MA3J4 - Modelling with PDE

Mathematical Model

A simplified description, especially a mathematical one, of a system or process, to assist calculations or predictions

Microscopic vs Macroscopic

• (typically ODEs), the behaviour of each object is described explicitly

• describe the behaviour of the distribution of the objects (usually with respect to space and time, i.e., PDEs).

Continuous vs Discrete

• involve variables that depend continuously on time or/and space, and are described by either ordinary or partial differential equations accordingly

• track time steps or/and grid points and are expressed through difference equations

Deterministic vs Probabilistic (Stochastic)

• every set of variables is completely determined by the parameters of the model

• randomness is present and the state of a variable corresponds to a probability distribution rather than a specific value

Abstract model

A map of the general implicit form M(x,p) = 0

Where M : X × P → Y is a map between Banach spaces. We refer to x ∈ X as the variables and p ∈ P the parameters or data of the model.

Direct problem

If M and p are given and x is the unknown;

Inverse problem

If M and x are given and the data p is the unknown

Identification problem

if x and p are given, but M is unknown

Well-posedness (Hadamard)

The direct problem is ___ for data P ⊆ P if:

1. for every p ∈ P, a solution x ∈ X exists

2. for every p ∈ P, the solution x ∈ X is unique

3. the solution depends continuously on the data on P.

Admissible data

We call____ a set P ⊆ P for which the model satisfies the first two conditions. (existence and uniqueness)

Depends continuously

Let P be admissible. Then the solution ___ on the data on P if for every p ∈ P and ϵ > 0 there exists δ > 0 such that for any p˜ ∈ P ∥p˜− p∥_P < δ ⇒ ∥xp˜ − xp∥_X < ϵ .

Condition Number

Let P be admissible and the map p→x_p be Lipschitz continuous on P. For any p∈P, the _____ is defined as K(p) = lim sup_p˜→p ∥xp˜ − xp∥_X /∥p˜− p∥_P