Least Squares Formulas - The Swiss Army Knife of Numerical Integration?a - The Swiss Army Knife of Numerical ...
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Least Squares Formulas - The Swiss Army Knife of Numerical Integration?a 2021 SIAM Annual Meeting (AN21) Jan Glaubitz1,2 Joint work Philipp Öffner3 and Anne Gelb1 July 19, 2021 1 Department of Mathematics, Dartmouth College, NH, USA 2 Max Planck Institute for Mathematics, Bonn, Germany 3 Institute of Mathematics, Johannes Gutenberg-University Mainz, Germany a This work was partially funded by DFG #SO 363/15-1 (Glaubitz), AFOSR #F9550-18-1-0316, ONR #N00014-20-1-2595 (Glaubitz and Gelb), and SNF #175784 (Öffner)
Outline
1. LS-QFs: Problem Statement & The Basic Idea
2. Past and Recent Developments
3. LS-CFs: Construction of provable positive and exact cubature formulas
4. Concluding Remarks
1Problem Statement (1D)
Consider the weighted integral
Zb
I [f ] := f (x)é(x) dx, é : [a, b ] → + (Riemann integrable)
a
Problem (Numerical Integration for Experimental Data)
Given N distinct data points x1 , . . . , xN ∈ [a, b ], find a positive and high-order
quadrature formula (QF) such that
N
¼
QN [f ] := wn f (xn ) ≈ I [f ]
n =1
for all Riemann integrable (or at least continuous) functions f .
• QN is positive ⇐⇒ wn > 0 for all n = 1, . . . , N
• High-order means that QN should be exact for polynomials of arbitrary
degree if N is sufficiently large. 2Least Squares Quadrature Formulas (LS-QFs)
• Let d () = span{ï0 , . . . , ïd }. We want a QF that is exact for all f ∈ d (),
QN [ïk ] = I [ïk ], k = 0, . . . , d . (1)
• If d + 1 < N , then (1) yields an under-determined linear system, Ðw = m,
ï0 (x1 ) . . . ï0 (xN ) w1 I [ï0 ]
. .. .. = ..
.. . . . (2)
ï (x ) . . . ï (x ) w I [ï ]
d 1 d N N d
which solutions form a (N − d − 1)-dimensional affine linear subspace of N
Definition (LS-QFs)
The LS-QF is the unique QF which weights correspond to the LS solution of (2):
N
¼
QNLS wnLS f (xn ), wLS = arg min kwk2
=
n =1 Ðw=m
3Some Past and Recent Developments
• Wilson (1970, Math. Comp.): Introduces the idea of LS-QFs (“nearest point
quadrature”)
• Wilson (1970, SINUM): For é ≡ 1 and equidistant data points, wLS > 0 if
N ≥ d (d + 2)
• Huybrechs (2009, Comp. Appl. Math.): Describes extension to general positive
weight functions é and scattered data points
• Glaubitz (2020, SINUM): Stable high-order LS-QFs for general é (allowed to
have mixed signs)
• Glaubitz (2020, arXiv): Provable positive and exact LS cubature formulas for
bounded domains Ò ⊂ s in arbitrary dimensions.
4Provable positive and exact LS cubature formulas (1/2)
Goal
Given N distinct data points x1 , . . . , xN ∈ Ò ⊂ s , we want to find a cubature
formula (CF) CN such that
• CN is positive, i. e., wn > 0 for all n = 1, . . . , N
• CN is exakt on F ⊂ C (Ò), i. e.,
Z
CN [f ] = f (x)é(x) dx ∀f ∈ F
Ò
Assumptions:
(A1) Ò ⊂ s is bounded with boundary of measure zero
(A2) é : Ò → + is Riemann integrable and positive almost everywhere
(A3) F is spanned by a basis of continuous and bounded function, {ïk }Kk=1 .
Furthermore, F contains constants. 5Provable positive and exact LS cubature formulas (2/2)
Main Result [Corollary 3.4 in (Glaubitz, arXiv:2009.11981 2020)]
Let (xn )n∈ ⊂ Ò be an equidistributed sequence with é(xn ) > 0 for all n.
Assuming that (A1) to (A3) hold, there exists an N0 ∈ such that for all N ≥ N0
and diagonal weight matrix
√ √ é(xn )N
R = diag r1 , . . . , rN , rn = , n = 1, . . . , N ,
vol(Ò)
the (weighted) LS-CF
N
¼
CNLS [f ] = wnLS f (xn ), wLS = arg min kR wk2
n =1 Ðw=m
is positive and exact for all f ∈ F .
vol(Ò) ´N
R
(xn )n∈ ⊂ Ò equidistributed ⇐⇒ limN →∞ N n =1 g(xn ) = Ò g(x) dx for all
measurable bounded functions g that are continuous almost everywhere
6Resulting Procedure & Numerical Results
Proposed Procedure:
• Consider d (s ) with total degree d that is successively increased.
• In every step (for fixed d ), construct an LS-CF that is positive and exact for
d (s ). This is done as follows.
• Starting from N = dim(d (s )), increase N (e. g. double it) until the
corresponding LS-CF is positive.
y
2 10 0
Ò
1
10 -5
-1 1 2x
10 2 10 3
-1
(b) Errors for é ≡ 1 and
7
(a) Domain f (x1 , x2 ) = exp(−x12 − x22 )Concluding Remarks
• The concept of least squares can be used to construct stable and high-order
quadrature and cubature formulas in many different situations
• In particular, we can construct provable positive and exact CF for quite
general domains and weight functions
Open Problems (Future Research)
• Can the condition that the data points come from an equidistributed
sequence be relaxed?
• Can we extend LS-CFs to general weight functions (potentially having mixed
signs) in multiple dimensions?
8References
This talk was based on the following works ([4, 3, 2, 1]):
J. Glaubitz.
Constructing positive interpolatory cubature formulas.
arXiv preprint arXiv:2009.11981, 2020.
J. Glaubitz.
Stable high-order cubature formulas for experimental data.
arXiv:2009.11981, 2020.
J. Glaubitz.
Stable high order quadrature rules for scattered data and general weight functions.
SIAM Journal on Numerical Analysis, 58(4):2144–2164, 2020.
J. Glaubitz and P. Öffner.
Stable discretisations of high-order discontinuous Galerkin methods on equidistant and scattered
points.
Applied Numerical Mathematics, 151:98–118, 2020.
Thank You!
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