Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids Partial Differential Equations II CS 205A: Mathematical Methods for Robotics, Vision, and Graphics Justin Solomon CS 205A: Mathematical Methods Partial Differential Equations II 1 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Partial Differential Equations II
CS 205A:Mathematical Methods for Robotics, Vision, and Graphics
Justin Solomon
CS 205A: Mathematical Methods Partial Differential Equations II 1 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Almost Done!
I Homework 7: 12/2 (two days late!)
I Homework 8: 12/9 (optional)
I Section: 12/6 (final review)
I Final exam: 12/12, 12:15pm (Gates B03)
Go to office hours!
CS 205A: Mathematical Methods Partial Differential Equations II 2 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Course Reviews
On Axess!Additional comments: [email protected]
CS 205A: Mathematical Methods Partial Differential Equations II 3 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Request for Help
CS 205A notesyour help!7−−−−−−−→ Textbook
I Review text
I Write reference implementations
I Solidify your CS205A knowledge
CS 205A: Mathematical Methods Partial Differential Equations II 4 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Final Exam
I CumulativeI Similar format to midterm
I Two sheets of notes
CS 205A: Mathematical Methods Partial Differential Equations II 5 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
This Week
Couple relationships between derivatives.
I Pressure gradient determining fluid flow
I Image operators using x and y derivatives
Partial Differential Equations (PDE)
CS 205A: Mathematical Methods Partial Differential Equations II 6 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Boundary Value Problems
I Dirichlet conditions: Value of f(~x) on ∂Ω
I Neumann conditions: Derivatives of f(~x) on ∂Ω
I Mixed or Robin conditions: Combination
CS 205A: Mathematical Methods Partial Differential Equations II 7 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Second-Order Model Equation
∑ij
aij∂f
∂xi∂xj+∑i
bi∂f
∂xi+ cf = 0
(∇>A∇ +∇ ·~b + c)f = 0
CS 205A: Mathematical Methods Partial Differential Equations II 8 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Classification of Second-Order PDE
(∇>A∇+∇ ·~b+ c)f = 0
I If A is positive or negative definite, system is elliptic.
I If A is positive or negative semidefinite, the systemis parabolic.
I If A has only one eigenvalue of different sign fromthe rest, the system is hyperbolic.
I If A satisfies none of the criteria, the system isultrahyperbolic.
CS 205A: Mathematical Methods Partial Differential Equations II 9 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Derivative Operator Matrix
h2 ~w = L1~y
−2 11 −2 1
1 −2 1. . . . . . . . .
1 −2 11 −2
Dirichlet
CS 205A: Mathematical Methods Partial Differential Equations II 10 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
What About First Derivative?
I Potential for asymmetry at boundary
I Centered differences: Fencepost problem
I Possible resolution: Imitate leapfrog
CS 205A: Mathematical Methods Partial Differential Equations II 11 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Fencepost Problem
CS 205A: Mathematical Methods Partial Differential Equations II 12 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Big Idea
Derivatives : Functions :: Matrices : Vectors
CS 205A: Mathematical Methods Partial Differential Equations II 13 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Elliptic PDE
Lf = g 7−→ L~y = ~b
Example: Laplace’s equation on a line
CS 205A: Mathematical Methods Partial Differential Equations II 14 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Elliptic PDE
Lf = g 7−→ L~y = ~bExample: Laplace’s equation on a line
CS 205A: Mathematical Methods Partial Differential Equations II 14 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Common Theme
Elliptic PDE 7→ Positive definite matrix
L = −D>D,D =
1−1 1−1 1
. . . . . .−1 1−1
Review: Name two ways to solve.
CS 205A: Mathematical Methods Partial Differential Equations II 15 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Common Theme
Elliptic PDE 7→ Positive definite matrix
L = −D>D,D =
1−1 1−1 1
. . . . . .−1 1−1
Review: Name two ways to solve.
CS 205A: Mathematical Methods Partial Differential Equations II 15 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Common Theme
Elliptic PDE 7→ Positive definite matrix
L = −D>D,D =
1−1 1−1 1
. . . . . .−1 1−1
Review: Name two ways to solve.
CS 205A: Mathematical Methods Partial Differential Equations II 15 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Time Dependence
Choice:
1. Treat t separate from ~x (“semidiscrete”)
2. Treat all variables democratically
(“fully discrete”)
CS 205A: Mathematical Methods Partial Differential Equations II 16 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Semidiscrete Heat Equation
ft = fxx
7−→ ft = Lf
Stability for elliptic spatialoperator (parabolic PDE)
CS 205A: Mathematical Methods Partial Differential Equations II 17 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Semidiscrete Heat Equation
ft = fxx 7−→ ft = Lf
Stability for elliptic spatialoperator (parabolic PDE)
CS 205A: Mathematical Methods Partial Differential Equations II 17 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Semidiscrete Heat Equation
ft = fxx 7−→ ft = Lf
Stability for elliptic spatialoperator (parabolic PDE)
CS 205A: Mathematical Methods Partial Differential Equations II 17 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Semidiscrete Time Stepping
Left with a multivariable ODE problem!
I Forward/backward Euler, RK, and friends
I Implicit vs. explicit (vs. symplectic)
I Alternative: Eigenvector methods
(low-frequency approximation)
CS 205A: Mathematical Methods Partial Differential Equations II 18 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Semidiscrete Time Stepping
Left with a multivariable ODE problem!
I Forward/backward Euler, RK, and friends
I Implicit vs. explicit (vs. symplectic)
I Alternative: Eigenvector methods
(low-frequency approximation)
CS 205A: Mathematical Methods Partial Differential Equations II 18 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Semidiscrete Time Stepping
Left with a multivariable ODE problem!
I Forward/backward Euler, RK, and friends
I Implicit vs. explicit (vs. symplectic)
I Alternative: Eigenvector methods
(low-frequency approximation)
CS 205A: Mathematical Methods Partial Differential Equations II 18 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Fully Discrete PDE
I Discretize ~x and t simultaneously
I Can create larger linear algebra problems
I Philosophical point: What is “fully” discrete?
CS 205A: Mathematical Methods Partial Differential Equations II 19 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Gradient Domain Inpainting
http://groups.csail.mit.edu/graphics/classes/CompPhoto06/html/lecturenotes/10_Gradient.pdf
CS 205A: Mathematical Methods Partial Differential Equations II 20 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Gradient Domain
Pipeline for image I(x, y):
1. Compute gradient: ~v(x, y) = ∇I(x, y)
2. Edit: ~v 7→ ~v′
3. Reconstruct: ∇g ?= ~v′
CS 205A: Mathematical Methods Partial Differential Equations II 21 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Gradient Domain Reconstruction
ming
∫Ω
‖∇g − ~v′‖22 dA
7→ ∇2g = ∇ · ~v′Elliptic!
CS 205A: Mathematical Methods Partial Differential Equations II 22 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Gradient Domain Reconstruction
ming
∫Ω
‖∇g − ~v′‖22 dA
7→ ∇2g = ∇ · ~v′Elliptic!
CS 205A: Mathematical Methods Partial Differential Equations II 22 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Incompressible Navier-Stokes
ρ
(∂~v
∂t+ ~v · ∇~v
)= −∇p+ µ∇2~v + ~f
I t ∈ [0,∞): Time
I ~v(t) : Ω→ R3: Velocity
I ρ(t) : Ω→ R: Density
I p(t) : Ω→ R: Pressure
I ~f(t) : Ω→ R3: External forces (e.g. gravity)
CS 205A: Mathematical Methods Partial Differential Equations II 23 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Lagrangian vs. Eulerian
I Lagrangian: Track parcels of fluid
I Eulerian: Fluid flows past a point in space
CS 205A: Mathematical Methods Partial Differential Equations II 24 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Marker-and-Cell (MAC) Grid
http://students.cs.tamu.edu/hrg/image/MAC.bmp
CS 205A: Mathematical Methods Partial Differential Equations II 25 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Splitting for Incompressible Flow
∇ · ~u = 0 (divergence-free)
ρt + ~u · ∇ρ = 0 (density advection)
~ut + ~u · ∇~u +∇pρ
= ~g (velocity advection)
http://www.stanford.edu/class/cs205b/lectures/lecture17.pdf
CS 205A: Mathematical Methods Partial Differential Equations II 26 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Steps for Flow (on board)
1. Adjust ∆t
2. Advect velocity
3. Apply forces
4. Solve for pressure: ∇ · ∇pρ = ∇ · ~u;
divergence-free projection
5. Advect density
http://www.proxyarch.com/util/techpapers/papers/Fluidflowfortherestofus.pdf
CS 205A: Mathematical Methods Partial Differential Equations II 27 / 28
Reminders Review Numerical PDEs Gradient Domain Inpainting Fluids
Semilagrangian Advection
ecmwf.int/newsevents/training/rcourse_notes/NUMERICAL_METHODS/NUMERICAL_METHODS/Numerical_methods6.html
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CS 205A: Mathematical Methods Partial Differential Equations II 28 / 28