We'll demonstrate inverse problems using 1D steady-state heat conduction:

with boundary conditions: \( T(0) = T_0 \), \( T(1) = T_1 \)

Physical meaning:

• \( T(x) \) = temperature distribution
• \( k \) = thermal diffusivity (unknown parameter we want to find)
• \( f(x) \) = heat source term (known)
• Boundary conditions specify temperatures at the ends

Why This Problem?

1. Physical relevance: Material property identification is crucial in engineering
2. Mathematical simplicity: 1D steady-state allows clear visualization
3. Exact solution available: We can verify our parameter recovery
4. Well-studied: Benchmark for inverse methods

The Challenge

Scenario: You're an engineer testing a new material. You can:

• Apply known heat sources \( f(x) \)
• Control boundary temperatures
• Measure temperature at a few points
• Cannot directly measure thermal diffusivity \( k \)

Goal: Determine \( k \) from sparse, noisy temperature measurements