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Proximal Gradient-based Low Rank Tensor Decomposition for State Dependent Riccati Equation

This paper proposes a proximal gradient-based low-rank tensor decomposition method using sparse optimization to derive reduced-order models from large-scale PDE control systems, enabling the efficient solution of reduced state-dependent Riccati equations.

Original authors: Jiahua Jiang, Carmeliza Navasca

Published 2026-05-22

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Jiahua Jiang, Carmeliza Navasca

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to steer a massive, chaotic ship through a storm. The ship is so big and the ocean so complex that calculating the perfect steering angle for every single wave in real-time would take a supercomputer years to figure out. This is the problem the authors are tackling: how to control huge, complex systems (like fluid flows or chemical reactions) without getting bogged down by the sheer amount of math required.

Here is a simple breakdown of their solution, using everyday analogies:

1. The Problem: The "Too Big to Handle" Ship

The paper deals with systems that have thousands of moving parts (called "dimensions"). To control them, you usually need to solve a very difficult math puzzle called the Riccati Equation.

The Analogy: Imagine trying to navigate that massive ship by calculating the exact physics of every single drop of water around it. The computer would crash because there are too many drops to count. The math gets so expensive it's "prohibitively expensive," meaning it's practically impossible to do in real-time.

2. The Solution: Taking a "Snapshot" and Finding the Pattern

Instead of looking at every drop of water, the authors suggest taking a series of "snapshots" of the ship's movement under different conditions. They stack these snapshots into a giant 3D block of data, which they call a Tensor.

The Analogy: Think of a Tensor like a thick photo album where every page is a different moment in time, and every photo shows the ship from a slightly different angle.

3. The Magic Trick: The "Proximal Gradient" Shrink-Ray

The authors use a special mathematical tool called Proximal Gradient-based Low Rank Tensor Decomposition. This sounds scary, but here is what it does:

The Analogy: Imagine your photo album is full of noise, static, and redundant pictures. You want to find the essential story.
- Low Rank Decomposition: This is like realizing that even though you have 1,000 photos, the ship is actually just doing three main things: rolling, pitching, and yawing. You can describe the whole album by just describing those three movements.
- Sparse Optimization (The Shrink-Ray): The authors use a "shrink-ray" (mathematically called a regularization parameter) to force the math to ignore the tiny, unimportant details. It asks, "What is the smallest number of movements needed to explain the ship's behavior?"
- The Result: They find a tiny, simplified version of the ship's behavior (a "Reduced Order Model") that captures the essence without the clutter.

4. The Control: Steering the Simplified Ship

Once they have this tiny, simplified model, they solve the steering puzzle (the Riccati Equation) on this small version instead of the giant one.

The Analogy: Instead of calculating the physics of the whole ocean, you calculate the physics of just the three main movements. It's like steering a toy boat instead of an aircraft carrier.
The Outcome: Because the math is now tiny, the computer can calculate the steering angle almost instantly. The paper claims this new method stabilizes the system (steers the ship to safety) much faster than the old, heavy methods.

5. The Proof: The Race

The authors tested this on a famous math model called the Allen-Cahn equation (which describes how patterns form in materials, like how ice crystals grow).

The Race: They compared their "Shrink-Ray" method against the standard, heavy method.
The Winner: Their method won by a landslide.
- Speed: It stabilized the system in a fraction of the time.
- Cost: The "fuel cost" (computational power) was nearly zero compared to the full model. In their data, the cost was so small it looked like a number with 29 zeros after the decimal point.

Summary

The paper proposes a way to take a massive, complicated control problem, compress it into a tiny, essential "core" using a smart mathematical filter (Proximal Gradient), and then solve the steering problem on that tiny core. The result is a control system that is incredibly fast, cheap to run, and highly effective at stabilizing complex systems.

What the paper does NOT claim:

It does not claim this works on biological systems, medical devices, or AI robots yet (though it mentions AI as a general field).
It does not promise to solve every control problem, only those that can be represented by the specific math models they tested (like the Allen-Cahn equation).
It focuses strictly on the math and the computer simulation results, not on real-world hardware deployment.

Technical Summary: Proximal Gradient-based Low Rank Tensor Decomposition for State Dependent Riccati Equation

Problem Statement
The paper addresses optimal control problems arising from partial differential equations (PDEs) that result in large-scale discrete dimensional control systems. The standard approach to finding nonlinear control feedback strategies involves solving the Hamilton-Jacobi-Bellman (HJB) equations, which is computationally prohibitive for high-dimensional systems. While the State-Dependent Riccati Equation (SDRE) offers an alternative formulation for constructing nonlinear control feedback, solving the SDRE for full-order models remains expensive, with a computational complexity of $O(N_t N_x^3)$ , where $N_t$ is the number of time steps and $N_x$ is the state dimension. The authors propose a method to reduce the order of these control systems to enable efficient solution of the SDRE while maintaining stability.

Methodology
The proposed approach utilizes tensor computation to derive reduced-order models (ROMs) from snapshot data of the system's solution profiles. The methodology proceeds through the following stages:

Tensor Data Construction: The authors construct a third-order tensor dataset from the solution profiles of the uncontrolled state dynamics (specifically the Allen-Cahn equation) under various control parameters. The tensor dimensions correspond to spatial discretization, time discretization, and control parameter values.
Low-Rank Tensor Decomposition: To reduce the dimensionality, the authors employ Canonical Polyadic (CP) decomposition to approximate the tensor data. The goal is to find a low-rank representation $T \approx \sum_{r=1}^R \alpha_r x_r \circ y_r \circ z_r$ .
Sparse Optimization for Rank Determination: A key component of the method is the use of a Proximal Gradient method for Sparsity (PGS) to determine the optimal tensor rank $R$ . This is formulated as a sparse optimization problem minimizing the reconstruction error plus an $\ell_1$ -norm regularization term on the coefficients $\alpha$ . To solve this, the authors utilize a flexible hybrid method combining the Proximal Gradient approach with Golub-Kahan bidiagonalization, replacing the $\ell_1$ penalty with a sequence of $\ell_2$ penalties to handle parameter tuning practically.
Reduced Order Model (ROM) Construction: Once the rank $R$ is identified, the factor matrices ( $X, Y, Z$ ) are refined using an Alternating Least-Squares (ALS) algorithm. The reduced state dynamics are derived by projecting the full-order system onto the subspace spanned by the left singular vectors of the product of the factor matrices. This yields reduced matrices $A_{red}$ and $B_{red}$ .
Control Synthesis: The reduced-order optimal control problem is solved using the SDRE framework. Because the dimension of the reduced system is significantly smaller than the full system, the SDRE can be solved efficiently at each time step.

Key Contributions
The paper outlines three primary contributions:

Application of Proximal Gradient to Control Data: The authors apply the proximal gradient method to nonlinear control datasets to determine a sharp tensor rank, utilizing flexible hybrid methods for regularization.
Rank-Dictated Dimensionality Reduction: The determined tensor rank $R$ directly dictates the reduced order dimension of the state system, ensuring the reduction is data-driven rather than heuristic.
Stabilizing Control Feedback: The method generates algorithms that find reduced basis elements to construct control feedback capable of stabilizing the state dynamics.

Numerical Results
The authors tested the method on the Allen-Cahn equation with a full-order dimension of $d=101$ .

Rank Determination: The PGS algorithm identified a tensor rank of $R=2$ , significantly smaller than the state dimension.
Stabilization Speed: In numerical experiments, the control feedback derived from the PGS-reduced model converged to zero faster than those from the full model or models using only ALS. Specifically, the PGS-controlled state converged at $t=0.1531$ , while other dynamics had not yet converged.
Computational Efficiency: The relative CPU time for the reduced model using PGS was approximately $1.965 \times 10^{-29}$ of the full model cost ( $J_{full}$ ). While there is an offline cost to determine the rank, the online computational complexity for the reduced SDRE drops from $O(N_t N_x^3)$ to $O(n_t R^3)$ , where $n_t$ is the time steps required for the reduced system to stabilize.

Significance and Claims
The paper claims that its proposed method offers significant advantages for large-scale optimal control problems:

Fast Stabilization: The approximated control feedback exhibits rapid stabilization of the control system.
Computational Savings: The online computational complexity is substantially lower than that of the standard state-dependent Riccati equation.
Cost Efficiency: The objective cost function associated with the control feedback from the reduced models is a small fraction of the cost of the full model.
Ease of Implementation: Tensor data structures are easily constructed from frontal slices of solution profiles, making the approach practical for implementation.

The authors conclude that by combining CP decomposition with proximal gradient sparsity and Golub-Kahan bidiagonalization, they can effectively estimate low ranks that dictate state dimension reduction, leading to fast control feedback stabilization and huge computational savings.