修士論文 要旨

Sparse grids capturing exponential decay and smoothness in Besov spaces(ベゾフ空間における関数の指数減衰と滑らかさを考慮したスパースグリッド)
数理情報学専攻 48216214
指導教員

1 Introduction
Many problems in computational science and engineering involve high-dimensional approximations.
Such functions appear when describing complex, nonlinear models and phenomena. In general, approximation of high-dimensional functions with linear
combination of basis functions require a large numberof basis functions. In some cases, the number of basis
functions grows exponentially with respect to dimension. A method called sparse grid is known as a technique to mitigate this computational diﬃculty. The
fundamental idea of sparse grids is to choose a ﬁxed
number of basis functions so that approximation error becomes as small as possible. Especially, smoothness and decay of functions are important to determine which basis functions should be used in sparsegrids. In this study, by considering newly deﬁnedfunctions spaces called exponentially weighted Besov
δ,w(Rd ), wespaces with dominating smoothness V Bp,qgive constructions of sparse grids capturing exponential decay and smoothness. Speciﬁcally, by obtainδ,w
ing wavelet characterization of V Bp,q(Rd ), the sparsegrids are obtained.
The motivation to consider sparse grids of this type
is to solve eigenvalue problems of the following Hamilton operator H.
NN X
KX1XZν
H := −∆i −2 α=1|xi − aν |α=1 ν=1+12NX
α,β=1α̸=β1.
|xi − xj |田中健一郎 准教授s̄δ,wness M Bp,q(Rd ). V Bp,q(Rd ) is a weighted space
whose weight function w increases or decreases exponentially at most, and has ﬂexibility in smoothness. Especially, this ﬂexibility includes smoothss̄
ness of Bp,q(Rd ), M Bp,q(Rd ) and their interpolations.
Deﬁnitions of these two Besov spaces are given by thefollowing quasinorms.
1/q∞Xs
f Bp,q(Rd ) := 2sjq ||ϕj (D)f ||qLp (2)j=0s̄
f M Bp,q(Rd ) := X

1/q2s̄·j̄q ||Φj̄ (D)f ||qLp j̄∈Nd0(3)where {ϕj }j∈N0 and {Φj̄ }j̄∈Nd0 are dyadic resolutions of unity, and ϕj (D) = F −1 ϕj F and Φj̄ (D) =F −1 Φj̄ F (F is Fourier transform). Real numbers s inss̄
(Rd ) are smoothness paramBp,q(Rd ) and s̄ in M Bp,qeters, and control the convergence rate of Lp -norm offrequency components. These parameters are generalized by using norms δ on Nd0 instead. Exponentially weighted Besov spaces with dominating mixed
δ,wsmoothness V Bp,q(Rd ) are deﬁned by the following
quasinorms.
1/q
∞X
δ,w
 .
2δ(j̄)q ∥Ψj̄ ∗ f ∥qLw
f V Bp,q(Rd ) := pj=0(1)

The Hamilton operator H, appears in theSchrödinger equation where multiple electronsinteracts. The dimension of this system is 3N ,where N denotes the number of electrons. Thus, thedimension of problems are high when the number ofelectrons N are large. However, the eigenfunctionsof above Hamilton operator H are known to have exponential decay and dominating mixed smoothness.
There are rooms to improve complexity of eigenvalue
problems of H by considering sparse grids capturing
exponential decay and smoothness.

2 Exponentially weighted Besov spaces withδ,w
dominating smoothness V Bp,q(Rd )δ,wV Bp,q(Rd ),小暮祥弘

Besov spaces
which are newly deﬁned in this study, is an extension of normal Besov
sspace Bp,q(Rd ) and Besov space with mixed smooth-(4)Note that, unlike quasinorms of non-weighted Besov
spaces (2) and (3) , weighted Lp -norms are used in(4). Furthermore, in (4), the dyadic resolution of
δ,wunity can not be used to deﬁne V Bp,q(Rd ), thus different test functions {Ψj̄ }j̄∈Nd0 are used. The reason
for this is technical. For more details, see our paper.
δ,w
3 Wavelet characterization of V Bp,q(Rd )and construction of sparse grids

Wavelet is a collection of orthogonal functions{ψ j̄,m̄ }j̄∈Nd0 ,m̄∈Zd which are derived from scaling andtranslation of only two functions. A function f ∈δ,w
V Bp,q(Rd ) allow an expansion by wavelets.
X Xλj̄,m̄ ψ j̄,m̄ (x)f (x) =m̄∈Zdj̄∈Nd0where λj̄,m̄ are real numbers and coeﬃcients of ψ j̄,m̄ .
δ,wNote that quasinorms of f |V Bp,q(Rd ) are characterized by {λj̄,m̄ }j̄∈Nd0 ,m̄∈Zd . Under certain assumptions, we prove the following relations(Theorem 3 inSparse grids capturing exponential decay and smoothness in Besov spaces (小暮祥弘)

Section 4.1).
XA∥f ∥V Bp,qδ,w ≤ X⃗

2δ(j)qq

λj̄,m̄ χj̄,m̄m̄∈Zd⃗j∈Nd01/qLpw≤ B∥f ∥V Bp,qδ,wwhre A and B are some positive constants, and χ⃗j,m⃗is an indicator function on Qj̄,m̄ = [m̄/2j̄ , (m̄+ 1̄)/2j̄ ].
This relation is called wavelet characterization ofδ,w
δ1 ,w
V Bp,q(Rd ). Sparse grids on V Bp,q(Rd ), which cancapture exponential decay and mixed smoothness offunction, can be easily derived from wavelet characδ,wterization of V Bp,q(Rd ). Consider an approximationof f by the following ﬁnite wavelet expansion.
X Xf≈λ⃗j,m̄ ψj̄,m̄(5)These choices are due to the regularity results in [1].
Results of numerical experiments are shown in ﬁgure
2. Minimum eigenvalue problems are solved when
N = 2, 4, 6. The results show we obtain asymptotic
convergence of minimum eigenvalue Emin when N =2, 4, but we can not reach asymptotic convergencewhen N = 6 with number of basis functions around
two hundred thousand.

m̄∈Mj̄
j̄∈Nd0where ﬁnite M⃗j are non-empty so that summation in
the left side is a ﬁnite summation. An approximation
error of (5) can be estimated by wavelet characterization, i.e.,f−図 1. Plots of center of wavelets in 2 dimension

X X

2when w(x) = eb/p|x|1 , b/p = 4, δ2 (·) =0.25| · |1 + 0.75| · |∞ , δ1 (·) = 0λ⃗j,m⃗ ψj̄,m̄m̄∈Mj̄
j̄∈Nd0δ ,11V Bp,q≲ max 2−(δ2 (j̄)−δ1 (j̄))
j̄∈Nd0|Q⃗j,m̄ |max

!1/p w(Qj̄,m̄ )m̄∈Zd \Mj̄×∥f ∥V B δ2 ,w .
p,q


(6)

Sparse grids, a collection of sets {M⃗j }j̄∈Nd0 , are obtained by minimizing approximation error in (6).
Figure 1 shows an example of obtained sparse grids
when dimension is 2.

4 Numerical Experiments
Consider the following problem to ﬁnd minimum
eigenvalue of H
HΨ = Emin Ψ
where Emin is a minimum eigenvalue, Ψ is a corresponding eigenfunctrion and H is 1-dimensional
Hamiltonian deﬁned by
H=−NN XK
X1X
∆α −Zν |xα − aν |2 α=1α=1 ν=11−22

where ∆α = ∂x∂2α,1N
X|xi − xj |α,β=1α̸=β2+ · · · + ∂x∂2α,d′and d′ = 1. In

this numerical experiments, we use sparse grids bysetting δ1 = 0, δ2 (j̄) = θ|j̄|1 + |j̄|∞ and p = 2 in (6).

図 2. Results of eigenvalue of problems of H.

∆E/E is an absolute value of diﬀerenceof successive eigenvalue over previous minimum eigenvalue.

参考文献[1] Hans-Christian Kreusler and Harry Yserentant. The
mixed regularity of electronic wave functions in fractional order and weighted sobolev spaces. Numerische
Mathematik, 121(4):781–802, 2012.