What are Inverse Problems?

You have the answer but want to know how you get their?

Mathematically speaking this is what is called an Inverse Problem. For example, you know where an aircraft crashed but need to know the parameter settings that caused this to happen. Another example might be known pollution. You can measure pollution at a location but want to know where did it come from. These are inverse problems.

Inverse Problems could be described as problems where the answer is known, but not the question. Or where the results, or consequences are known, but not the cause.

Calculus programming solves Inverse Problems

If a solution exists, Calculus programming can find it. Inverse Problems are kind of a half-step between Simulation and Optimization. Say y = f(x; a, b, c) and you know Y at a location X but need to find your parameters a, b, & c. Calculus programming has a find statement that handles this quite nicely. A few runs should convince you on the solution or that your math model needs improving.


Inverse Problem Examples:

Problem-Solving Applications include:

CurvFit: a curve fitting program with Lorentzian, Sine, Exponential and Power series are available models to match your data.

Match-n-Freq: a Matched Filter program used to filter signals and slim pulses.

Industry Problem-Solving Descriptions include:

Electrical Filter Design: find the transfer function's poles & zeros; H(s) = Yout(s) / Yin(s).

Pulse Slimming to minimize InterSymbol Interference: via Arbitrary Equalization with Simple LC Structures to reduce errors.

Voice Coil Motor: basically an electromagnetic transducer in which a coil placed in a magnetic pole gap experiences a force proportional to the current passing through the coil.

AC Motor Design: a simulation program for A.C. motor design that was reapplied as a constrained optimization problem with 12 unknown parameters and 7 constraints.

Digitized Signal from Magnetic Recording: Magnetic recording of transitions written onto a computer disc drive may produce an isolated pulse as shown.

PharmacoKinetics: an open-two- compartment model with first order absorption into elimination from central compartment is presented here.


 
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Textbooks - Parameter Estimation 4 ODE/PDE - Signal Analysis / Spectral Estimation - Body Plasma - Solar Cell
Increasing Productivity Examples: AC Motor Design - Matched Filters - Pulse Slimming / InterSymbol Interference - Pilot (safe) Ejection - PharmacoKinetics Simulation - Poisson's (Differential) Equation - Schrodinger (Differential) Equation - BVP 4 PDE Equations - Implicit (Differential) Equations