Forward Problem, Overdetermined Problems, Penalty Functional, Adjoint-Representer and Representer Equations.

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Module 1

Module 2

Linear One-Dimensional Problem

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Print Lesson

Topic 1: The Forward Problem

Discrete Analogue

The Forward Problem

Well Posedness

Existence

Uniqueness

Well Posedness Proved!

Topic 2: Overdetermined Problems

Overdetermined Problems

Discrete Analogue

The Data

Discrete Analogue

Failure of the Forward Solution

Topic 3: Penalty Functional - Least-squares Fitting

Discrete Analogue

Penalty Functional - Least-squares Fitting

Discrete Analogue

The Calculus of Variations

Derivation of the First Variation the Long Way

Derivation of the First Variation the Short Way

Discrete Analogue

Euler-Lagrange Equations: Integration by Parts

Discrete Analogue

Untangling to Obtain Euler-Lagrange (EL) Equations

Topic 4: Adjoint-Representer and Representer Equations

Discrete Analogue

Adjoint and Representer Equations

Topic 5: Determining the Representer Coefficients

Determining the Representer Coefficients

Determining an Explicit Expression for the Optimal Solution

Topic 6: Naive Summary of Steps

Naive Summary of Steps

Topic 7: Regularity of the Inverse Estimate: Realizability

Regularity of the Inverse Estimate - Realizability

Nondiagonal Weighting: Kernal Inverses of Weights

Minimize as Before

Overview of Convolutions

Discrete Analogue

Space Convolutions

Discrete Analogue

Time Convolutions

Discrete Analogue

Space and Time Convolutions

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