An inverse problem approach to approximating sensor data in cyber physical systems

This paper presents a new matrix algebraic approach to the direct solution of inverse boundary value problems (IBVP). The synthesis of admissible functions and differentiating matrices is given particular attention. All the necessary mathematical elements are derived from basic principles. The method yields a linear operator for the solution of IBVPs. A single matrix multiplication is required at run-time to determine the solution. The number of FLOPS required is constant and known a-priory making the solution suitable for use in embedded real-time systems. The concept of discrete basis function design is introduced for the first time. The method enables the design of special discrete basis functions which yield optimal noise performance and numerical efficiency for specific tasks. All the methods are also verified in a laboratory test system and compared with results from an optical reference measurement.