betekintés - Computational Complexity - # High-dimensional Kolmogorov backward equation

Efficient Tensor Network Approach for Solving High-Dimensional Kolmogorov Backward Equations

Q: What are the limitations of the functional hierarchical tensor representation, and how can it be further improved to capture more complex correlation structures in high-dimensional problems

The functional hierarchical tensor (FHT) representation, while effective for capturing correlations in high-dimensional problems, has certain limitations that can be addressed for further improvement: Limited Representation Power: The FHT ansatz may struggle to capture highly nonlinear or complex correlation structures present in some high-dimensional problems. To improve this, more flexible tensor network architectures can be explored, allowing for a richer representation of correlations. Scalability: As the dimensionality of the problem increases, the scalability of the FHT representation may become a challenge. Enhancements in the hierarchical tensor structure and sketching algorithms can help improve the scalability of the FHT ansatz for larger dimensions. Handling Irregular Domains: The FHT representation may face difficulties in capturing correlations in irregularly shaped domains. Developing techniques to adapt the FHT ansatz to irregular domains or incorporating domain-specific information can enhance its applicability in such cases. Incorporating Non-Gaussian Distributions: The FHT representation is primarily designed for Gaussian distributions. Extending the approach to handle non-Gaussian distributions can broaden its applicability to a wider range of probability distributions encountered in high-dimensional problems. By addressing these limitations through advancements in tensor network structures, sketching algorithms, and adaptability to different types of distributions and domains, the FHT representation can be further improved to capture more complex correlation structures in high-dimensional problems.

Q: Given the connection between the Kolmogorov backward equation and stochastic optimal control, how can the tensor network approach be adapted to solve high-dimensional stochastic control problems

The tensor network approach can be adapted to solve high-dimensional stochastic control problems by integrating the principles of stochastic optimal control theory with the hierarchical tensor structure and sketching algorithm. Here's how the approach can be tailored for stochastic control problems: Incorporating Control Variables: Extend the tensor network representation to include control variables that influence the system dynamics. By incorporating the control inputs into the tensor network structure, the approach can model the impact of control actions on the system evolution. Dynamic Programming Formulation: Adapt the tensor network approach to solve the Bellman equation associated with stochastic control problems. By formulating the dynamic programming recursion in terms of the Markov operator and using the FHT ansatz for value functions, the method can efficiently compute optimal control policies. Policy Iteration: Implement a policy iteration scheme within the tensor network framework to iteratively update the control policy based on the estimated value functions. By leveraging the hierarchical tensor structure for policy evaluation and improvement, the approach can converge to optimal control strategies in high-dimensional systems. Handling Constraints: Modify the tensor network approach to incorporate constraints on control inputs or system states. By integrating constraint handling mechanisms into the tensor network representation, the method can address feasibility and optimality considerations in stochastic control problems. By customizing the tensor network approach to accommodate the unique characteristics of stochastic control problems, such as control variables, dynamic programming formulations, policy iteration, and constraint handling, the method can effectively solve high-dimensional stochastic control problems with improved efficiency and accuracy.

Alapfogalmak

A tensor network approach is introduced to efficiently solve high-dimensional Kolmogorov backward equations by directly approximating the Markov operator.

Kivonat

The key highlights and insights of the content are:

The authors focus on solving the high-dimensional Kolmogorov backward equation, which arises from the discretization of an infinite-dimensional Markov chain into a large finite dimension.
They propose a tensor network approach to approximate the Markov operator, which is the key to solving the Kolmogorov backward equation. The Markov operator is obtained under a functional hierarchical tensor (FHT) ansatz using a hierarchical sketching algorithm.
When the terminal condition admits an FHT ansatz, the proposed operator-based approach allows efficient computation of the PDE solution through tensor network contraction. It also provides an efficient way to solve the Kolmogorov forward equation when the initial distribution is in an FHT ansatz.
The authors apply the proposed approach successfully to two challenging time-dependent Ginzburg-Landau models with hundreds of variables, demonstrating its effectiveness in overcoming the curse of dimensionality.
The tensor network structure and hierarchical sketching algorithm are crucial components that enable the efficient representation and computation of the high-dimensional Markov operator.

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arxiv.org

Statisztikák

The authors apply the proposed approach to two time-dependent Ginzburg-Landau models, one in 1D with 128 dimensions and one in 2D with hundreds of dimensions.

Idézetek

"Solving high-dimensional partial differential equations necessitates methods free of exponential scaling in the dimension of the problem."
"The learned operator allows an efficient way to obtain the approximate value of u(x, T −t) for arbitrary point x ∈Ωand terminal condition f."
"Using sophisticated numerical techniques, we are able to solve the Ginzburg-Landau model with d in the hundreds."

Főbb Kivonatok

Solving high-dimensional Kolmogorov backward equations with operator-valued functional hierarchical tensor for Markov operators

by Xun Tang,Lea... : arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.08823.pdf

Solving high-dimensional Kolmogorov backward equations with operator-valued functional hierarchical tensor for Markov operators

Mélyebb kérdések

How can the proposed tensor network approach be extended to solve other types of high-dimensional partial differential equations beyond the Kolmogorov backward equation

The proposed tensor network approach for solving the Kolmogorov backward equation can be extended to solve other types of high-dimensional partial differential equations by adapting the hierarchical tensor structure and sketching algorithm to the specific characteristics of the new equations. Here are some ways to extend the approach:

Different PDE Formulations: The tensor network approach can be applied to various types of PDEs, such as the heat equation, wave equation, or diffusion equation. By adjusting the basis functions, bipartition structures, and sketching techniques to suit the specific properties of these equations, the method can be extended to solve a wide range of high-dimensional PDEs.

Incorporating Different Boundary Conditions: The approach can be modified to handle different boundary conditions, initial conditions, and domain geometries. By customizing the tensor network structure and sketching algorithm to accommodate these variations, the method can effectively solve PDEs with diverse boundary and initial conditions.

Incorporating Different Dynamics: The tensor network approach can be adapted to solve PDEs arising from various dynamic systems beyond Langevin dynamics. By adjusting the representation of the Markov operator and the sketching algorithm to match the dynamics of the specific system, the method can be extended to solve high-dimensional PDEs in different dynamic settings.

Incorporating Nonlinear Terms: To handle PDEs with nonlinear terms, the tensor network approach can be enhanced to capture the nonlinear correlations and interactions present in the system. This may involve developing more sophisticated tensor network structures and sketching techniques to accurately represent the nonlinear dynamics in the equations.

By customizing the tensor network approach to the specific requirements of different types of high-dimensional PDEs, it can be effectively extended to solve a wide range of complex partial differential equations beyond the Kolmogorov backward equation.

What are the limitations of the functional hierarchical tensor representation, and how can it be further improved to capture more complex correlation structures in high-dimensional problems

The functional hierarchical tensor (FHT) representation, while effective for capturing correlations in high-dimensional problems, has certain limitations that can be addressed for further improvement:

Limited Representation Power: The FHT ansatz may struggle to capture highly nonlinear or complex correlation structures present in some high-dimensional problems. To improve this, more flexible tensor network architectures can be explored, allowing for a richer representation of correlations.

Scalability: As the dimensionality of the problem increases, the scalability of the FHT representation may become a challenge. Enhancements in the hierarchical tensor structure and sketching algorithms can help improve the scalability of the FHT ansatz for larger dimensions.

Handling Irregular Domains: The FHT representation may face difficulties in capturing correlations in irregularly shaped domains. Developing techniques to adapt the FHT ansatz to irregular domains or incorporating domain-specific information can enhance its applicability in such cases.

Incorporating Non-Gaussian Distributions: The FHT representation is primarily designed for Gaussian distributions. Extending the approach to handle non-Gaussian distributions can broaden its applicability to a wider range of probability distributions encountered in high-dimensional problems.

By addressing these limitations through advancements in tensor network structures, sketching algorithms, and adaptability to different types of distributions and domains, the FHT representation can be further improved to capture more complex correlation structures in high-dimensional problems.

Given the connection between the Kolmogorov backward equation and stochastic optimal control, how can the tensor network approach be adapted to solve high-dimensional stochastic control problems

The tensor network approach can be adapted to solve high-dimensional stochastic control problems by integrating the principles of stochastic optimal control theory with the hierarchical tensor structure and sketching algorithm. Here's how the approach can be tailored for stochastic control problems:

Incorporating Control Variables: Extend the tensor network representation to include control variables that influence the system dynamics. By incorporating the control inputs into the tensor network structure, the approach can model the impact of control actions on the system evolution.

Dynamic Programming Formulation: Adapt the tensor network approach to solve the Bellman equation associated with stochastic control problems. By formulating the dynamic programming recursion in terms of the Markov operator and using the FHT ansatz for value functions, the method can efficiently compute optimal control policies.

Policy Iteration: Implement a policy iteration scheme within the tensor network framework to iteratively update the control policy based on the estimated value functions. By leveraging the hierarchical tensor structure for policy evaluation and improvement, the approach can converge to optimal control strategies in high-dimensional systems.

Handling Constraints: Modify the tensor network approach to incorporate constraints on control inputs or system states. By integrating constraint handling mechanisms into the tensor network representation, the method can address feasibility and optimality considerations in stochastic control problems.

By customizing the tensor network approach to accommodate the unique characteristics of stochastic control problems, such as control variables, dynamic programming formulations, policy iteration, and constraint handling, the method can effectively solve high-dimensional stochastic control problems with improved efficiency and accuracy.