Fast transforms based on asymptotic expansions of special ...pi.math.cornell.edu/~ajt/presentations/FastTransforms.pdfPart II: Fast transforms based on asymptotic expansions Asymptotic

Fast algorithms based on asymptotic expansions of special functions

Alex Townsend

MIT

Cornell University, 23rd January 2015, CAM Colloquium

Introduction

Two trends in the computing era

Alex Townsend @MIT

Tabulations Software

Logarithms

Gauss quadrature

Special functions

1/22

Introduction

Two trends in the computing era

Alex Townsend @MIT

Classicalanalysis

Numericalanalysis

= ~~

Fast multipole method

Butterfly schemes

Iterative algorithms

Finite element method

Tabulations Software

Logarithms

Gauss quadrature

Special functions

1/22

Introduction

A selection of my research

Alex Townsend @MIT

𝑝(𝑥, 𝑦)

𝑞(𝑥, 𝑦)= 0

[Nakatsukasa, Noferini, & T., 2015]

2/22

𝑓(𝑥, 𝑦) ≈

𝑗=1

𝐾

𝑐𝑗 𝑦 𝑟𝑗(𝑥)

[Trefethen & T., 2014]

Algebraic geometry

Approx. theory

Introduction

A selection of my research

Alex Townsend @MIT

Algebraic geometry

𝑝(𝑥, 𝑦)

𝑞(𝑥, 𝑦)= 0

[Nakatsukasa, Noferini, & T., 2015]

Approx. theory

[Battels & Trefethen, 2004]

𝑓(𝑥, 𝑦) ≈

𝑗=1

𝐾

𝑐𝑗 𝑦 𝑟𝑗(𝑥)

[Trefethen & T., 2014]

2/22

Introduction

Many special functions are trigonometric-like

Alex Townsend @MIT

Trigonometric functionscos(ωx), sin(𝜔𝑥)

Chebyshev polynomialsTn(x)

Legendre polynomialsPn(x)

Bessel functionsJv(z)

Airy functionsAi(x)

Also, Jacobi polynomials, Hermite polynomials, parabolic cylinder functions, etc.

3/22

Introduction


Alex Townsend @MIT





Airy functionsAi(x)


3/22

Introduction


Alex Townsend @MIT





Airy functionsAi(x)


3/22

Introduction


Alex Townsend @MIT





Airy functionsAi(x)


3/22

Introduction


Alex Townsend @MIT





Airy functionsAi(x)


3/22

Introduction


Alex Townsend @MIT





Airy functionsAi(x)


3/22

Introduction

Asymptotic expansions of special functions

Alex Townsend @MIT

Legendre polynomials

Bessel functions

Thomas Stieltjes

Hermann Hankel

𝑃𝑛(cos 𝜃) ~ 𝐶𝑛

𝑚=0

𝑀−1

ℎ𝑚,𝑛

cos 𝑚 + 𝑛 +12𝜃 − 𝑚 +

12𝜋2

(2 sin 𝜃)𝑚+12

, 𝑛 → ∞

𝐽0 𝑧 ~2

𝜋𝑧(cos 𝑧 −

𝜋

4

𝑚=0

𝑀−1−1 𝑚𝑎2𝑚𝑧2𝑚

− sin(𝑧 −𝜋

4)

𝑚=0

𝑀−1(−1)𝑚𝑎2𝑚+1𝑧2𝑚+1

)

𝑧 → ∞

4/22

Introduction

Numerical pitfalls of an asymptotic expansionist

Alex Townsend @MIT

Fix M. Where is the asymptotic expansion accurate?

𝐽0 𝑧 =2

𝜋𝑧(cos 𝑧 −

𝜋

4

𝑚=0

𝑀−1−1 𝑚𝑎2𝑚𝑧2𝑚

− sin(𝑧 −𝜋

4)

𝑚=0

𝑀−1(−1)𝑚𝑎2𝑚+1𝑧2𝑚+1

) + 𝑅𝑀(𝑧)

5/22

Introduction

Talk outline

Alex Townsend @MIT

Using asymptotics expansions of special functions:

I. Computing Gauss quadrature rules. [Hale & T., 2013], [T., Trodgon, & Olver, 2015]

II. Fast transforms based on asymptotic expansions. [Hale & T., 2014], [T., 2015]

6/22

Part I: Computing Gauss quadrature rules

Alex Townsend @MIT


Pn(x) = Legendre polynomial of degree n

7/22


Definition

Alex Townsend @MIT

n-point quadrature rule: Gauss quadrature rule:

−1

1

𝑓 𝑥 𝑑𝑥 ≈

𝑘=1

𝑛

𝑤𝑘𝑓 𝑥𝑘𝑥𝑘 = 𝑘𝑡ℎ zero of 𝑃𝑛

𝑤𝑘 = 2(1 − 𝑥𝑘2)-1 [𝑃𝑛

′(𝑥𝑘)]-2

1. Exactly integrates polynomials of

degree ≤ 2n - 1. Best possible.

2. Computed by the Golub–Welsch

algorithm. [Golub & Welsch, 1969]

3. In 2004: no scheme for

computing Gauss rules in O(n). [Bailey, Jeyabalan, & Li, 2004]

8/22


Gauss quadrature experiments

Alex Townsend @MIT

Quadrature error Execution time

Qu

ad

ratu

re e

rro

r

Ex

ecu

tio

n t

ime

The Golub–Welsch algorithm is beautiful, but not the state-of-the-art.

There is another way...

n n

9/22


Newton’s method

Alex Townsend @MIT

Newton’s method

Isaac Newton

1. Evaluation scheme for Pn(𝑐𝑜𝑠 θ) (𝑠𝑖𝑛 θ)1/2

2. Evaluation scheme for 𝑑

𝑑𝜃Pn( 𝑐𝑜𝑠 θ) (𝑠𝑖𝑛 θ)1/21/2

3. Initial guesses for cos-1(xk)

Solve Pn(𝑐𝑜𝑠 θ) (𝑠𝑖𝑛 θ)1/2 = 0, θ ∈ (0, π).

(with asymptotics and a change of variables)

10/22



𝑑𝜃Pn( 𝑐𝑜𝑠 θ) (𝑠𝑖𝑛 θ)1/2)1/2



Newton’s method… with asymptotics and a change of variables

Alex Townsend @MIT

Newton’s method

Isaac Newton


Pn(x)


10/22



Alex Townsend @MIT

Newton’s method

Isaac Newton



𝑑𝜃Pn( 𝑐𝑜𝑠 θ) (𝑠𝑖𝑛 θ)1/2)1/2



Pn(x)


10/22



Alex Townsend @MIT

Isaac Newton

Pn(𝑐𝑜𝑠 θ) (𝑠𝑖𝑛 θ)1/2



𝑑𝜃Pn( 𝑐𝑜𝑠 θ) (𝑠𝑖𝑛 θ)1/2


Newton’s method



10/22



Alex Townsend @MIT

Isaac Newton






Newton’s method



10/22



Alex Townsend @MIT

Isaac Newton






Newton’s method



10/22


Evaluation scheme for Pn

Alex Townsend @MIT

Interior (in grey region):Pn(𝑐𝑜𝑠 θ) (𝑠𝑖𝑛 θ)1/2

Boundary:

[Baratella & Gatteschi, 1988]

11/22

𝑃𝑛(cos 𝜃)(sin 𝜃)1/2 ≈ 𝜃

12(𝐽0(𝜌𝜃)

𝑚=0

𝑀−1𝐴𝑚(𝜃)

𝜌2𝑚+ 𝜃𝐽1(𝜌𝜃)

𝑚=0

𝑀−1𝐵𝑚(𝜃)

𝜌2𝑚+1), 𝜌 = 𝑛 +

1

2

𝑃𝑛(cos 𝜃)(sin 𝜃)1/2

≈ 𝐶𝑛

𝑚=0

𝑀−1

ℎ𝑚,𝑛

cos 𝑚 + 𝑛 +12𝜃 − 𝑚 +

12𝜋2

2(2 sin 𝜃)𝑚

Theorem

For 𝜖 > 0, 𝑛 ≥ 200, and θ𝑘𝑖𝑛𝑖𝑡= (𝑘 −

1

2)𝜋

𝑛, Newton converges and the nodes are

computed to an accuracy of 𝒪(𝜖) in 𝒪(𝑛log(1/𝜖)) operations.



Alex Townsend @MIT


Boundary:



12(𝐽0(𝜌𝜃)

𝑚=0



𝑚=0


𝜌2𝑚+1), 𝜌 = 𝑛 +

1

2


≈ 𝐶𝑛

𝑚=0

𝑀−1

ℎ𝑚,𝑛

cos 𝑚 + 𝑛 +12𝜃 − 𝑚 +

12𝜋2

2(2 sin 𝜃)𝑚

11/22

Theorem

For 𝜖 > 0, 𝑛 ≥ 200, and θ𝑘𝑖𝑛𝑖𝑡= (𝑘 −

1

2)𝜋

𝑛, Newton converges and the nodes are

computed to an accuracy of 𝒪(𝜖) in 𝒪(𝑛log(1/𝜖)) operations.



Alex Townsend @MIT


Boundary:

Theorem

For 𝑛 ≥ 200 and θ𝑘𝑖𝑛𝑖𝑡= (𝑘 −

1

2)𝜋

𝑛, Newton converges to the nodes in 𝒪(𝑛) operations.


11/22


12(𝐽0(𝜌𝜃)

𝑚=0



𝑚=0


𝜌2𝑚+1), 𝜌 = 𝑛 +

1

2


≈ 𝐶𝑛

𝑚=0

𝑀−1

ℎ𝑚,𝑛

cos 𝑚 + 𝑛 +12𝜃 − 𝑚 +

12𝜋2

2(2 sin 𝜃)𝑚


Initial guesses for the nodes

Alex Townsend @MIT

Max error in the initial guesses

There are now better initial guesses. [Bogaert, 2014]

Interior: Use Tricomi’s

initial guesses. [Tricomi, 1950]

Boundary: Use Olver’s

initial guesses. [Olver, 1974]

Ignace Bogaert

Ma

x e

rro

r

12/22


The race to compute high-order Gauss quadrature

Alex Townsend @MIT

“The race to computehigh-order Gauss quadrature”

[T., 2015]

13/22


Applications

Alex Townsend @MIT

1. Psychological: Gauss quadrature

rules are cheap.

2. Experimental mathematics: High

precision quadratures. [Bailey, Jeyabalan,

& Li, 2004]

3. Cosmic microwave background:

Noise removal by L2-projection.

Generically, composite rules converge algebraically.

−11 1

1+1002𝑥2dxQ

ua

dra

ture

err

or

14/22

Part II: Fast transforms based on asymptotic expansions

Alex Townsend @MIT

Part II: Fast transforms based on asymptoticexpansions

𝑷𝑵𝑐 =𝑃0 cos 𝜃0 ⋯ 𝑃𝑁−1 cos 𝜃0⋮ ⋱ ⋮

𝑃0 cos 𝜃𝑁−1 ⋯ 𝑃𝑁−1 cos 𝜃𝑁−1

𝑐1⋮𝑐𝑁−1

, 𝜃𝑘=𝑘𝜋

𝑁 − 1

15/22

𝜃0 𝜃1


Fast transforms as evaluation schemes

Alex Townsend @MIT 16/22

DFT 𝑓𝑘 = 𝑛=0𝑁−1𝑐𝑛𝑧𝑘𝑛 , 𝑧𝑘 = 𝑒

−2𝜋𝑖𝑘/𝑁

DCT 𝑓𝑘 = 𝑛=0𝑁−1𝑐𝑛𝑐𝑜𝑠(𝑛𝜃𝑘) , 𝜃𝑘=

𝑘𝜋

𝑁−1

DCT 𝑓𝑘 = 𝑛=0𝑁−1𝑐𝑛𝑇𝑛(𝑐𝑜𝑠𝜃𝑘)

DLT 𝑓𝑘 = 𝑛=0𝑁−1𝑐𝑛𝑃𝑛(𝑐𝑜𝑠𝜃𝑘)

DHT 𝑓𝑘 = 𝑛=0𝑁−1𝑐𝑛𝐽𝑣 𝑗𝑣,𝑛𝑟𝑘 , 𝑟𝑘 =

𝑗𝑣,𝑛

𝑗𝑣,𝑁, 𝑗𝑣,𝑛 = 𝑛𝑡ℎ 𝑟𝑜𝑜𝑡 𝑜𝑓 𝐽𝑣 𝑧

𝒛𝟎

𝒛𝟏𝒛𝟐







𝑘𝜋

𝑁−1




𝑗𝑣,𝑛


𝜃0 𝜃1

𝒛𝟎

𝒛𝟏𝒛𝟐







𝑘𝜋

𝑁−1




𝑗𝑣,𝑛


𝒛𝟎

𝒛𝟏𝒛𝟐

𝜃0 𝜃1



Alex Townsend @MIT




𝑘𝜋

𝑁−1




𝑗𝑣,𝑛


16/22

𝒛𝟎

𝒛𝟏𝒛𝟐

𝜃0 𝜃1


Related work

Alex Townsend @MIT

Many others: Candel, Cree, Johnson, Mori, Orszag, Suda, Sugihara, Takami, etc.

Fast transforms are very important.

Emmanuel Candes

Boag, Demanet,

Michielssen,

O’Neil, Ying

James Cooley

Ahmed, Tukey, Gauss,

Gentleman, Natarajan,

Rao

Leslie Greengard

Alpert, Barnes, Hut,

Martinsson, Rokhlin,

Mohlenkamp, Tygert

Daniel Potts

Driscoll, Healy,

Steidl, Tasche

17/22

𝑃𝑛(cos 𝜃𝑘) = 𝐶𝑛

𝑚=0

𝑀−1

ℎ𝑚,𝑛

cos 𝑚 + 𝑛 +12𝜃𝑘 − 𝑚 +

12𝜋2

(2 sin 𝜃𝑘)𝑚 + 1/2 + 𝑅𝑀,𝑛(𝜃𝑘)


Asymptotic expansions as a matrix decomposition

Alex Townsend @MIT

The asymptotic expansion

gives a matrix decomposition (sum of diagonally scaled DCTs and DSTs):

𝑷𝑵 𝑷𝑵𝑨𝑺𝒀 𝑹𝑴,𝑵= +

𝑷𝑵=

𝑚=0

𝑀−1

𝐷𝑢𝑚𝑪𝑵𝐷𝐶ℎ𝑚 + 𝐷𝑣𝑚

0 0 00 𝑺𝑵−𝟐 00 0 0

𝐷𝐶ℎ𝑚 + 𝑹𝑴,𝑵

18/22


Be careful and stay safe

Alex Townsend @MIT

Error curve: 𝑅𝑀,𝑛 𝜃𝑘 = 𝜖

Fix M. Where is the asymptotic expansion accurate?

𝑹𝑴,𝑵 =

𝑅𝑀,𝑛(𝜃𝑘) ≤2𝐶𝑛ℎ𝑀,𝑛

(2 sin 𝜃𝑘)𝑀+1/2

19/22


Partitioning and balancing competing costs

Alex Townsend @MIT

Theorem

The matrix-vector product 𝑓 = PN𝑐 can

be computed in 𝒪 (𝑁((log𝑁)2/ log log𝑁)

operations.

α too small

20/22

0 1

α too small



Alex Townsend @MIT

Theorem


be computed in 𝒪(𝑁((log𝑁)2/ log log𝑁)

operations.

20/22

0 1



Alex Townsend @MIT

The matrix-vector product 𝑓 = 𝐏𝐍𝑐 can


operations.

α too small

Theorem

20/22

0 1



Alex Townsend @MIT

α too large

Theorem



operations.

20/22

0 1



Alex Townsend @MIT

α = O (1/ log𝑁)

PN(j)𝑐 PN

EVAL𝑐

Theorem

𝒪 (N (log N)

2

log log 𝑁) 𝒪 (

N (log N)2

log log 𝑁)



operations.

20/22

0 1


Numerical results


105

No precomputation.

0.15

0.1

0.05

0

-0.05

-0.1

-0.15

-0.2

-0.250.3 0.32 0.34

x

Modification of rough signals

−1

1

𝑃𝑛 𝑥 𝑒−𝑖𝜔𝑡𝑑𝑥 = 𝑖𝑚

2𝜋

−𝜔𝐽𝑚+12(−𝜔)

Direct approachUsing asymptotics

10210-3

10-2

10-1

100

103 104

Ex

ecu

tio

n t

ime

N

Future work

Fast spherical harmonic transform


Spherical harmonic transform:

𝑓 𝜃, 𝜙 =

𝑙=0

𝑁−1

𝑚=−𝑙

𝑙

𝛼𝑙𝑚𝑃𝑙𝑚 (cos 𝜃)𝑒𝑖𝑚𝜙

[Mohlenkamp, 1999]

[Rokhlin & Tygert, 2006]

[Tygert, 2008]

10-2

100

102

104

104E

xec

uti

on

tim

eN

105

PrecomputationFast transformDirect approach

A new generation of fast algorithms with no precomputation.

Existing approaches

103

Alex Townsend @MIT

Thank you

References

Alex Townsend @MIT

• D. H. Bailey, K. Jeyabalan, and X. S. Li, A comparison of three high-precision quadrature schemes, 2005.

• P. Baratella and I. Gatteschi, The bounds for the error term of an asymptotic approximation of Jacobipolynomials, Lecture notes in Mathematics, 1988.

• Z. Battels and L. N. Trefethen, An extension of Matlab to continuous functions and operators, SISC,2004.

• I. Bogaert, Iteration-free computation of Gauss–Legendre quadrature nodes and weights, SISC, 2014.

• G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature rules, Math. Comput., 1969.

• N. Hale and A. Townsend, Fast and accurate computation of Gauss–Legendre and Gauss–Jacobiquadrature nodes and weights, SISC, 2013.

• N. Hale and A. Townsend, A fast, simple, and stable Chebyshev–Legendre transform using an asymptoticformula, SISC, 2014.

• Y. Nakatsukasa, V. Noferini, and A. Townsend, Computing the common zeros of two bivariate functionsvia Bezout resultants, Numerische Mathematik, 2014.

References

Alex Townsend @MIT

• F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, 1974.

• A. Townsend and L. N. Trefethen, An extension of Chebfun to two dimensions, SISC, 2013.

• A. Townsend, T. Trogdon, and S. Olver, Fast computation of Gauss quadrature nodes and weights on thewhole real line, to appear in IMA Numer. Anal., 2015.

• A. Townsend, The race to compute high-order Gauss quadrature, SIAM News, 2015.

• A. Townsend, A fast analysis-based discrete Hankel transform using asymptotics expansions, submitted,2015.

• F. G. Tricomi, Sugli zeri dei polinomi sferici ed ultrasferici, Ann. Mat. Pura. Appl., 1950.

Fast transforms based on asymptotic expansions of special ...pi.math.cornell.edu/~ajt/presentations/FastTransforms.pdfPart II: Fast transforms based on asymptotic expansions Asymptotic

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