检索结果-内蒙古大学图书馆

Solving cluster moment relaxation with hierarchical matrix

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Wang, Yi Khoo, Yuehaw Department of Statistics University of Chicago United States Department of Statistics and Committee on Computational and Applied Mathematics University of Chicago United States

Convex relaxation methods are powerful tools for studying the lowest energy of many-body problems. By relaxing the representability conditions for marginals to a set of local constraints, along with a global semidefinite constraint, a polynomial-time solvable semidefinite program (SDP) that provides a lower bound for the energy can be derived. In this paper, we propose accelerating the solution of such an SDP relaxation by imposing a hierarchical structure on the positive semidefinite (PSD) primal and dual variables. Furthermore, these matrices can be updated efficiently using the algebra of the compressed representations within an augmented Lagrangian method. We achieve quadratic and even near-linear time per-iteration complexity. Through experimentation on the quantum transverse field Ising model, we showcase the capability of our approach to provide a sufficiently accurate lower bound for the exact ground-state energy. Copyright © 2024, The Authors. All rights reserved.

关键词： Ising model

TIME-DEPENDENT REGULARIZED 13-MOMENT EQUATIONS WITH ONSAGER BOUNDARY CONDITIONS IN THE LINEAR REGIME

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Lin, Bo Wang, Haoxuan Yang, Siyao Cai, Zhenning Department of Mathematics National University of Singapore 119076 Singapore Committee on Computational and Applied Mathematics Department of Statistics University of Chicago ChicagoIL60637 United States

We develop the time-dependent regularized 13-moment equations for general elastic collision models under the linear regime. Detailed derivation shows the proposed equations have super-Burnett order for small Knudsen numbers, and the moment equations enjoy a symmetric structure. A new modification of Onsager boundary conditions is proposed to ensure stability as well as the removal of undesired boundary layers. Numerical examples of one-dimensional channel flows is conducted to verified our *** Codes 76P05, 82C40 Copyright © 2024, The Authors. All rights reserved.

关键词： Channel flow

Noise-induced tipping under periodic forcing: Preferred tipping phase in a non-adiabatic forcing regime

学校读者我要写书评

暂无评论

arXiv 2018年

作者： Chen, Yuxin Gemmer, John A. Silber, Mary Volkening, Alexandria Department of Engineering Sciences and Applied Mathematics Northwestern University Department of Mathematics and Statistics Wake Forest University Committee on Computational and Applied Mathematics Department of Statistics University of Chicago Mathematical Biosciences Institute Ohio State University

We consider a periodically-forced 1-D Langevin equation that possesses two stable periodic solutions in the absence of noise. We ask the question: is there a most likely noise-induced transition path between these periodic solutions that allows us to identify a preferred phase of the forcing when tipping occurs? The quasistatic regime, where the forcing period is long compared to the adiabatic relaxation time, has been well studied;our work instead explores the case when these timescales are comparable. We compute optimal paths using the path integral method incorporating the Onsager-Machlup functional and validate results with Monte Carlo simulations. Results for the preferred tipping phase are compared with the deterministic aspects of the problem. We identify parameter regimes where nullclines, associated with the deterministic problem in a 2-D extended phase space, form passageways through which the optimal paths transit. As the nullclines are independent of the relaxation time and the noise strength, this leads to a robust deterministic predictor of preferred tipping phase in a regime where forcing is neither too fast, nor too slow.68Q25, 60G17, 37J50 Copyright © 2018, The Authors. All rights reserved.

关键词： Relaxation time

RE-ANCHORING QUANTUM MONTE CARLO WITH TENSOR-TRAIN SKETCHING

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Yu, Ziang Zhang, Shiwei Khoo, Yuehaw Committee on Computational and Applied Mathematics University of Chicago United States Center for Computational Quantum Physics Flatiron Institute United States Department of Statistics University of Chicago United States

We propose a novel algorithm for calculating the ground-state energy of quantum many-body systems by combining auxiliary-field quantum Monte Carlo (AFQMC) with tensor-train sketching. In AFQMC, having a good trial wavefunction to guide the random walk is crucial for avoiding sign problems. Typically, this trial wavefunction is fixed throughout the simulation. Our proposed method iterates between determining a new trial wavefunction in the form of a tensor train, derived from the current walkers, and using this updated trial wavefunction to anchor the next phase of AFQMC. Numerical results demonstrate that our algorithm is highly accurate for large spin systems, achieving a relative error of 10−5 in estimating ground-state energies. Additionally, the overlap between our estimated trial wavefunction and the ground-state wavefunction achieves a high-fidelity. We provide a convergence proof, highlighting how an effective trial wavefunction can reduce the variance in the AFQMC energy *** Codes 15A69, 62G07, 65C05 Copyright © 2024, The Authors. All rights reserved.

关键词： Tensors

SOLVING CALDEIRA-LEGGETT MODEL BY INCHWORM METHOD WITH FROZEN GAUSSIAN APPROXIMATION

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Wang, Geshuo Yang, Siyao Cai, Zhenning Department of Mathematics National University of Singapore Singapore119076 Singapore Committee on Computational and Applied Mathematics Department of Statistics University of Chicago ChicagoIL60637 United States

We propose an algorithm that combines the inchworm method and the frozen Gaussian approximation to simulate the Caldeira-Leggett model in which a quantum particle is coupled with thermal harmonic baths. In particular, we are interested in the real-time dynamics of the reduced density operator. In our algorithm, we use frozen Gaussian approximation to approximate the wave function as a wave packet in integral form. The desired reduced density operator is then written as a Dyson series, which is the series expression of path integrals in quantum mechanics of interacting systems. To compute the Dyson series, we further approximate each term in the series using Gaussian wave packets, and then employ the idea of the inchworm method to accelerate the convergence of the series. The inchworm method formulates the series as an integro-differential equation of "full propagators", and rewrites the infinite series on the right-hand side using these full propagators, so that the number of terms in the sum can be significantly reduced, and faster convergence can be achieved. The performance of our algorithm is verified numerically by various *** Codes 45J02, 81V99 Copyright © 2024, The Authors. All rights reserved.

关键词： Wave functions

Solving Caldeira-Leggett Model by Inchworm Method with Frozen Gaussian Approximation

学校读者我要写书评

暂无评论

SSRN

SSRN 2024年

We propose an algorithm that combines the inchworm method and the frozen Gaussian approximation to simulate the Caldeira-Leggett model in which a quantum particle is coupled with thermal harmonic baths. In particular, we are interested in the real-time dynamics of the reduced density operator. In our algorithm, we use frozen Gaussian approximation to approximate the wave function as a wave packet in integral form. The desired reduced density operator is then written as a Dyson series, which is the series expression of path integrals in quantum mechanics of interacting systems. To compute the Dyson series, we further approximate each term in the series using Gaussian wave packets, and then employ the idea of the inchworm method to accelerate the convergence of the series. The inchworm method formulates the series as an integro-differential equation of ``full propagators'', and rewrites the infinite series on the right-hand side using these full propagators, so that the number of terms in the sum can be significantly reduced, and faster convergence can be achieved. The performance of our algorithm is verified numerically by various experiments. © 2024, The Authors. All rights reserved.

关键词： Wave functions

Analysis of a computational Framework for Bayesian Inverse Problems: Ensemble Kalman Updates and MAP Estimators Under Mesh Refinement

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Sanz-Alonso, Daniel Waniorek, Nathan Department of Statistics University of Chicago ChicagoIL United States Committee on Computational and Applied Mathematics University of Chicago ChicagoIL United States

This paper analyzes a popular computational framework to solve infinite-dimensional Bayesian inverse problems, discretizing the prior and the forward model in a finite-dimensional weighted inner product space. We demonstrate the benefit of working on a weighted space by establishing operator-norm bounds for finite element and graph-based discretizations of Matérn-type priors and deconvolution forward models. For linear-Gaussian inverse problems, we develop a general theory to characterize the error in the approximation to the posterior. We also embed the computational framework into ensemble Kalman methods and MAP estimators for nonlinear inverse problems. Our operator-norm bounds for prior discretizations guarantee the scalability and accuracy of these algorithms under mesh *** Codes 65M32 (Primary) 68Q25, 35Q62 62F15 (Secondary) © 2023, CC BY.

关键词： Inverse problems

Path properties of atmospheric transitions: illustration with a low-order sudden stratospheric warming model

学校读者我要写书评

暂无评论

arXiv 2020年

作者： Finkel, Justin Abbot, Dorian S. Weare, Jonathan Committee on Computational and Applied Mathematics University of Chicago Department of the Geophysical Sciences University of Chicago Courant Institute of Mathematical Sciences New York University

Many rare weather events, including hurricanes, droughts, and floods, dramatically impact human life. To accurately forecast these events and characterize their climatology requires specialized mathematical techniques to fully leverage the limited data that are available. Here we describe transition path theory (TPT), a framework originally developed for molecular simulation, and argue that it is a useful paradigm for developing mechanistic understanding of rare climate events. TPT provides a method to calculate statistical properties of the paths into the event. As an initial demonstration of the utility of TPT, we analyze a low-order model of sudden stratospheric warming (SSW), a dramatic disturbance to the polar vortex which can induce extreme cold spells at the surface in the midlatitudes. SSW events pose a major challenge for seasonal weather prediction because of their rapid, complex onset and development. Climate models struggle to capture the long-term statistics of SSW, owing to their diversity and intermittent nature. We use a stochastically forced Holton-Mass-type model with two stable states, corresponding to radiative equilibrium and a vacillating SSW-like regime. In this stochastic bistable setting, from certain probabilistic forecasts TPT facilitates estimation of dominant transition pathways and return times of transitions. These "dynamical statistics" are obtained by solving partial differential equations in the model’s phase space. With future application to more complex models, TPT and its constituent quantities promise to improve the predictability of extreme weather events, through both generation and principled evaluation of forecasts. Copyright © 2020, The Authors. All rights reserved.

关键词： Phase space methods

Ensemble Kalman Filters with Resampling

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Al-Ghattas, Omar Bao, Jiajun Sanz-Alonso, Daniel Department of Statistics University of Chicago ChicagoIL United States Committee on Computational and Applied Mathematics University of Chicago ChicagoIL United States

Filtering is concerned with online estimation of the state of a dynamical system from partial and noisy observations. In applications where the state of the system is high dimensional, ensemble Kalman filters are often the method of choice. These algorithms rely on an ensemble of interacting particles to sequentially estimate the state as new observations become available. Despite the practical success of ensemble Kalman filters, theoretical understanding is hindered by the intricate dependence structure of the interacting particles. This paper investigates ensemble Kalman filters that incorporate an additional resampling step to break the dependency between particles. The new algorithm is amenable to a theoretical analysis that extends and improves upon those available for filters without resampling, while also performing well in numerical examples. © 2023, CC BY.

关键词： Wiener filtering