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Dimension reduction for large-scale stochastic systems with non-zero initial states and controlled diffusion

Martin Redmann

TL;DR

The paper tackles dimension reduction for large-scale stochastic systems with non-zero initial states and controlled diffusion. It introduces two MOR strategies: a transformation-based approach yielding a priori guarantees via a new Gramian pair and Hankel singular values, and a decoupled dynamics approach that uses two Gramian pairs to derive an a posteriori HSV-based bound. Key contributions include the construction of Gramians for controlled diffusion, explicit a priori error bounds tied to truncated HSVs, and a complementary two-gramian method for decoupled control and initial-state dynamics with an HSV-based bound, all framed within a unified stochastic MOR theory. The results guarantee stable reduced models and extend deterministic inhomogeneous reduction ideas to stochastic systems with non-zero initial data, enabling scalable simulation and control design for high-dimensional SDEs.

Abstract

In this paper, we establish new strategies to reduce the dimension of large-scale controlled stochastic differential equations with non-zero initial states. The first approach transforms the original setting into a stochastic system with zero initial states. This transformation naturally leads to equations with controlled diffusion. A detailed analysis of dominant subspaces and bounds for the reduction error is provided in this controlled diffusion framework. Subsequently, we introduce a reduced system for the original framework and prove an a-priori error bound for the first ansatz. This bound involves so-called Hankel singular values that are linked to a new pair of Gramians. A second strategy is presented that is based on the idea of reducing control and initial state dynamics separately. Here, different Gramians are used in order to derive a reduced model and their relation to dominant subspaces are pointed out. We also show an a posteriori error bound for the second approach involving two types of Hankel singular values.

Dimension reduction for large-scale stochastic systems with non-zero initial states and controlled diffusion

TL;DR

The paper tackles dimension reduction for large-scale stochastic systems with non-zero initial states and controlled diffusion. It introduces two MOR strategies: a transformation-based approach yielding a priori guarantees via a new Gramian pair and Hankel singular values, and a decoupled dynamics approach that uses two Gramian pairs to derive an a posteriori HSV-based bound. Key contributions include the construction of Gramians for controlled diffusion, explicit a priori error bounds tied to truncated HSVs, and a complementary two-gramian method for decoupled control and initial-state dynamics with an HSV-based bound, all framed within a unified stochastic MOR theory. The results guarantee stable reduced models and extend deterministic inhomogeneous reduction ideas to stochastic systems with non-zero initial data, enabling scalable simulation and control design for high-dimensional SDEs.

Abstract

In this paper, we establish new strategies to reduce the dimension of large-scale controlled stochastic differential equations with non-zero initial states. The first approach transforms the original setting into a stochastic system with zero initial states. This transformation naturally leads to equations with controlled diffusion. A detailed analysis of dominant subspaces and bounds for the reduction error is provided in this controlled diffusion framework. Subsequently, we introduce a reduced system for the original framework and prove an a-priori error bound for the first ansatz. This bound involves so-called Hankel singular values that are linked to a new pair of Gramians. A second strategy is presented that is based on the idea of reducing control and initial state dynamics separately. Here, different Gramians are used in order to derive a reduced model and their relation to dominant subspaces are pointed out. We also show an a posteriori error bound for the second approach involving two types of Hankel singular values.
Paper Structure (12 sections, 11 theorems, 79 equations)

This paper contains 12 sections, 11 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Setting, notation and goal
  3. First approach
  4. Second approach
  5. Model reduction for \ref{['original_system']} with $x_0=0$
  6. Gramians and dominant subspaces
  7. Balancing transformation and reduced order model
  8. Model reduction for \ref{['original_system']} with $u\equiv 0$
  9. Model reduction for general systems \ref{['original_system']}
  10. The approach according to Subsection \ref{['approach1']}
  11. The approach according to Subsection \ref{['approach2']}
  12. Supporting lemma

Key Result

Proposition 3.2

Let the solution of the uncontrolled equation stochstatenew be mean square asymptotically stable. Then, gramP has a solution $P>0$ with $U(P^{-1})>0$.

Theorems & Definitions (30)

  • Remark 2.1
  • Remark 2.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Definition 3.5
  • Proposition 3.6
  • ...and 20 more