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At least 253 records · Page 14

Kernel learning backward SDE filter for data assimilation

In this paper, we develop a kernel learning backward SDE filter method to estimate the state of a stochastic dynamical system based on its partial noisy observations. A system of forward backward stochastic differential equations is used to propagate the state of the target dynamical model, and Bayesian inference is applied to incorporate the observational information. Further, to characterize the dynamical model in the entire state space, we introduce a kernel learning method to learn a continuous global approximation for the conditional probability density function of the target state by using discrete approximated density values as training data. Numerical experiments demonstrate that the kernel learning backward SDE is highly effective.

97 MATHEMATICS AND COMPUTING↗

Risk-Constrained Reinforcement Learning for Inverter-Dominated Power System Controls

Here, this paper develops a risk-aware controller for grid-forming inverters (GFMs) to minimize large frequency oscillations in GFM inverter-dominated power systems. To tackle the high variability from loads/renewables, we incorporate a mean-variance risk constraint into the classical linear quadratic regulator (LQR) formulation for this problem. The risk constraint aims to bound the time-averaged cost of state variability and thus can improve the worst-case performance for large disturbances. The resulting risk-constrained LQR problem is solved through the dual reformulation to a minimax problem, by using a reinforcement learning (RL) method termed as stochastic gradient-descent with max-oracle (SGDmax). In particular, the zero-order policy gradient (ZOPG) approach is used to simplify the gradient estimation using simulated system trajectories. Numerical tests conducted on the IEEE 68-bus system have validated the convergence of our proposed SGDmax for GFM model and corroborate the effectiveness of the risk constraint in improving the worst-case performance while reducing the variability of the overall control cost.

Frequency control↗

Advanced Dynamically Adaptive Algorithms for Stochastic Simulations on Extreme Scales

The focus of the project is the development of mathematical methods and high-performance computational tools for stochastic simulations, with a particular emphasis on computations on extreme scales. The core of the project revolves around the design of highly efficient and scalable numerical algorithms that can adaptively and accurately, in high dimensional spaces, resolve stochastic problems with limited smoothness, even containing discontinuities.

97 MATHEMATICS AND COMPUTING↗

Force-Free Identification of Minimum-Energy Pathways and Transition States for Stochastic Electronic Structure Theories

Here, the accurate mapping of potential energy surfaces (PESs) is crucial to our understanding of the numerous physical and chemical processes mediated by atomic rearrangements, such as conformational changes and chemical reactions, and the thermodynamic and kinetic feasibility of these processes. Stochastic electronic structure theories, e.g., Quantum Monte Carlo (QMC) methods, enable highly accurate total energy calculations that in principle can be used to construct the PES. However, their stochastic nature poses a challenge to the computation and use of forces and Hessians, which are typically required in algorithms for minimum-energy pathway (MEP) and transition state (TS) identification, such as the nudged elastic band (NEB) algorithm and its climbing image formulation. Here, we present strategies that utilize the surrogate Hessian line-search method, previously developed for QMC structural optimization, to efficiently identify MEP and TS structures without requiring force calculations at the level of the stochastic electronic structure theory. By modifying the surrogate Hessian algorithm to operate in path-orthogonal subspaces and at saddle points, we show that it is possible to identify MEPs and TSs by using a force-free QMC approach. We demonstrate these strategies via two examples, the inversion of the ammonia (NH 3 ) molecule and the nucleophilic substitution (S N 2) reaction F – + CH 3 F → FCH 3 + F – . We validate our results using Density Functional Theory (DFT)- and Coupled Cluster (CCSD, CCSD(T))-based NEB calculations. We then introduce a hybrid DFT-QMC approach to compute thermodynamic and kinetic quantities, free energy differences, rate constants, and equilibrium constants that incorporates stochastically optimized structures and their energies, and show that this scheme improves upon DFT accuracy. Our methods generalize straightforwardly to other systems and other high-accuracy theories that similarly face challenges computing energy gradients, paving the way for highly accurate PES mapping, transition state determination, and thermodynamic and kinetic calculations at significantly reduced computational expense.

Iyer, Gopal R.↗

S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Models

While projection-based reduced order models can reduce the dimension of full order solutions, the resulting reduced models may still contain terms that scale with the full order dimension. Hyper-reduction techniques are sampling-based methods that further reduce this computational complexity by approximating such terms with a much smaller dimension. The goal of this work is to introduce the points selection algorithm developed by Shin and Xiu as a hyper-reduction method. The selection algorithm was originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity $\mathcal{S}$ that measures both the column orthogonality and the determinant, we refer to the algorithm as S-OPT. Numerical examples are provided to demonstrate the performance of S-OPT and to compare its performance with a gappy proper orthogonal decomposition (POD) algorithm. Here, we found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy than gappy POD especially when the number of sampling points is small, although we note that S-OPT shows slow asymptotic convergence with respect to the number of samples for some applications, e.g., Lagrangian hydrodynamics.

97 MATHEMATICS AND COMPUTING↗

Resilient Transmission Hardening Planning in a High Renewable Penetration Erra

Hardening components in transmission systems is a practice to improve system resilience against possible disturbances caused by natural disasters. In a power system with a very high penetration of renewable energy, the system hardening will be further complicated by the uncertainty and variability of renewable energy. In this paper, we study the transmission line hardening planing problem in the context of probabilistic power flows injected by the high penetration of renewable energy. We assume that the probabilistic information of renewable energy is incomplete and ambiguous and propose a data-driven approach to approximate the renewable uncertainty sets. We then extend the N - 1 security criteria to multiple simultaneous contingencies and seek to prepare a hardening plan for the worst-case scenarios. A two-stage data-driven stochastic model is formulated by considering the joint worst-case wind output distribution and transmission line contingencies. Then, we apply the Column-and-Constraints generation method to solve the proposed model. To test the effectiveness of the proposed approach, we conduct experiments on 24-bus and 118-bus test systems. We numerically show that the data-driven approach can effectively address the uncertainty ambiguity and the proposed approach can produce effective hardening plans that improve the system resilience.

Column-and-Constraint generation↗

A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations

We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable discontinuous Galerkin (HDG) discretization of elliptic partial differential equations (PDEs), which allows us to obtain high-order accurate solutions of the governing PDE; (2) the reduced basis method for a new HDG discretization of the underlying PDE to enable real-time solution of the parameterized PDE in the presence of stochastic parameters; and (3) a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the reduced basis approximations. Furthermore, we develop a posteriori error estimates for our approximations of the statistical outputs. Based on these error estimates, we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of Monte Carlo samples to achieve a given error tolerance. We provide numerical examples to demonstrate the performance of the proposed method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Stochastic pre-event preparation for enhancing resilience of distribution systems

Extreme weather events are the common causes for power supply interruptions and power outages in electrical distribution systems. Improving the distribution system and enhancing its resilience is becoming crucial due to the increased frequency of extreme weather events. Preparation and allocation of multiple flexible resources, such as mobile resources, fuel resources, and labor resources before extreme weather events can mitigate the effects of extreme weather events and enhance the resilience of power distribution systems. Here, in this paper, a two-stage stochastic mixed-integer linear programming (SMILP) is proposed to optimize the preparation and resource allocation process for upcoming extreme weather events, which leads to faster and more efficient post-event restoration. The objective of the proposed two-stage SMILP is to maximize the served load and minimize the operating cost of flexible resources. The first stage in the optimization problem selects the amounts and locations of different resources. The second stage considers the operational constraints of the distribution system and repair crew scheduling constraints. The proposed stochastic pre-event preparation model is solved by a scenario decomposition method, Progressive Hedging (PH), to ease the computational complexity introduced by a large number of scenarios. Furthermore, to show the impact of solar photovoltaic (PV) generation on system resilience, three types of PV systems are considered during a power outage and the resilience improvements with different PV penetration levels are compared. Numerical results from simulations on a large-scale (more than 10,000 nodes) distribution feeder have been used to validate the effectiveness and scalability of the proposed method.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Scalable branching on dual decomposition of stochastic mixed-integer programming problems

In this work, we present a scalable branching method for the dual decomposition of stochastic mixed-integer programming. Our new branching method is based on the branching method proposed by Caroe and Schultz that creates branching disjunctions on first-stage variables only. We propose improvements to the process for creating branching disjunctions, including (1) branching on the optimal solutions of the Dantzig-Wolfe reformulation of the restricted master problem and (2) using a more comprehensive (yet simple) measure for the dispersions associated with subproblem solution infeasibility. We prove that the proposed branching process leads to an algorithm that terminates finitely, and we provide conditions under which globally optimal solutions can be identified after termination. We have implemented our new branching method, as well as the Caroe-Schultz method and a branch-and-price method, in the open-source software package DSP. Using SIPLIB test instances, we present extensive numerical results to demonstrate that the proposed branching method significantly reduces the number of node subproblems and solution times.

97 MATHEMATICS AND COMPUTING↗

Stochastic Inversion of Gaussian Random Media Using Transverse Coherence Functions for Reflected Waves: Theory and Method

The transverse coherence functions (TCFs) of phase and amplitude fluctuations of a seismic wave are powerful to estimate the spatial distribution, length scales, and strength of random heterogeneities. However, TCFs have been formulated for transmitted waves only, not for reflected waves. In this paper, we derive reflection TCFs for Gaussian random media. Furthermore, we propose to invert for Gaussian random media using the reflection TCFs based on the grid search. We validate the new reflection TCF formulas using 2D finite-difference numerical experiments. The numerical example also illustrates the feasibility and efficiency of the inversion. The stochastic inversion using reflected waves can be used in both exploration and global seismology.

58 GEOSCIENCES↗

Advanced Dynamically Adaptive Algorithms for Stochastic Simulations on Extreme Scales

The focus of the project is the development of mathematical methods and high-performance com- putational tools for stochastic simulations, with a particular emphasis on computations on extreme scales. The core of the project revolves around the design of highly e cient and scalable numer- ical algorithms that can adaptively and accurately, in high dimensional spaces, resolve stochastic problems with limited smoothness, even containing discontinuities.

97 MATHEMATICS AND COMPUTING↗

Multiscale Modeling and Uncertainty Quantification for Nuclear Fuel Performance

In this project, we will address the challenges associated with constructing high fidelity multiscale models of nuclear fuel performance. We (*) propose a novel approach for coupling mesoscale and macroscale models, (*) devise efficient numerical methods for simulating the coupled system, and (*) devise and analyze effective numerical approaches for error and uncertainty quantification for the coupled multiscale system. As an integral part of the project, we will carry out analysis of the effects of upscaling and downscaling, investigate efficient methods for stochastic sensitivity analysis of the individual macroscale and mesoscale models, and carry out a posteriori error analysis for computed results. We will pursue development and implementation of solutions in software used at Idaho National Laboratories on models of interest to the Nuclear Energy Advanced Modeling and Simulation (NEAMS) program.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Weak Convergence of a Mass-Structured Individual-Based Model

We propose a model of chemostat where the bacterial population is individually-based, each bacterium is explicitly represented and has a mass evolving continuously over time. The substrate concentration is represented as a conventional ordinary differential equation. These two components are coupled with the bacterial consumption. Mechanisms acting on the bacteria are explicitly described (growth, division and washout). Bacteria interact via consumption. We set the exact Monte Carlo simulation algorithm of this model and its mathematical representation as a stochastic process. We prove the convergence of this process to the solution of an integro-differential equation when the population size tends to infinity. Finally, we propose several numerical simulations.

60 APPLIED LIFE SCIENCES↗

An alternative realization of the exact non-Markovian stochastic Schrödinger equation

Based on the path integral approach, we derive a new realization of the exact non-Markovian stochastic Schrödinger equation (SSE). The main difference from the previous non-Markovian quantum state diffusion (NMQSD) method is that the complex Gaussian stochastic process used for the forward propagation of the wave function is correlated, which may be used to reduce the amplitude of the non-Markovian memory term at high temperatures. The new SSE is then written into the recently developed hierarchy of pure states scheme, in a form that is more closely related to the hierarchical equation of motion approach. Numerical simulations are then performed to demonstrate the efficiency of the new method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Negative fluxes and cell-miss errors in the random ray method

The random ray method is a recently developed stochastic method for solving neutral particle transport problems based on the method of characteristics. Perhaps surprisingly for a characteristics-based method using flat sources, we note that the random ray method can produce negative fluxes which may be numerically troublesome in several situations. These occur most severely in fixed source problems where the source is in a region with a small cross section. Additionally, we briefly discuss another source of bias which can occur in similar situations, namely a ray missing a mesh with a strong source and small cross section, resulting in the entirety of the source being unphysically deposited locally. This paper describes the mechanism by which negative fluxes may occur and several different methods to mitigate their effects. These fixes are tested on an eigenvalue problem, a ‘fusion-like’ shielding problem, and a shielding problem featuring an adjoint calculation. Even when extremely coarse random ray quadratures are used such that 20%–30% of cells are missed during a given iteration, use of the preferred fix technique ensures local flux tally errors remain trivial (below 1%). The preferred fix is now the default option in SCONE and OpenMC.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Optimization problems governed by systems of PDEs with uncertainties

This paper reviews current theoretical and numerical approaches to optimization problems governed by partial differential equations (PDEs) that depend on random variables or random fields. Such problems arise in many engineering, science, economics and societal decision-making tasks. This paper focuses on problems in which the governing PDEs are parametrized by the random variables/fields, and the decisions are made at the beginning and are not revised once uncertainty is revealed. Examples of such problems are presented to motivate the topic of this paper, and to illustrate the impact of different ways to model uncertainty in the formulations of the optimization problem and their impact on the solution. A linear–quadratic elliptic optimal control problem is used to provide a detailed discussion of the set-up for the risk-neutral optimization problem formulation, study the existence and characterization of its solution, and survey numerical methods for computing it. Different ways to model uncertainty in the PDE-constrained optimization problem are surveyed in an abstract setting, including risk measures, distributionally robust optimization formulations, probabilistic functions and chance constraints, and stochastic orders. Furthermore, approximation-based optimization approaches and stochastic methods for the solution of the large-scale PDE-constrained optimization problems under uncertainty are described. Some possible future research directions are outlined.

Heinkenschloss, Matthias [Rice Univ., Houston, TX ↗

Stochastic noise can be helpful for variational quantum algorithms

Saddle points constitute a crucial challenge for first-order gradient descent algorithms. In notions of classical machine learning, they are avoided, for example, by means of stochastic gradient descent methods. In this work, we provide evidence that the saddle-points problem can be naturally avoided in variational quantum algorithms by exploiting the presence of stochasticity. We prove convergence guarantees and present practical examples in numerical simulations and on quantum hardware. We argue that the natural stochasticity of variational algorithms can be beneficial for avoiding strict saddle points, i.e., those saddle points with at least one negative Hessian eigenvalue. This insight that some levels of shot noise could help is expected to add a new perspective to notions of near-term variational quantum algorithms. Published by the American Physical Society 2025

Liu, Junyu↗

A Two-Stage Decomposition Approach for AC Optimal Power Flow

The alternating current optimal power flow (AC-OPF) problem is critical to power system operations and planning, but it is generally hard to solve due to its nonconvex and large-scale nature. Furthermore, this paper proposes a scalable decomposition approach in which the power network is decomposed into a master network and a number of subnetworks, where each network has its own AC-OPF subproblem. This formulates a two-stage optimization problem and requires only a small amount of communication between the master and subnetworks. The key contribution is a smoothing technique that renders the response of a subnetwork differentiable with respect to the input from the master problem, utilizing properties of the barrier problem formulation that naturally arises when subproblems are solved by a primal-dual interior-point algorithm. Consequently, existing efficient nonlinear programming solvers can be used for both the master problem and the subproblems. The advantage of this framework is that speedup can be obtained by processing the subnetworks in parallel, and it has convergence guarantees under reasonable assumptions. The formulation is readily extended to instances with stochastic subnetwork loads. Numerical results show favorable performance and illustrate the scalability of the algorithm which is able to solve instances with more than 11 million buses.

24 POWER TRANSMISSION AND DISTRIBUTION↗