Beyond Chebyshev technology
Strathclyde, 25th June 2015
26th Biennial Numerical Analysis Conference
Alex Townsend
MIT
Joint work with Nick Hale and Sheehan Olver
Beyond Chebyshev technology
Strathclyde, 25th June 2015
26th Biennial Numerical Analysis Conference
Alex Townsend
MIT
Joint work with Nick Hale and Sheehan Olver
Beyond Chebyshev technology
Strathclyde, 25th June 2015
26th Biennial Numerical Analysis Conference
Alex Townsend
MIT
Joint work with Nick Hale and Sheehan Olver
Introduction
Chebfun stands for Chebyshev fun
Chebfun = Chebyshev fun
Chebyshev technology: A powerful approach for numerically computing with
functions. It is based on (piecewise) polynomial interpolation at Chebyshev
points and Chebyshev polynomials:
xj “ cospjπ{nq,
0 ď j ď n,
Tk px q “ cospk cos´1 x q
There exist fast, accurate algorithms for integration, differentiation, rootfinding,
minimization, solution of ODEs, etc.
Alex Townsend @ MIT
1/17
Introduction
Chebyshev technology is powerful
Chebyshev interpolant of fpxq:
n
ÿ
f px q « pn px q “
ck T k p x q ,
pn pxj q “ f pxj q,
xj “ chebpts
k “0
Powerful concepts: FFT, barycentric formula, colleague matrix, collocation
SIAM digit challenge problem
3
2
2
0.5
v
0
y
-2
2
−0.5
0
0.5
[Trefethen, 02]
Alex Townsend @ MIT
2
0
0
-2
2
-2
2
0
−1
2
−2
−0.5
2
1
0
−1
−1
Computing spherical choreographies
Inverse transform sampling
1
1
−3
−3
-2 -2
−2
−1
2
0
x
1
2
2
3
-2 -2
[Olver & T., 14]
-2 -2
[Montanelli & Gushterov, 15]
2/17
Introduction
Chebyshev technology can also bring us together
Chebyshev technology can also bring us together. My friends:
« 45 peer-reviewed publications, « 5 essays, and a book called ATAP.
Alex Townsend @ MIT
3/17
Introduction
Characters: Standard orthogonal polynomials on r´1, 1s
Name
Notation
w px q
Chebyshev polynomials (of 1st kind)
Tk px q
p1 ´ x 2 q´1{2
Chebyshev polynomials (of 2nd kind)
Uk px q
p1 ´ x 2 q1{2
Legendre polynomials
Pk px q
1
Ultraspherical polynomials
Ck px q
pλq
p1 ´ x 2 qλ´1{2
Fast transforms now available for Legendre and ultraspherical technology
[Hale & T., 2014], [Hale & T., 2015]
Alex Townsend @ MIT
4/17
Introduction
Talk overview and a trivial example
Fact: Chebyshev polynomials satisfy:
ż1
´1
$
’
&π,
Tj px qTk px q
a
dx “ π{2,
’
%
1 ´ x2
0,
j “ k “ 0,
j “ k ą 0,
j , k.
Inconvenience: The 1st coeff of a Chebyshev series is treated as special:
ż
8
ÿ
1
2 1 f px qTk px q
a
f px q “ a0 T0 px q `
dx.
ak T k p x q ,
ak “
2
π ´1
1 ´ x2
k “1
Alternative: Chebyshev polynomials of the 2nd kind:
#
ż1
a
π{2,
Uj px qUk px q 1 ´ x 2 dx “
0,
´1
j “ k,
j , k.
Results: Saves your sanity and avoids mistakes.
Alex Townsend @ MIT
5/17
Introduction
Talk overview and a trivial example
Fact: Chebyshev polynomials satisfy:
ż1
´1
$
’
&π,
Tj px qTk px q
a
dx “ π{2,
’
%
1 ´ x2
0,
j “ k “ 0,
j “ k ą 0,
j , k.
Inconvenience: The 1st coeff of a Chebyshev series is treated as special:
ż
8
ÿ
1
2 1 f px qTk px q
a
f px q “ a0 T0 px q `
dx.
ak T k p x q ,
ak “
2
π ´1
1 ´ x2
k “1
Alternative: Chebyshev polynomials of the 2nd kind:
#
ż1
a
π{2,
Uj px qUk px q 1 ´ x 2 dx “
0,
´1
j “ k,
j , k.
Results: Saves your sanity and avoids mistakes.
Alex Townsend @ MIT
5/17
Introduction
Talk overview and a trivial example
Fact: Chebyshev polynomials satisfy:
ż1
´1
$
’
&π,
Tj px qTk px q
a
dx “ π{2,
’
%
1 ´ x2
0,
j “ k “ 0,
j “ k ą 0,
j , k.
Inconvenience: The 1st coeff of a Chebyshev series is treated as special:
ż
8
ÿ
1
2 1 f px qTk px q
a
f px q “ a0 T0 px q `
dx.
ak T k p x q ,
ak “
2
π ´1
1 ´ x2
k “1
Alternative: Chebyshev polynomials of the 2nd kind:
#
ż1
a
π{2,
Uj px qUk px q 1 ´ x 2 dx “
0,
´1
j “ k,
j , k.
Results: Saves your sanity and avoids mistakes.
Alex Townsend @ MIT
5/17
Introduction
Talk overview and a trivial example
Fact: Chebyshev polynomials satisfy:
ż1
´1
$
’
&π,
Tj px qTk px q
a
dx “ π{2,
’
%
1 ´ x2
0,
j “ k “ 0,
j “ k ą 0,
j , k.
Inconvenience: The 1st coeff of a Chebyshev series is treated as special:
ż
8
ÿ
1
2 1 f px qTk px q
a
f px q “ a0 T0 px q `
dx.
ak T k p x q ,
ak “
2
π ´1
1 ´ x2
k “1
Alternative: Chebyshev polynomials of the 2nd kind:
#
ż1
a
π{2,
Uj px qUk px q 1 ´ x 2 dx “
0,
´1
j “ k,
j , k.
Results: Saves your sanity and avoids mistakes.
Alex Townsend @ MIT
5/17
Introduction
Talk overview and a trivial example
Fact: Chebyshev polynomials satisfy:
ż1
´1
$
’
&π,
Tj px qTk px q
a
dx “ π{2,
’
%
1 ´ x2
0,
j “ k “ 0,
j “ k ą 0,
j , k.
Inconvenience: The 1st coeff of a Chebyshev series is treated as special:
ż
8
ÿ
1
2 1 f px qTk px q
a
f px q “ a0 T0 px q `
dx.
ak T k p x q ,
ak “
2
π ´1
1 ´ x2
k “1
Alternative: Chebyshev polynomials of the 2nd kind:
#
ż1
a
π{2,
Uj px qUk px q 1 ´ x 2 dx “
0,
´1
j “ k,
j , k.
Results: Saves your sanity and avoids mistakes.
Alex Townsend @ MIT
5/17
Introduction
Talk overview and a trivial example
Fact: Chebyshev polynomials satisfy:
ż1
´1
$
’
&π,
Tj px qTk px q
a
dx “ π{2,
’
%
1 ´ x2
0,
j “ k “ 0,
j “ k ą 0,
j , k.
Inconvenience: The 1st coeff of a Chebyshev series is treated as special:
ż
8
ÿ
1
2 1 f px qTk px q
a
f px q “ a0 T0 px q `
dx.
ak T k p x q ,
ak “
2
π ´1
1 ´ x2
k “1
Alternative: Chebyshev polynomials of the 2nd kind:
#
ż1
a
π{2,
Uj px qUk px q 1 ´ x 2 dx “
0,
´1
j “ k,
j , k.
Results: Saves your sanity and avoids mistakes.
Alex Townsend @ MIT
5/17
Introduction
Talk overview and a trivial example
Fact: Chebyshev polynomials satisfy:
ż1
´1
$
’
&π,
Tj px qTk px q
a
dx “ π{2,
’
%
1 ´ x2
0,
j “ k “ 0,
j “ k ą 0,
j , k.
Inconvenience: The 1st coeff of a Chebyshev series is treated as special:
ż
8
ÿ
1
2 1 f px qTk px q
a
f px q “ a0 T0 px q `
dx.
ak T k p x q ,
ak “
2
π ´1
1 ´ x2
k “1
Alternative: Chebyshev polynomials of the 2nd kind:
#
ż1
a
π{2,
Uj px qUk px q 1 ´ x 2 dx “
0,
´1
j “ k,
j , k.
Results: Saves your sanity and avoids mistakes.
Alex Townsend @ MIT
5/17
Introduction
Talk overview and a trivial example
Fact: Chebyshev polynomials satisfy:
ż1
´1
$
’
&π,
Tj px qTk px q
a
dx “ π{2,
’
%
1 ´ x2
0,
j “ k “ 0,
j “ k ą 0,
j , k.
Inconvenience: The 1st coeff of a Chebyshev series is treated as special:
ż
8
ÿ
1
2 1 f px qTk px q
a
f px q “ a0 T0 px q `
dx.
ak T k p x q ,
ak “
2
π ´1
1 ´ x2
k “1
Alternative: Chebyshev polynomials of the 2nd kind:
#
ż1
a
π{2,
Uj px qUk px q 1 ´ x 2 dx “
0,
´1
j “ k,
j , k.
Results: Saves your sanity and avoids mistakes.
Alex Townsend @ MIT
5/17
Introduction
Talk overview and a trivial example
Fact: Chebyshev polynomials satisfy:
ż1
´1
$
’
&π,
Tj px qTk px q
a
dx “ π{2,
’
%
1 ´ x2
0,
j “ k “ 0,
j “ k ą 0,
j , k.
Inconvenience: The 1st coeff of a Chebyshev series is treated as special:
ż
8
ÿ
1
2 1 f px qTk px q
a
f px q “ a0 T0 px q `
dx.
ak T k p x q ,
ak “
2
π ´1
1 ´ x2
k “1
Alternative: Chebyshev polynomials of the 2nd kind:
#
ż1
a
π{2,
Uj px qUk px q 1 ´ x 2 dx “
0,
´1
j “ k,
j , k.
Results: Saves your sanity and avoids mistakes.
Alex Townsend @ MIT
5/17
Example 1
Convolution
Fact: The Fourier transform of Tk px q is complicated:
ż1
Tk px qe ´iωx dx “ complicated,
[Fokas & Smitheman, 12].
´1
Inconvenience: Convolution cannot be based on the convolution theorem:
ż minp1,x `1q
h px q “ pf ˚ g qpx q “
f pt qg px ´ t qdt.
1
ÝÑ
maxp´1,x ´1q
t
ÝÑ
x
´1
´2
Alex Townsend @ MIT
Quadrature is expensive.
0
2
6/17
Example 1
Convolution
Fact: The Fourier transform of Tk px q is complicated:
ż1
Tk px qe ´iωx dx “ complicated,
[Fokas & Smitheman, 12].
´1
Inconvenience: Convolution cannot be based on the convolution theorem:
ż minp1,x `1q
h px q “ pf ˚ g qpx q “
f pt qg px ´ t qdt.
1
ÝÑ
maxp´1,x ´1q
t
ÝÑ
x
´1
´2
Alex Townsend @ MIT
Quadrature is expensive.
0
2
6/17
Example 1
Convolution
Fact: The Fourier transform of Tk px q is complicated:
ż1
Tk px qe ´iωx dx “ complicated,
[Fokas & Smitheman, 12].
´1
Inconvenience: Convolution cannot be based on the convolution theorem:
ż minp1,x `1q
h px q “ pf ˚ g qpx q “
f pt qg px ´ t qdt.
1
ÝÑ
maxp´1,x ´1q
t
ÝÑ
x
´1
´2
Alex Townsend @ MIT
Quadrature is expensive.
0
2
6/17
Example 1
Convolution
Fact: The Fourier transform of Tk px q is complicated:
ż1
Tk px qe ´iωx dx “ complicated,
[Fokas & Smitheman, 12].
´1
Inconvenience: Convolution cannot be based on the convolution theorem:
ż minp1,x `1q
h px q “ pf ˚ g qpx q “
f pt qg px ´ t qdt.
1
ÝÑ
maxp´1,x ´1q
t
ÝÑ
x
´1
´2
Alex Townsend @ MIT
Quadrature is expensive.
0
2
6/17
Example 1
Convolution
Fact: The Fourier transform of Tk px q is complicated:
ż1
Tk px qe ´iωx dx “ complicated,
[Fokas & Smitheman, 12].
´1
Inconvenience: Convolution cannot be based on the convolution theorem:
ż minp1,x `1q
h px q “ pf ˚ g qpx q “
f pt qg px ´ t qdt.
1
ÝÑ
maxp´1,x ´1q
t
ÝÑ
x
´1
´2
Alex Townsend @ MIT
Quadrature is expensive.
0
2
6/17
Example 1
Convolution (cont.)
Alternative: The Fourier transform of Pk px q is convenient [Hale & T., 2014]:
ż1
Pk px qe ´iωx dx “ 2p´i qk jk pωq,
rDLMF, p18.17.19qs,
´1
where jk pωq “ spherical Bessel function.
Theorem (Convolution theorem)
Let m and n be integers. Then,
ż
2p´i qm`n 8
pPm ˚ Pn qpx q “
jm pωqjn pωqe iωx ,
π
´8
f and g in
Pk basis
f ˚ g in Pk basis
Alex Townsend @ MIT
Fourier
transform
jm pωq “ spherical Bessel.
f̂ and ĝ in
jk basis
Inverse
f̂ ˆ ĝ in jk basis
transform
7/17
Example 1
Convolution (cont.)
Alternative: The Fourier transform of Pk px q is convenient [Hale & T., 2014]:
ż1
Pk px qe ´iωx dx “ 2p´i qk jk pωq,
rDLMF, p18.17.19qs,
´1
where jk pωq “ spherical Bessel function.
Theorem (Convolution theorem)
Let m and n be integers. Then,
ż
2p´i qm`n 8
pPm ˚ Pn qpx q “
jm pωqjn pωqe iωx ,
π
´8
f and g in
Pk basis
f ˚ g in Pk basis
Alex Townsend @ MIT
Fourier
transform
jm pωq “ spherical Bessel.
f̂ and ĝ in
jk basis
Inverse
f̂ ˆ ĝ in jk basis
transform
7/17
Example 1
Convolution (cont.)
Alternative: The Fourier transform of Pk px q is convenient [Hale & T., 2014]:
ż1
Pk px qe ´iωx dx “ 2p´i qk jk pωq,
rDLMF, p18.17.19qs,
´1
where jk pωq “ spherical Bessel function.
Theorem (Convolution theorem)
Let m and n be integers. Then,
ż
2p´i qm`n 8
pPm ˚ Pn qpx q “
jm pωqjn pωqe iωx ,
π
´8
f and g in
Pk basis
f ˚ g in Pk basis
Alex Townsend @ MIT
Fourier
transform
jm pωq “ spherical Bessel.
f̂ and ĝ in
jk basis
Inverse
f̂ ˆ ĝ in jk basis
transform
7/17
Example 1
Convolution (cont.)
Alternative: The Fourier transform of Pk px q is convenient [Hale & T., 2014]:
ż1
Pk px qe ´iωx dx “ 2p´i qk jk pωq,
rDLMF, p18.17.19qs,
´1
where jk pωq “ spherical Bessel function.
Theorem (Convolution theorem)
Let m and n be integers. Then,
ż
2p´i qm`n 8
pPm ˚ Pn qpx q “
jm pωqjn pωqe iωx ,
π
´8
f and g in
Pk basis
f ˚ g in Pk basis
Alex Townsend @ MIT
Fourier
transform
jm pωq “ spherical Bessel.
f̂ and ĝ in
jk basis
Inverse
f̂ ˆ ĝ in jk basis
transform
7/17
Example 1
Convolution (cont.)
Alternative: The Fourier transform of Pk px q is convenient [Hale & T., 2014]:
ż1
Pk px qe ´iωx dx “ 2p´i qk jk pωq,
rDLMF, p18.17.19qs,
´1
where jk pωq “ spherical Bessel function.
Theorem (Convolution theorem)
Let m and n be integers. Then,
ż
2p´i qm`n 8
pPm ˚ Pn qpx q “
jm pωqjn pωqe iωx ,
π
´8
f and g in
Pk basis
f ˚ g in Pk basis
Alex Townsend @ MIT
Fourier
transform
jm pωq “ spherical Bessel.
f̂ and ĝ in
jk basis
Inverse
f̂ ˆ ĝ in jk basis
transform
7/17
Example 1
Convolution (cont.)
Results:
Computational timings
3
10
Mollification of rough signals
Quadrature
New
2
Computation time
10
OpN 3 q
1
10
OpN 2 q
0
10
−1
10
−2
10
1
10
2
10
3
10
N
4
10
5
10
Now used in the conv(f, g) command in Chebfun.
Alex Townsend @ MIT
8/17
Example 2
Solving differential equations
Fact: The derivative of a Chebyshev polynomial satisfies:
# řk ´1
2k j odd Tj px q,
k “ even,
řk ´1
Tk1 px q “
2k j even Tj px q ´ 1, k “ odd.
Inconvenience: The Chebyshev-tau spectral method leads to dense matrices.
For example,
du
` 4xu “ 0
up´1q “ c,
dx
¨
ˆ
ˆ
˚
˚
˚
˚ ˆ
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
..
ˆ
˛¨
u0
˛
ˆ
...
..
.
‹˚
ˆ ‹˚
‹˚
˚
ˆ ‹
‹˚
.. ‹ ˚
˚
. ‹
‹˚
‹˚
‹˚
‹˚
‹˚
ˆ ‹˚
‚˝
u1
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹“˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‚ ˝
.
..
ˆ
.
ˆ
..
.
ˆ
ˆ
Alex Townsend @ MIT
¨¨¨
ˆ
ˆ
¨
ˆ
ˆ
u2
..
.
..
.
un´1
c
˛
‹
0 ‹
‹
0 ‹
‹
‹
‹
.. ‹ .
. ‹
.. ‹
‹
. ‹
‹
‚
0
9/17
Example 2
Solving differential equations
Fact: The derivative of a Chebyshev polynomial satisfies:
# řk ´1
2k j odd Tj px q,
k “ even,
řk ´1
Tk1 px q “
2k j even Tj px q ´ 1, k “ odd.
Inconvenience: The Chebyshev-tau spectral method leads to dense matrices.
For example,
du
` 4xu “ 0
up´1q “ c,
dx
¨
ˆ
ˆ
˚
˚
˚
˚ ˆ
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
..
ˆ
˛¨
u0
˛
ˆ
...
..
.
‹˚
ˆ ‹˚
‹˚
˚
ˆ ‹
‹˚
.. ‹ ˚
˚
. ‹
‹˚
‹˚
‹˚
‹˚
‹˚
ˆ ‹˚
‚˝
u1
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹“˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‚ ˝
.
..
ˆ
.
ˆ
..
.
ˆ
ˆ
Alex Townsend @ MIT
¨¨¨
ˆ
ˆ
¨
ˆ
ˆ
u2
..
.
..
.
un´1
c
˛
‹
0 ‹
‹
0 ‹
‹
‹
‹
.. ‹ .
. ‹
.. ‹
‹
. ‹
‹
‚
0
9/17
Example 2
Solving differential equations
Fact: The derivative of a Chebyshev polynomial satisfies:
# řk ´1
2k j odd Tj px q,
k “ even,
řk ´1
Tk1 px q “
2k j even Tj px q ´ 1, k “ odd.
Inconvenience: The Chebyshev-tau spectral method leads to dense matrices.
For example,
du
` 4xu “ 0
up´1q “ c,
dx
¨
ˆ
ˆ
˚
˚
˚
˚ ˆ
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
..
ˆ
˛¨
u0
˛
ˆ
...
..
.
‹˚
ˆ ‹˚
‹˚
˚
ˆ ‹
‹˚
.. ‹ ˚
˚
. ‹
‹˚
‹˚
‹˚
‹˚
‹˚
ˆ ‹˚
‚˝
u1
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹“˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‚ ˝
.
..
ˆ
.
ˆ
..
.
ˆ
ˆ
Alex Townsend @ MIT
¨¨¨
ˆ
ˆ
¨
ˆ
ˆ
u2
..
.
..
.
un´1
c
˛
‹
0 ‹
‹
0 ‹
‹
‹
‹
.. ‹ .
. ‹
.. ‹
‹
. ‹
‹
‚
0
9/17
Example 2
Solving differential equations
Fact: The derivative of a Chebyshev polynomial satisfies:
# řk ´1
2k j odd Tj px q,
k “ even,
řk ´1
Tk1 px q “
2k j even Tj px q ´ 1, k “ odd.
Inconvenience: The Chebyshev-tau spectral method leads to dense matrices.
For example,
du
` 4xu “ 0
up´1q “ c,
dx
¨
ˆ
ˆ
˚
˚
˚
˚ ˆ
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
..
ˆ
˛¨
u0
˛
ˆ
...
..
.
‹˚
ˆ ‹˚
‹˚
˚
ˆ ‹
‹˚
.. ‹ ˚
˚
. ‹
‹˚
‹˚
‹˚
‹˚
‹˚
ˆ ‹˚
‚˝
u1
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹“˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‚ ˝
.
..
ˆ
.
ˆ
..
.
ˆ
ˆ
Alex Townsend @ MIT
¨¨¨
ˆ
ˆ
¨
ˆ
ˆ
u2
..
.
..
.
un´1
c
˛
‹
0 ‹
‹
0 ‹
‹
‹
‹
.. ‹ .
. ‹
.. ‹
‹
. ‹
‹
‚
0
9/17
Example 2
Solving differential equations (cont.)
Alternative: Let differentiation convert to ultraspherical bases. [Olver & T.,
2013]
p1 q
p2q
Tk1 px q “ kCk ´1 px q,
p3q
Tk2 px q “ 2kCk ´2 px q,
Tk3 px q “ 8kCk ´3 px q.
For example,
du
` 4xu “ 0
dx
¨
1
˚
˚
˚
˚ 2
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
´1
1
2
´1
´1
p´1qn´1
..
´1
3
.
..
´1
.
1
..
.
n´3
1
´1
n´2
1
Alex Townsend @ MIT
¨¨¨
´1
2
1
1
up´1q “ c,
n´1
¨
˛¨
u0
˛
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‚˝
u1
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹“˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‚ ˝
u2
..
.
..
.
un´1
c
˛
‹
0 ‹
‹
0 ‹
‹
‹
‹
‹
.. ‹ .
. ‹
.. ‹
‹
. ‹
‹
‹
‚
0
10/17
Example 2
Solving differential equations (cont.)
Alternative: Let differentiation convert to ultraspherical bases. [Olver & T.,
2013]
p1 q
p2q
Tk1 px q “ kCk ´1 px q,
p3q
Tk2 px q “ 2kCk ´2 px q,
Tk3 px q “ 8kCk ´3 px q.
For example,
du
` 4xu “ 0
dx
¨
1
˚
˚
˚
˚ 2
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
´1
1
2
´1
´1
p´1qn´1
..
´1
3
.
..
´1
.
1
..
.
n´3
1
´1
n´2
1
Alex Townsend @ MIT
¨¨¨
´1
2
1
1
up´1q “ c,
n´1
¨
˛¨
u0
˛
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‚˝
u1
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹“˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‚ ˝
u2
..
.
..
.
un´1
c
˛
‹
0 ‹
‹
0 ‹
‹
‹
‹
‹
.. ‹ .
. ‹
.. ‹
‹
. ‹
‹
‹
‚
0
10/17
Example 2
Solving differential equations (cont.)
Alternative: Let differentiation convert to ultraspherical bases. [Olver & T.,
2013]
p1 q
p2q
Tk1 px q “ kCk ´1 px q,
p3q
Tk2 px q “ 2kCk ´2 px q,
Tk3 px q “ 8kCk ´3 px q.
For example,
du
` 4xu “ 0
dx
¨
1
˚
˚
˚
˚ 2
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˚
˝
´1
1
2
´1
´1
p´1qn´1
..
´1
3
.
..
´1
.
1
..
.
n´3
1
´1
n´2
1
Alex Townsend @ MIT
¨¨¨
´1
2
1
1
up´1q “ c,
n´1
¨
˛¨
u0
˛
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‹˚
‚˝
u1
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹“˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‹ ˚
‚ ˝
u2
..
.
..
.
un´1
c
˛
‹
0 ‹
‹
0 ‹
‹
‹
‹
‹
.. ‹ .
. ‹
.. ‹
‹
. ‹
‹
‹
‚
0
10/17
Example 2
Solving differential equations (cont.)
Results:
[Olver & T., 2013], [T. & Olver, 2015]
u1 px q ` x 3 upx q “ 100 sinp20,000x 2 q,
∇2 u ` 5000p2 ´ y qu “ 0,
up´1q “ 0.
u|BΩ “ 1.
1.2
1
u(x)
0.8
degree(u) = 20391
time “ 1.21s
0.6
0.4
time = 15.5s
0.2
0
−0.2
−1
Alex Townsend @ MIT
−0.5
0
x
0.5
1
11/17
Example 2
Solving differential equations (future work)
Results (in the pipeline, perhaps?):
∇2 u ` 500p1 ´ y qu “ ´1,
Ω “ penrose,
u |B Ω “ 0
Using an extension of Hierarchical Poincare–Steklov scheme [Martinsson 2012].
Alex Townsend @ MIT
12/17
Example 3
Best low rank approximation
Fact: Chebyshev polynomials are not orthogonal in the L 2 inner-product:
ż1
Tj px qTk px qdx , 0,
j , k.
´1
Inconvenience: Tasks connected to L 2 -orthogonality are awkward.
Best least squares poly. approx.: }f px q ´
n
ÿ
ck Tk px q}L 2 “ min.
k “0
Best low-rank approximation: }f px, y q ´
r
ÿ
σj uj py qvj px q}L 2 “ min.
j “1
Alex Townsend @ MIT
13/17
Example 3
Best low rank approximation
Fact: Chebyshev polynomials are not orthogonal in the L 2 inner-product:
ż1
Tj px qTk px qdx , 0,
j , k.
´1
Inconvenience: Tasks connected to L 2 -orthogonality are awkward.
Best least squares poly. approx.: }f px q ´
n
ÿ
ck Tk px q}L 2 “ min.
k “0
Best low-rank approximation: }f px, y q ´
r
ÿ
σj uj py qvj px q}L 2 “ min.
j “1
Alex Townsend @ MIT
13/17
Example 3
Best low rank approximation
Fact: Chebyshev polynomials are not orthogonal in the L 2 inner-product:
ż1
Tj px qTk px qdx , 0,
j , k.
´1
Inconvenience: Tasks connected to L 2 -orthogonality are awkward.
Best least squares poly. approx.: }f px q ´
n
ÿ
ck Tk px q}L 2 “ min.
k “0
Best low-rank approximation: }f px, y q ´
r
ÿ
σj uj py qvj px q}L 2 “ min.
j “1
Alex Townsend @ MIT
13/17
Example 3
Best low rank approximation
Fact: Chebyshev polynomials are not orthogonal in the L 2 inner-product:
ż1
Tj px qTk px qdx , 0,
j , k.
´1
Inconvenience: Tasks connected to L 2 -orthogonality are awkward.
Best least squares poly. approx.: }f px q ´
n
ÿ
ck Tk px q}L 2 “ min.
k “0
Best low-rank approximation: }f px, y q ´
r
ÿ
σj uj py qvj px q}L 2 “ min.
j “1
Alex Townsend @ MIT
13/17
Example 3
Best low rank approximation (cont.)
Alternative: The Pk px q basis is orthogonal wrt. the L 2 inner-product:
ż1
Pj px qPk px qdx “ 0,
j , k.
´1
For best least squares poly. approx. of degree n:
8
ÿ
f px q “
leg
ak Pk px q
ñ
k “0
}f ´
n
ÿ
leg
ak Pk px q}L 2 “ min.
k “0
For best low-rank approximation:
f px, y q “
m ÿ
n
ÿ
j “0 k “0
Cjk Pj py qPk px q
ñ
f px, y q “
r
ÿ
σj uj py qvj px q
jloooooooomoooooooon
“1
Computed via the discrete svd of C
Alex Townsend @ MIT
14/17
Example 3
Best low rank approximation (cont.)
Alternative: The Pk px q basis is orthogonal wrt. the L 2 inner-product:
ż1
Pj px qPk px qdx “ 0,
j , k.
´1
For best least squares poly. approx. of degree n:
8
ÿ
f px q “
leg
ak Pk px q
ñ
k “0
}f ´
n
ÿ
leg
ak Pk px q}L 2 “ min.
k “0
For best low-rank approximation:
f px, y q “
m ÿ
n
ÿ
j “0 k “0
Cjk Pj py qPk px q
ñ
f px, y q “
r
ÿ
σj uj py qvj px q
jloooooooomoooooooon
“1
Computed via the discrete svd of C
Alex Townsend @ MIT
14/17
Example 3
Best low rank approximation (cont.)
Alternative: The Pk px q basis is orthogonal wrt. the L 2 inner-product:
ż1
Pj px qPk px qdx “ 0,
j , k.
´1
For best least squares poly. approx. of degree n:
8
ÿ
f px q “
leg
ak Pk px q
ñ
k “0
}f ´
n
ÿ
leg
ak Pk px q}L 2 “ min.
k “0
For best low-rank approximation:
f px, y q “
m ÿ
n
ÿ
j “0 k “0
Cjk Pj py qPk px q
ñ
f px, y q “
r
ÿ
σj uj py qvj px q
jloooooooomoooooooon
“1
Computed via the discrete svd of C
Alex Townsend @ MIT
14/17
Example 3
Best low rank function approximation (cont.)
Results:
Best least squares approx. of degree 15
f px q “
Alex Townsend @ MIT
Best rank 3 approx.
1
1`100x 2
15/17
Bonus example
Gauss quadrature
The race for high order Gauss–Legendre quadrature, in SIAM News [T., 14]
tic, [x, w] = legpts( 10 ); toc
Elapsed time is 0.001256 seconds.
tic, [x, w] = legpts( 100 ); toc
Elapsed time is 0.001289 seconds.
tic, [x, w] = legpts( 1000000 ); toc
Elapsed time is 0.140018 seconds.
Similar advances in jacpts(), hermpts(), and lagpts().
[Hale & T., 14], [T., Trogdon, & Olver, 15], [Glaser, Lui, & Rokhlin, 07]
Ignace Bogaert
Alex Townsend @ MIT
16/17
Conclusion
and a thank you using Legendre technology
“80% of the time use
Tk ,
otherwise use
something else”
Alex Townsend @ MIT
17/17
Conclusion
and a thank you using Legendre technology
“80% of the time use
Tk ,
otherwise use
something else”
Alex Townsend @ MIT
Rank 1
Rank 2
Rank 3
Rank 4
Rank 5
Rank 15
17/17