetration, compared with above 90 percent for WECC and approximately 67 to 68 percent for EI. Current IBR shares in the U.S. portions of EI, WECC, and ERCOT are 16 percent, 33 percent, and 44 percent, respectively, indicating varying proximity to critical inertia thresholds. These findings highlight the importance of full dynamic simulations to accurately estimate critical inertia and guide transmission planning under high renewable penetration scenarios.

eess.SY↗
arXiv2026

Budget-Aware LLM Discovery via Cost-Calibrated Frontier Utility

Large language models increasingly support scientific and algorithmic discovery through inference-time search over evaluated candidates. Existing adaptive discovery controllers assign credit based only on score progress, even though prompt length, retries, and guidance calls cause search actions to incur different token costs. We prove that cost-blind credit can forfeit all but a vanishing fraction of attainable quality as frontiers multiply and costs diverge. Under a fixed search-side token budget, the controller must decide which frontier is improving and whether its gain justifies the realized cost before the budget is exhausted. We introduce \textbf{CostAda}, a cost-calibrated adaptive controller built around \emph{cost-calibrated frontier utility}. The utility values frontier progress relative to realized action cost and conditions that credit on the remaining budget. CostAda uses this signal to control local exploration intensity, frontier allocation, and budgeted tactic intervention. Cost and remaining budget therefore shape the search rather than serving only as accounting variables or a stopping rule. CostAda reaches the strongest baseline's full-budget quality with at most half the budget on twelve of sixteen benchmark--backbone pairs while achieving the strongest mean final quality on all eight benchmarks under GLM-5 and GPT-5.4.

cs.LG↗
arXiv2026

Frequency Response of Windowed DFT Phasor Estimation: Impact on Oscillation Observability

Phasor measurement units (PMUs) are widely used for sub-synchronous oscillation monitoring, yet the effect of windowed discrete Fourier transform (DFT)-based phasor estimation on oscillation observability is not fully characterized. This letter derives the complete complex-valued frequency response of the windowed DFT phasor estimator under both magnitude and phase modulation. The analysis shows that the estimation window introduces both frequency-dependent magnitude attenuation and phase shift to oscillation components, governed by the complex gain. A simple recovery method is also proposed to restore the true oscillation amplitude and phase from PMU data using the analytically known complex gain. The results are validated through time-domain simulations and provide guidance for industry practitioners on interpreting PMU-based oscillation measurements and selecting appropriate window lengths.

eess.SP↗
arXiv2026

Quality-Diversity Optimization as Multi-Objective Optimization

The Quality-Diversity (QD) optimization aims to discover a collection of high-performing solutions that simultaneously exhibit diverse behaviors within a user-defined behavior space. This paradigm has stimulated significant research interest and demonstrated practical utility in domains including robot control, creative design, and adversarial sample generation. A variety of QD algorithms with distinct design principles have been proposed in recent years. Instead of proposing a new QD algorithm, this work introduces a novel reformulation by casting the QD optimization as a multi-objective optimization (MOO) problem with a huge number of optimization objectives. By establishing this connection, we enable the direct adoption of well-established MOO methods, particularly set-based scalarization techniques, to solve QD problems through a collaborative search process. We further provide a theoretical analysis demonstrating that our approach inherits theoretical guarantees from MOO while providing desirable properties for the QD optimization. Experimental studies across several QD applications confirm that our method achieves performance competitive with state-of-the-art QD algorithms.

cs.LG↗
arXiv2025

Understanding Regional Inertia Dynamics in CAISO from Real Grid Disturbances

The shift from synchronous generators to inverter-based resources has caused power system inertia to be unevenly distributed across power grids. As a result, certain grid regions are more vulnerable to high rate-of-change of frequency (RoCoF) during disturbances. This paper presents a measurement-based framework for estimating grid inertia in CAISO (California Independent System Operator) region using real disturbance-driven frequency data from the Frequency Monitoring Network (FNET/GridEye). By analyzing confirmed disturbances from 2013 to 2024, we identify trends in regional inertia and frequency dynamics, highlighting their relationship with renewable generation and the evolving duck curve. Regional RoCoF values were up to six times higher than interconnection-wide values, coinciding with declining inertia. Recent recovery in inertia is attributed to the increased deployment of battery energy storage systems with synthetic inertia capabilities. These findings underscore the importance of regional inertia monitoring, strategic resource planning, and adaptive operational practices to ensure grid reliability amid growing renewable integration.

eess.SY↗
arXiv2025

FoMEMO: Towards Foundation Models for Expensive Multi-objective Optimization

Expensive multi-objective optimization is a prevalent and crucial concern in many real-world scenarios, where sample-efficiency is vital due to the limited evaluations to recover the true Pareto front for decision making. Existing works either involve rebuilding Gaussian process surrogates from scratch for each objective in each new problem encountered, or rely on extensive past domain experiments for pre-training deep learning models, making them hard to generalize and impractical to cope with various emerging applications in the real world. To address this issue, we propose a new paradigm named FoMEMO (Foundation Models for Expensive Multi-objective Optimization), which enables the establishment of a foundation model conditioned on any domain trajectory and user preference, and facilitates fast in-context optimization based on the predicted preference-wise aggregated posteriors. Rather than accessing extensive real-world domain experiments for training, we demonstrate that pre-training the foundation model with a diverse set of hundreds of millions of synthetic data can lead to superior generalization and optimization performance to unknown problems, without necessitating any subsequent model training or updates in the following optimization process.

cs.LG↗
arXiv2025

EoH-S: Evolution of Heuristic Set using LLMs for Automated Heuristic Design

Automated Heuristic Design (AHD) using Large Language Models (LLMs) has achieved notable success in recent years. Despite the effectiveness of existing approaches, they only design a single heuristic to serve all problem instances, often inducing poor generalization across different distributions or settings. To address this issue, we propose Automated Heuristic Set Design (AHSD), a new formulation for LLM-driven AHD. The aim of AHSD is to automatically generate a small-sized complementary heuristic set to serve diverse problem instances, such that each problem instance could be optimized by at least one heuristic in this set. We show that the objective function of AHSD is monotone and supermodular. Then, we propose Evolution of Heuristic Set (EoH-S) to apply the AHSD formulation for LLM-driven AHD. With two novel mechanisms of complementary population management and complementary-aware memetic search, EoH-S could effectively generate a set of high-quality and complementary heuristics. Comprehensive experimental results on three AHD tasks with diverse instances spanning various sizes and distributions demonstrate that EoH-S consistently outperforms existing state-of-the-art AHD methods and achieves up to 60\% performance improvements.

cs.AI↗
arXiv2024

Practical Power System Inertia Monitoring Based on Pumped Storage Hydropower Operation Signature

This paper proposes a practical method to monitor power system inertia using Pumped Storage Hydropower (PSH) switching-off events. This approach offers real-time system-level inertia estimation with minimal expenses, no disruption, and the inclusion of behind-the-meter inertia. First, accurate inertia estimation is achieved through improved RoCoF calculation that accounts for pre-event RoCoF, reducing common random frequency fluctuations in practice. Second, PSH field data is analyzed, highlighting the benefits of using switching-off events for grid inertia estimation. Third, an event detection trigger is designed to capture pump switching-off events based on local and system features. Fourth, the method is validated on the U.S. Eastern Interconnection model with over 60,000 buses, demonstrating very high accuracy (3%-5% error rate). Finally, it is applied to the U.S. Western Interconnection, with field validation showing a 9.9% average absolute error rate. Despite challenges in practical power system inertia estimation, this method enhances decision-making for power grid reliability and efficiency, addressing challenges posed by renewable energy integration.

eess.SY↗
arXiv2024

Few for Many: Tchebycheff Set Scalarization for Many-Objective Optimization

Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.

cs.LG↗
arXiv2022

Build Smart Grids on Artificial Intelligence -- A Real-world Example

Power grid data are going big with the deployment of various sensors. The big data in power grids creates huge opportunities for applying artificial intelligence technologies to improve resilience and reliability. This paper introduces multiple real-world applications based on artificial intelligence to improve power grid situational awareness and resilience. These applications include event identification, inertia estimation, event location and magnitude estimation, data authentication, control, and stability assessment. These applications are operating on a real-world system called FNET-GridEye, which is a wide-area measurement network and arguably the world-largest cyber-physical system that collects power grid big data. These applications showed much better performance compared with conventional approaches and accomplished new tasks that are impossible to realized using conventional technologies. These encouraging results demonstrate that combining power grid big data and artificial intelligence can uncover and capture the non-linear correlation between power grid data and its stabilities indices and will potentially enable many advanced applications that can significantly improve power grid resilience.

eess.SY↗