Implicit Runge-Kutta Method for Uncertainty Propagation
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Solution Overview
Problem
Traditional algorithms for propagating an ensemble of states in uncertainty propagation require significant computational resources, as they solve each initial value problem independently, failing to exploit proximity of initial conditions to reduce computational cost.
Innovation Solution
The implementation of an implicit-Runge-Kutta-based method that records the trajectory of one initial value problem and uses it to solve subsequent problems with reduced computational effort, leveraging proximity to enhance convergence and avoid re-estimating errors.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If traditional algorithms solve each initial value problem independently, then each problem is solved with full computational resources, but the total computational cost of propagating uncertainty increases significantly
Solution Approach 1:
The patent performs preliminary computation by solving one initial value problem completely first, storing its trajectory and intermediate results. This preliminary solution is then reused as a foundation for solving subsequent problems with similar initial conditions, avoiding redundant computations and significantly reducing total computational cost while maintaining accuracy.
Solution Approach 2:
The patent creates a copy of the trajectory and computational results from the first solved problem. This copied information is then adapted and reused for solving subsequent problems with proximity initial conditions, eliminating the need to recompute identical or similar portions and thereby improving computational efficiency.
2Productivity
If traditional algorithms do not exploit proximity of initial conditions, then implementation is simpler, but computational resources are wasted on redundant calculations
Solution Approach 1:
The patent applies local quality by treating problems with proximity initial conditions differently from completely independent problems. When initial conditions are close, the algorithm exploits this local similarity by reusing computational results, while maintaining the ability to handle completely independent problems through full computation. This localized optimization improves efficiency without requiring complete algorithmic restructuring.
Data Source
AI summary
Embodiments of the present invention are directed to reducing the computational cost of propagating the uncertainty in the state of an object, which requires the numerical solution of an initial value problem (IVP) ensemble, by means of a computer. The present invention uses an implicit-Runge-Kutta-based (IRK) method that exploits the proximity of the initial conditions within the IVP ensemble. More specifically, one of the IVPs can be solved over a given time span consisting of one or more time steps with an IRK method, and the resulting trajectory can be recorded. The system of equations that arises on each time step of the IRK method can be solved using various iterative methods. The remaining IVPs can be solved over the same timespan using the same IRK method and time steps, together with the recorded trajectory from the first propagation.