Explore the fascinating world of working backwards from effects to causes
An inverse problem is when we have observed data and need to determine the model parameters that produced it. Unlike forward problems where we know the parameters and predict the observations, inverse problems require us to work backwards from data to parameters.
Example: Drop height/angle → Physics → Landing position
Known: Model parameters (θ)
Find: Observed data (d)
Example: Landing position → Physics → Drop height/angle
Known: Observed data (d)
Find: Model parameters (θ)
d: Observed data (measurements)
G: Forward model (physics/process)
θ: Model parameters (unknowns)
ε: Measurement noise/errors
θ_est: Estimated parameters
||G(θ) - d||²: Data misfit (how well parameters explain data)
λ: Regularization parameter
R(θ): Regularization term (prior knowledge/constraints)
Data: X-ray measurements
Parameters: Tissue densities
Data: Seismic waves
Parameters: Rock velocities
Data: 2D images
Parameters: 3D structure
Data: Temperature now
Parameters: Initial temp
Observe how we estimate model parameters (signal frequency and amplitude) from noisy observed data
Frequency: 0.02, Amplitude: 50
Calculating...
Understand how model parameters produce observed data. Develop mathematical model: d = G(θ) where d is data and θ are parameters
Set up equation to find parameters θ given data d. Often involves solving: θ = G⁻¹(d)
Handle non-uniqueness (multiple parameters fit data), instability (small data changes cause large parameter changes)
Add constraints or prior knowledge about parameters to stabilize the solution
Check if estimated parameters produce observed data when run through forward model
Non-uniqueness: Multiple causes can produce the same effect
Noise sensitivity: Small measurement errors cause large solution errors
Computational cost: Solutions often require iterative algorithms
Incomplete data: Limited measurements constrain reconstruction quality
Minimize ||Ax - b||²
Add penalty term λ||x||²
Use probability and priors
Gradual refinement