Augmented Lagrangian Method for Coupled-Cluster Convergence Improvement

Core Concepts

Improving convergence properties of the single-reference coupled cluster method through an augmented Lagrangian formalism.

Abstract

The article introduces an augmented Lagrangian method (alm-CC) to enhance the stability and convergence of the coupled-cluster (CC) approach. It addresses issues with conventional CC methods, such as non-convergence and convergence to unphysical states. The proposed method minimizes an energy term alongside solving CC equations, transforming root-finding into a minimization problem. Numerical experiments demonstrate superior convergence performance with comparable computational costs to traditional CC methods. The content is structured around Introduction, Previous Works, Contribution, Optimizing Coupled Cluster, The Cluster Matrices, Numerics, Agreement with Quasi-Newton Method, Global Convergence Behavior sections.

Introduction:

• Describes the long-standing challenge of describing electrons in quantum many-body systems.

Previous Works:

• Discusses numerical methods in computational chemistry and physics community for tackling large problems efficiently.

Contribution:

• Proposes an augmented Lagrangian formulation (alm-CC) to address convergence issues in CC methods by minimizing energy terms alongside solving CC equations.

Optimizing Coupled Cluster:

• Introduces optimization framework as an alternative to conventional CC method using augmented Lagrangian approach.

The Cluster Matrices:

• Details the mathematical structures underlying the CC ansatz focusing on fermionic creation and annihilation matrices.

Numerics:

• Compares alm-CCSD with conventional CCSD for various scenarios like helium atom and dissociation processes showing improved convergence stability.

Agreement with Quasi-Newton Method:

• Demonstrates agreement between alm-CCSD and conventional CCSD for well-known scenarios where CCSD works reliably.

Global Convergence Behavior:

• Investigates multiple roots issue in H4 model system showcasing enhanced convergence stability of alm over Newton's method.

Stats

The number K is related to the number of basis functions used for discretizing the electronic Schrödinger equation.
For more complex systems we may easily obtain matrices of size 280 × 280 which would naively require about 10^22 GB to merely store the matrix.

Quotes

"An augmented Lagrangian formalism can help address these issues without a significant performance overhead."
"We propose to minimize an energy term in addition to solving the CC equations."

Key Insights Distilled From

Augmented Lagrangian method for coupled-cluster

by Fabian M. Fa... at arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.16381.pdf

Deeper Inquiries

How does the proposed augmented Lagrangian method compare with other optimization techniques

The proposed augmented Lagrangian method offers a unique approach to optimizing the coupled cluster (CC) method in quantum chemistry. In comparison to other optimization techniques like the conventional (quasi) Newton method, the augmented Lagrangian formulation addresses issues related to convergence stability and finding the ground state solution more effectively. By introducing an energy minimization component alongside solving the CC equations, the augmented Lagrangian method provides a more stable convergence towards the ground state and is less susceptible to local minima.

What are potential drawbacks or limitations of employing an augmented Lagrangian approach in quantum chemistry

While the augmented Lagrangian approach shows promise in improving convergence properties and stability in quantum chemistry calculations, there are potential drawbacks and limitations to consider. One limitation is that implementing this method may introduce additional computational complexity due to the need for iterative optimization loops within each iteration of solving CC equations. This could potentially increase computational costs for large systems or complex molecules. Additionally, fine-tuning hyperparameters in the augmented Lagrangian framework may require expertise and careful calibration to ensure optimal performance.

How might advancements in optimization algorithms impact broader applications beyond quantum chemistry

Advancements in optimization algorithms have far-reaching implications beyond quantum chemistry applications. The development of efficient optimization techniques can benefit various fields such as machine learning, finance, logistics, healthcare analytics, and many others where complex mathematical models need to be optimized for better decision-making processes. Improved convergence methods can enhance algorithm efficiency, reduce computation time, and enable more accurate predictions or solutions across diverse domains. As optimization algorithms continue to evolve with innovations like augmented Lagrangian methods, their impact on broader applications will likely lead to enhanced problem-solving capabilities and advancements in scientific research and technological innovation.