Abstract Geometrical Space for System Approximation
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Solution Overview
Problem
Existing system design and optimization methods face challenges in dealing with complex, nonlinear interdependencies among parameters, leading to inefficient discrete approximation of systems like filters and controllers, as naive approaches fail to account for the geometric structure of these systems.
Innovation Solution
The method involves embedding systems into an abstract geometrical space with a metric, allowing for the determination of a nearest discrete point that represents a substantially optimal discrete approximation, using techniques such as Riemannian geometry and fair sampling to reduce computational complexity.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of operation
If naive optimization approaches are used that assume linear proximity of parameter values corresponds to linear proximity of system behaviors, then the optimization process is simple, but the approximation quality is poor
Solution Approach 1:
The patent transforms the optimization problem from the original parameter space to a geometric space where systems are represented as points with defined metrics. This dimensional transformation allows the use of geometric concepts (distance, neighbors, paths) to navigate the design space, resolving the contradiction by providing both systematic structure and improved approximation quality without naive linearity assumptions
Solution Approach 2:
The patent changes the fundamental parameters of the optimization approach by introducing a metric space framework where distance between systems is defined geometrically rather than through simple parameter differences. This parameter change enables accurate discrete approximation while maintaining computational tractability through geometric algorithms
2Manufacturing precision
If systematic design methods based on optimization are used to handle complex interdependencies, then the approximation quality improves, but the computational complexity increases significantly
Solution Approach 1:
The patent segments the complex optimization problem into geometric components: defining a metric space, identifying discrete grid points, computing geodesic paths, and finding nearest neighbors. This segmentation transforms an intractable global optimization problem into a series of manageable geometric computations, reducing computational complexity while maintaining approximation quality
Solution Approach 2:
The patent replaces traditional optimization mechanics (gradient descent, iterative solving) with geometric mechanics (metric definitions, path finding, neighbor search). This substitution leverages the inherent geometric structure of the system space to achieve efficient computation without the iterative complexity of conventional optimization methods
3Ease of operation
If discrete coefficient filter design is achieved by rounding coefficients of infinite-precision design, then the process is simple, but the frequency response quality is insufficient
Solution Approach 1:
The patent performs preliminary geometric preparation by defining the metric space and discrete grid structure before attempting to find the optimal discrete filter. This preliminary action creates a framework that guides the search for the nearest discrete point, ensuring both simplicity and frequency response accuracy by avoiding post-hoc rounding of infinite-precision designs
Data Source
AI summary
System and method for approximating a system. A multi-parameter representation of a family of systems is stored. An embedding of the family into an abstract geometrical continuous space with a metric and defined by the parameters is determined. Coordinates of the space specify values for the parameters of systems of the family. The space includes a grid of points representing respective discrete approximations of the systems. A first point corresponding to a desired instance of a system is determined. The first point's coordinates specify values for the parameters of the instance. The space is sampled using a mapping of a well-distributed point set from a Euclidean space of the parameters to the abstract space. A nearest discrete point to the first point is determined which specifies values for parameters for an optimal discrete approximation of the desired instance, which are useable to implement the discrete approximation of the desired instance.