AM 205: lecture 16 - courses.seas.harvard.edu

AM 205: lecture 16

I Last time: hyperbolic PDEs

I Today: parabolic and elliptic PDEs, introduction tooptimization

The Heat Equation

The canonical parabolic equation is the heat equation

ut − αuxx = f (t, x),

where α models thermal diffusivity

In this section, we shall omit α for convenience

Note that this is an Initial-Boundary Value Problem:

I We impose an initial condition u(0, x) = u0(x)

I We impose boundary conditions on both sides of the domain

The Heat Equation

A natural idea would be to discretize uxx with a central difference,and employ the Euler method in time:

Un+1j − Un

j

∆t−

Unj−1 − 2Un

j + Unj+1

∆x2= 0

Or we could use backward Euler in time:

Un+1j − Un

j

∆t−

Un+1j−1 − 2Un+1

j + Un+1j+1

∆x2= 0

The Heat Equation

Or we could do something “halfway in between”:

Un+1j − Un

j

∆t− 1

2

Un+1j−1 − 2Un+1

j + Un+1j+1

∆x2− 1

2

Unj−1 − 2Un

j + Unj+1

∆x2= 0

This is called the Crank–Nicolson method1

In fact, it is common to consider a 1-parameter “family” ofmethods that include all of the above: the θ-method

Un+1j − Un

j

∆t− θ

Un+1j−1 − 2Un+1

j + Un+1j+1

∆x2− (1 − θ)

Unj−1 − 2Un

j + Unj+1

∆x2= 0

where θ ∈ [0, 1]

1From a paper by Crank and Nicolson in 1947, note: “Nicolson” is not atypo!

The Heat Equation

With the θ-method:

I θ = 0 =⇒ Euler

I θ = 12 =⇒ Crank–Nicolson

I θ = 1 =⇒ backward Euler

For the θ-method, we can

1. perform Fourier stability analysis

2. calculate the truncation error

The θ-Method: Stability

Fourier stability analysis: Set Unj (k) = λ(k)ne ik(j∆x) to get

λ(k) =1− 4(1− θ)µ sin2

(12k∆x

)1 + 4θµ sin2

(12k∆x

)where µ ≡ ∆t/(∆x)2

Here we cannot get λ(k) > 1, hence only concern is λ(k) < −1

Let’s find conditions for stability, i.e. we want λ(k) ≥ −1:

1− 4(1− θ)µ sin2

(1

2k∆x

)≥ −

[1 + 4θµ sin2

(1

2k∆x

)]


Or equivalently:

4µ(1− 2θ) sin2

(1

2k∆x

)≤ 2

For θ ∈ [0.5, 1] this inequality is always satisfied, hence theθ-method is unconditionally stable (i.e. stable independent of µ)

In the θ ∈ [0, 0.5) case, the “most unstable” Fourier mode is whenk = π/∆x , since this maximizes the factor sin2

(12k∆x

)


Note that this corresponds to the highest frequency mode that canbe represented on our grid, since with k = π/∆x we have

e ik(j∆x) = eπij = (eπi )j = (−1)j

The k = π/∆x mode:

0 1 2 3 4 5 6 7 8 9 10−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

j

eπij


This “sawtooth” mode is stable (and hence all modes are stable) if

4µ(1− 2θ) ≤ 2⇐⇒ µ ≤ 1

2(1− 2θ),

Hence for θ ∈ [0, 0.5), the θ-method is conditionally stable


0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

35

40

45

θ

µ

For θ ∈ [0, 0.5), θ-method is stable if µ is in the “green region,”i.e. approaches unconditional stability as θ → 0.5


Note that if we set θ to a value in [0, 0.5), then stability time-step

restriction is quite severe: ∆t ≤ (∆x)2

2(1−2θ)

Contrast this to the hyperbolic case where we had ∆t ≤ ∆xc

This is an indication that the system of ODEs that arise fromspatially discretizing the heat equation are stiff

The θ-Method: Accuracy

The truncation error analysis is fairly involved, hence we just givethe result:

T nj ≡

un+1j − un

j

∆t− θ

un+1j−1 − 2un+1

j + un+1j+1

∆x2− (1 − θ)

unj−1 − 2un

j + unj+1

∆x2

= [ut − uxx ] +

[(1

2− θ

)∆tuxxt −

1

12(∆x)2uxxxx

]+

[1

24(∆t)2uttt −

1

8(∆t)2uxxtt

]+

[1

12

(1

2− θ

)∆t(∆x)2uxxxxt −

2

6!(∆x)4uxxxxxx

]+ · · ·

The term ut − uxx in T nj vanishes since u solves the PDE

The θ-Method: Accuracy

Key point: This is a first order method, unless θ = 1/2, in whichcase we get a second order method!

θ-method gives us consistency (at least first order) and stability(assuming ∆t is chosen appropriately when θ ∈ [0, 1/2))

Hence, from Lax Equivalence Theorem, the method is convergent

The Heat Equation

Note that the heat equation models a diffusive process, hence ittends to smooth out discontinuities

Python demo: Heat equation with discontinous initial condition

0

2

4

6

8

10

0

1

2

3

0

0.2

0.4

0.6

0.8

1

xt

This is very different to hyperbolic equations, e.g. the advectionequation will just transport a discontinuity in u0

Elliptic PDEs

Elliptic PDEs

The canonical elliptic PDE is the Poisson equation

In one-dimension, for x ∈ [a, b], this is −u′′(x) = f (x) withboundary conditions at x = a and x = b

We have seen this problem already: Two-point boundary valueproblem!

(Recall that Elliptic PDEs model steady-state behavior, there is notime-derivative)

Elliptic PDEs

In order to make this into a PDE, we need to consider more thanone spatial dimension

Let Ω ⊂ R2 denote our domain, then the Poisson equation for(x , y) ∈ Ω is

uxx + uyy = f (x , y)

This is generally written more succinctly as ∆u = f

We again need to impose boundary conditions (Dirichlet,Neumann, or Robin) on ∂Ω (recall ∂Ω denotes boundary of Ω)

Elliptic PDEs

We will consider how to use a finite difference scheme toapproximate this 2D Poisson equation

First, we introduce a uniform grid to discretize Ω

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

Elliptic PDEs

Let h = ∆x = ∆y denote the grid spacing

Then,

I xi = ih, i = 0, 1, 2 . . . , nx − 1,

I yj = jh, j = 0, 1, 2, . . . , ny − 1,

I Ui ,j ≈ u(xi , yj)

Then, we need to be able to approximate uxx and uyy on this grid

Natural idea: Use central difference approximation!

Elliptic PDEs

We have

uxx(xi , yj) =u(xi−1, yj) − 2u(xi , yj) + u(xi+1, yj)

h2+ O(h2),

anduyy (xi , yj) =

u(xi , yj−1) − 2u(xi , yj) + u(xi , yj+1)

h2+ O(h2),

so that

uxx(xi , yj) + uyy (xi , yj) =

u(xi , yj−1) + u(xi−1, yj) − 4u(xi , yj) + u(xi+1, yj) + u(xi , yj+1)

h2+ O(h2)

Elliptic PDEs

Hence we define our approximation to the Laplacian as

Ui ,j−1 + Ui−1,j − 4Ui ,j + Ui+1,j + Ui ,j+1

h2

This corresponds to a “5-point stencil”

Elliptic PDEs

As usual, we represent the numerical solution as a vector U ∈ Rnxny

We want to construct a differentiation matrix D2 ∈ Rnxny×nxny

that approximates the Laplacian

Question: How many non-zero diagonals will D2 have?

To construct D2, we need to be able to relate the entries of thevector U to the “2D grid-based values” Ui ,j

Elliptic PDEs

Hence we need to number the nodes from 1 to nxny — we numbernodes along the “bottom row” first, then second bottom row, etc

Let G denote the mapping from the 2D indexing to the 1Dindexing. From the above figure we have:

G(i , j ; nx) = jnx + i , and hence UG(i ,j ;nx ) = Ui ,j

Elliptic PDEs

Let us focus on node (i , j) in our F.D. grid, this corresponds toentry G(i , j ; nx) of U

Due to the 5-point stencil, row G(i , j ; nx) of D2 will only havenon-zeros in columns

G(i , j − 1; nx) = G(i , j ; nx)− nx , (1)

G(i − 1, j ; nx) = G(i , j ; nx)− 1, (2)

G(i , j ; nx) = G(i , j ; nx), (3)

G(i + 1, j ; nx) = G(i , j ; nx) + 1, (4)

G(i , j + 1; nx) = G(i , j ; nx) + nx (5)

I (2), (3), (4), give the same tridiagonal structure that we’reused to from differentiation matrices in 1D domains

I (1), (5) give diagonals shifted by ±nx

Elliptic PDEs

For example, sparsity pattern of D2 when nx = ny = 6

Elliptic PDEs

Python demo: Solve the Poisson equation

∆u = − exp−(x − 0.25)2 − (y − 0.5)2

,

for (x , y) ∈ Ω = [0, 1]2 with u = 0 on ∂Ω

x

y

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

Nonlinear Equations and Optimization

Motivation: Nonlinear Equations

So far we have mostly focused on linear phenomena

I Interpolation leads to a linear system Vb = y (monomials) orIb = y (Lagrange polynomials)

I Linear least-squares leads to the normal equationsATAb = AT y

I We saw examples of linear physical models (Ohm’s Law,Hooke’s Law, Leontief equations) =⇒ Ax = b

I F.D. discretization of a linear PDE leads to a linear algebraicsystem AU = F

Motivation: Nonlinear EquationsOf course, nonlinear models also arise all the time

I Nonlinear least-squares, Gauss–Newton/Levenberg–Marquardt

I Countless nonlinear physical models in nature, e.g.non-Hookean material models2

I F.D. discretization of a non-linear PDE leads to a nonlinearalgebraic system

2Important in modeling large deformations of solids


Another example is computation of Gauss quadraturepoints/weights

We know this is possible via roots of Legendre polynomials

But we could also try to solve the nonlinear system of equationsfor (x1,w1), (x2,w2), . . . , (xn,wn)


e.g. for n = 2, we need to find points/weights such that allpolynomials of degree 3 are integrated exactly, hence

w1 + w2 =

∫ 1

−11dx = 2

w1x1 + w2x2 =

∫ 1

−1xdx = 0

w1x21 + w2x

22 =

∫ 1

−1x2dx = 2/3

w1x31 + w2x

32 =

∫ 1

−1x3dx = 0


We usually write a nonlinear system of equations as

F (x) = 0,

where F : Rn → Rm

We implicity absorb the “right-hand side” into F and seek a rootof F

In this Unit we focus on the case m = n, m > n gives nonlinearleast-squares


We are very familiar with scalar (m = 1) nonlinear equations

Simplest case is a quadratic equation

ax2 + bx + c = 0

We can write down a closed form solution, the quadratic formula

x =−b ±

√b2 − 4ac

2a


In fact, there are also closed-form solutions for arbitrary cubic andquartic polynomials, due to Ferrari and Cardano (∼ 1540)

Important mathematical result is that there is no general formulafor solving fifth or higher order polynomial equations

Hence, even for the simplest possible case (polynomials), the onlyhope is to employ an iterative algorithm

An iterative method should converge in the limit n→∞, andideally yields an accurate approximation after few iterations


There are many well-known iterative methods for nonlinearequations

Probably the simplest is the bisection method for a scalar equationf (x) = 0, where f ∈ C [a, b]

Look for a root in the interval [a, b] by bisecting based on sign of f


#!/usr/bin/python

from math import *

# Function to consider

def f(x):

return x*x-4*sin(x)

# Initial interval: assume f(a)<0 and f(b)>0

a=1

b=3

# Bisection search

while b-a>1e-8:

print a,b

c=0.5*(b+a)

if f(c)<0: a=c

else: b=c

print "# Root at",0.5*(a+b)


1 1.5 2 2.5 3−4

−2

0

2

4

6

8

10

1.932 1.933 1.934 1.935

−0.01

−0.005

0

0.005

0.01

Root in the interval [1.933716, 1.933777]


Bisection is a robust root-finding method in 1D, but it does notgeneralize easily to Rn for n > 1

Also, bisection is a crude method in the sense that it makes no useof magnitude of f , only sign(f )

We will look at mathematical basis of alternative methods whichgeneralize to Rn:

I Fixed-point iteration

I Newton’s method

Optimization

Motivation: Optimization

Another major topic in Scientific Computing is optimization

Very important in science, engineering, industry, finance,economics, logistics,...

Many engineering challenges can be formulated as optimizationproblems, e.g.:

I Design car body that maximizes downforce3

I Design a bridge with minimum weight

3A major goal in racing car design


Of course, in practice, it is more realistic to consider optimizationproblems with constraints, e.g.:

I Design car body that maximizes downforce, subject to aconstraint on drag

I Design a bridge with minimum weight, subject to a constrainton strength


Also, (constrained and unconstrained) optimization problems arisenaturally in science

Physics:

I many physical systems will naturally occupy a minimumenergy state

I if we can describe the energy of the system mathematically,then we can find minimum energy state via optimization


Biology:

I recent efforts in Scientific Computing have sought tounderstand biological phenomena quantitively via optimization

I computational optimization of, e.g. fish swimming or insectflight, can reproduce behavior observed in nature

I this jells with the idea that evolution has been “optimizing”organisms for millions of year


All these problems can be formulated as: Optimize (max. or min.)an objective function over a set of feasible choices, i.e.

Given an objective function f : Rn → R and a set S ⊂ Rn,we seek x∗ ∈ S such that f (x∗) ≤ f (x), ∀x ∈ S

(It suffices to consider only minimization, maximization isequivalent to minimizing −f )

S is the feasible set, usually defined by a set of equations and/orinequalities, which are the constraints

If S = Rn, then the problem is unconstrained


The standard way to write an optimization problem is

minx∈S

f (x) subject to g(x) = 0 and h(x) ≤ 0,

where f : Rn → R, g : Rn → Rm, h : Rn → Rp


For example, let x1 and x2 denote radius and height of a cylinder,respectively

Minimize the surface area of a cylinder subject to a constraint onits volume4 (we will return to this example later)

minx

f (x1, x2) = 2πx1(x1 + x2)

subject to g(x1, x2) = πx21x2 − V = 0

4Heath Example 6.2


If f , g and h are all affine, then the optimization problem is calleda linear program

(Here the term “program” has nothing to do with computerprogramming; instead it refers to logistics/planning)

Affine if f (x) = Ax + b for a matrix A, i.e. linear plus a constant5

Linear programming mayalready be familiar

Just need to check f (x) onvertices of the feasible region

5Recall that “affine” is not the same as ”linear”, i.e.f (x + y) = Ax + Ay + b and f (x) + f (y) = Ax + Ay + 2b


If the objective function or any of the constraints are nonlinear thenwe have a nonlinear optimization problem or nonlinear program

We will consider several different approaches to nonlinearoptimization in this Unit

Optimization routines typically use local information about afunction to iteratively approach a local minimum


In some cases this easily gives a global minimum

−1 −0.5 0 0.5 10

0.5

1

1.5

2

2.5

3


But in general, global optimization can be very difficult

0 0.2 0.4 0.6 0.8 1−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

We can get “stuck” in local minima!


And can get much harder in higher spatial dimensions

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−0.5

0

0.5

1

1.5

2

2.5


There are robust methods for finding local minimima, and this iswhat we focus on in AM205

Global optimization is very important in practice, but in generalthere is no way to guarantee that we will find a global minimum

Global optimization basically relies on heuristics:

I try several different starting guesses (“multistart” methods)

I simulated annealing

I genetic methods6

6Simulated annealing and genetic methods are covered in AM207

Root Finding: Scalar Case

Fixed-Point Iteration

Suppose we define an iteration

xk+1 = g(xk) (∗)

e.g. recall Heron’s Method from Assignment 0 for finding√a:

xk+1 =1

2

(xk +

a

xk

)This uses gheron(x) = 1

2 (x + a/x)


Suppose α is such that g(α) = α, then we call α a fixed point of g

For example, we see that√a is a fixed point of gheron since

gheron(√a) =

1

2

(√a + a/

√a)

=√a

A fixed-point iteration terminates once a fixed point is reached,since if g(xk) = xk then we get xk+1 = xk

Also, if xk+1 = g(xk) converges as k →∞, it must converge to afixed point: Let α ≡ limk→∞ xk , then7

α = limk→∞

xk+1 = limk→∞

g(xk) = g

(limk→∞

xk

)= g(α)

7Third equality requires g to be continuous


Hence, for example, we know if Heron’s method converges, it willconverge to

√a

It would be very helpful to know when we can guarantee that afixed-point iteration will converge

Recall that g satisfies a Lipschitz condition in an interval [a, b] if∃L ∈ R>0 such that

|g(x)− g(y)| ≤ L|x − y |, ∀x , y ∈ [a, b]

g is called a contraction if L < 1


Theorem: Suppose that g(α) = α and that g is a contractionon [α−A, α+A]. Suppose also that |x0−α| ≤ A. Then thefixed point iteration converges to α.

Proof:|xk − α| = |g(xk−1)− g(α)| ≤ L|xk−1 − α|,

which implies|xk − α| ≤ Lk |x0 − α|

and, since L < 1, |xk − α| → 0 as k →∞. (Note that|x0 − α| ≤ A implies that all iterates are in [α− A, α + A].)

(This proof also shows that error decreases by factor of L eachiteration)

AM 205: lecture 16 - courses.seas.harvard.edu

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