w:x&` k
Æ"-!,)&
.%0
Marco Iglesias Andrew M. Stuart
oJ% N ℄e X ℄y )TJ%Hve ℄ sxTh <Hv
P "%0TV3<e#: hFT*Pj m a( F>
w=V, VM w M m 1veTTg.TJ%iÆ X ℄
y )Tx*/q<< C oFT
A $ YoJ% . eTH3nT 2 t%?K - x7:>Fw?
2 t|TÆ b Z 'xR3VlÆ b TY } /7>&s _ n 87?V>
&T}b% $ 2 tnkT,,a X ℄T>\T7>&a X
℄T%?a X ℄ySP )7) 87?V>&:T ^ !q%7}b% $ R
1SP<FT e 1. *' Bayes (/$)1) | T 2 t Æ TÆ b u Æ: y Tr4Æ b xH | l w Z
H FizTM Æ:L>\Tx&℄T e7XJI (forward
problem), 2 t G /7e℄TÆ b u >& | TÆ: y. e78JI (inverse problem), 2 tL/ :eR1 w 1℄& SPTÆ: y TÆ b u T7d [6, 11];
| Ce$goJ%T#nH Hadamard (D ') V 4F6 T - _ i ( ℄yzL —— BHy=y℄enT yeTN&STnG3
J v = / 7N 2 t9 ez R>oJ%℄T vT ^ !7d>&e
3T X ℄yF )jhToJ%T Bayes rn ~ K - x3?LTr 6
KV ^ NNN ~ /?rn z x. z WL)oJ%T℄y
H Bayes rne_ > (u, y) : oJ%T℄L: u H℄ y z
T+P " vB> P(u|y) (?GeO> u|y—— #u). V+P " vT 1
Bayes <V :>
$SIAM News, Vol. 47 (2014), No. 6, UQ and a Model Inverse Problem: CO Capture/
c 2014 Society for
Storage, Marco Iglesias, Andrew M. Stuart, figure number 3. Copyright Industrial and Applied Mathematics. Reprinted with permission. All rights reserved. , +02;U9$H;y
Marco Iglesias ) Nottingham GCUq X Andrew Stuart ) Warwick
GiUX
H!w98 Stuart I,Jd
k Savannah mxU 2014 ,
;+02Y ^z
*2" (2014 SIAM Conference on Uncertainty Quantification) hUTm2U
1) Thomas Bayes, 1702–1761 = )1AESE / 2 6x 1742 = 2?)/E22
ABayes D I #& =UiI 1 8$;U Bayes ^ f8^J #& 1A
UmO4iU-f—— jv
2
· 11 ·
P(u|y) ∝ P(y|u)P(u).
CG>=R5 "v P(u|y), >H℄n y '7BYT u K-YQT
Y}W7 HD P(y|u) 3YT P(u), ~Ze7℄TÆb u, :℄&P
T y Tx : 4T P(u) L / K - HSPnZ H 7BY TY } V . K
-/ p'*/T bTsBtV  hFK - u,hT0
Bayes L= C */ $ Y
575 1 eKR v/#d
}b7 GE> e .
℄CZT AP G; in y /
76S =R v& d| b
H GE> e
" vYP'T
6 1 Bayes pfUY^z *
Vp>K - {) K
-TEjH :Z5? q eX℄yT>&e/n (bu KR), K-:#d
}bT>&/n (SP =R), K-x:&d|bTdUT>&
u3T>&aH3 X ℄y?8HV5℄eoJ%T Bayes rn ——
a HuT ' " vTNsq<e 0 T —— UW X ℄yo. n
6STh8TqQbT 'g xtPV X ℄yTUWH ^ !*/ —— H. #
PT( F > wV, T =H ` !M.TF x ~ /℄ 2 tJ
v nT*/ <T  —— eT=
2. #*' CO2 21
7) e*/#:T*P
{)8 > 6Se:℄*P
TJ%T4lqb`)K-u
,T{Br 7_.℄
TM Fk*TJ% [3]. a G n K - s R T Æ CO
u b YVTe?xwye,g
,eTJ%g\tT CO B
6 2 CO UZWCs
rY[*~~[T2 F )7$
q T ' (5 2). V z T 2 t1 / Æ CO u b YV vT \ oT ? ;vr
41
2tThÆbBrY[Tt3yCTY"ymad'zTBHV
Z TÆ: u b fTX| T& \=x : T+ Ern
GPS n TYvT& \17YV . [Yx)T%?& \ (Æ
:) 7d C "y (Æ b) TJ%j"h 3z X T7djxHgguT 2
2
2
2
· 12 ·
T8< g :q%Vl l =PBrY[T T y"NxwyVa ℄- R:
CO n O`T I HTYVL [ Ty\
NK - u HtKL | L Z T Bayes ℄ ! e'$| {)z V*/e3T
*PjS 0 [ H e Z p ,WJ% 2 t —— / g|_eg M yT
% ! (? ;vr4)—— T℄ ℄ Tt 3yTÆ b {J g T "
vH{J g T " v| C g VC|K - RHu,T{J g " 7> qb h' " vK - |6HoFTÆ bt 3y{J
gN ℄}GT ? ;vr4T*P TX ℄ *P:℄HF T*/e Za ?
℄-HST 10 <
etP7oJ%T Bayes rnTNFTwn [10] HQ
/NaT:xFT~rnr .E P h/
%>K - Tz%H*/ 5 TM.=Hrn % x=Tq<r . -=HT9>
B HN. z J *P Z 1BHu,"℄*/TJ%TOQ 2l =B
Y TYKOTJ% l =8vTz X qb h' " vTr Vz X jK -: 7uT T' " vTz X ?8VlQ 2 J%V(/T
Vz%|*/DT72 b -: 3p7,vPnT}Gy,
rnT>p ℄OnTbVg|Hgg"℄*/TY 2 tT L gV
91V < g T1|pevP (a 87 ? ;vr4TWJ% 2 t), (a ;v), "% (a ℄℄:+TY ") 0 (a > h'
"TnnzslTn) TLf:H*gL Vha|SP"℄
*/TQ 2 u T℄&
0P CO BrT K - u
eYV t 3yYY> a 5 3
5eJ5T.T “\t
y” H5e t 3yv,
TC BY# X ℄ 1YVT
.M'TDa= U `TDa 6 3 Q / u 4zUZH o 4 V aUw - 3
t3yH'9T\HeJ5e u (), w (3u (f), w -R3u (6)
t 3y,LV^ . T℄D X ℄V^ g VT[n)/K ?
℄L\TV g “\tT” . t 3yTY 2 tR y v.TRO`.
Tu iKaUiYY} ?Y2 t7~|" g t 3y1 g “\
t”, 7~-|u m a 5 3 (5) T C tVlJ%TUZ=S
E slVl 2 tT4G_T#uG_xH [9] eQP
M 0X ℄ Y " vWJ% 2 t' " vj1 Bayes L X ℄
Z 'K - jyh h' " T=7T B :\Monte Carlo-Markov
(' jx{) 7B MCMC, hVlJ%T Z Tni K S 10 g <℄
- etPH{JeVlrnH % | M g NFT [5]. Za ?A ,T
2
2
· 13 ·
87 Monte Carlo Trn<> 5W > C N T " G b [2], VC|
}Gy (* LDO*TII) 1NvT%?K - 3:tPoJ% L Vg'
Monte Carlo rn [8] ; Monte Carlo rn [2] TlN?8 Tgg 4
rn=
CrnT*/ [1, 4, 7, 12], aH [13] el0T7$x :yPoJ%T!
e-3Vlrn3x: Monte Carlo rnT}Gy3sRTbTxHG [14, 15]
= C #uG_eQPTdU_ j HVps <*/F Th:11 oF tjgT*/=|
>Z:Æ_Tq< Tvm <Th O _TVz%W=H
yR[ : Ws  E~ h y v}GT Bayes 2 tKP SkT:1$
ÆZe7WH N pTh*PT*/D V<Th <
YO GTu,o(4 Z vhA1 (Engineering and Physical
Sciences Research Council, EPSRC), > lhA1 (European Research Council, ERC)
= + rhk (Office of Naval Research, ONR) >X7GTh#T}p
Andraw M. Stuart W> EPSRC }pg 5 EQUIP TSvDhttp://www2.warwick.
−1/2
ac.uk/fac/sci/maths/research/grants/equip/.
T NWYO }MFvBT |h~U r D K - T#G#
: $T{NWH}MFtJT Andraw M. Stuart W E ! %
K - T#GK - h - 5^To
(+
[ 1 ] I. Babuska, R. Tempone, and G. Zouraris, Galerkin finite element approximations of
stochastic elliptic partial differential equations, SIAM J. Numer. Anal., 42 (2004),
800–825.
[ 2 ] R. Caflisch, Monte Carlo and quasi-Monte Carlo methods, Acta Numer., 7 (1998), 1–49.
[ 3 ] A. Chadwick, CCS: Between a rock and a hard place? Greenhouse Gases: Science and
Technology, 2 (2011), 99–101.
[ 4 ] A. Cohen, R. DeVore, and C. Schwab, Convergence rates of best N-term Galerkin
approximations for a class of elliptic sPDEs, Found. Comput. Math., 10 (2010), 615–
646.
[ 5 ] S. Cotter, G. Roberts, A. M. Stuart, and D. White, MCMC methods for functions:
Modifying old algorithms to make them faster, Stat. Sci., 28 (2013), 424–446.
[ 6 ] H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer,
Philadelphia, 1996.
[ 7 ] R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach,
Springer, New York, 1991.
[ 8 ] M. Giles, Multilevel Monte Carlo path simulation, Oper. Res., 56 (2008), 607–617.
[ 9 ] M. Iglesias, K. Lin, and A. M. Stuart, Well-posed Bayesian geometric inverse problems
arising in subsurface flow, arxiv.org/abs/1401.5571, to appear in Inverse Problems,
special issue on Bayesian inversion, 2014.
[10] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, 160,
Applied Mathematical Sciences, Springer-Verlag, New York, 2005.
[11] J. B. Keller, Inverse problems, Amer. Math. Monthly, 83 (1976), 107–118.
· 14 ·
[12] C. Schwab and C. J. Gittelson, Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs, Acta Numer., 20 (2011), 291–467.
[13] C. Schwab and A. M. Stuart, Sparse deterministic approximation of Bayesian inverse
problems, Inverse Problems, 28 (2012); arxiv.org/abs/1103.4522.
[14] A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numer., 19 (2010),
451–559.
[15] A. M. Stuart, Inverse Problems and Uncertainty Quantification, invited lecture, ICM
2014, Seoul; http://homepages.warwick.ac.uk/~masdr/TALKS/stuart SIAMUQ.pdf.
3; #
(
SCM i)
****************************************************************************
g Y 81 )
(
(+
[ 1 ] L. E. J. Brouwer, Leven, Kunst en Mystiek, J. Waltman Jr., Delft, 1905. English
translation with an introduction by W. P. van Stigt in Notre Dame Journal of Formal
Logic 37(3) (1996), 381–429.
[2]
, Collected Works, Volume 1: Philosophy and Foundations of Mathematics, A.
Heyting, editor, North-Holland, Amsterdam, 1975.
[3]
, Collected Works, Volume 2: Geometry, Analysis, Topology and Mechanics, H.
Freudenthal, editor, North-Holland, Amsterdam, 1976.
[ 4 ] J. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Harvard,
Cambridge, MA, 1979.
[ 5 ] H.-D. Ebbinghaus with V. Peckhaus, Ernst Zermelo: An Approach to His Life and
Work, Springer, Berlin, 2007.
[ 6 ] M. Georgiadou, Constantin Carathéodory: Mathematics and Politics in Turbulent
Times, Springer, Berlin, 2004.
[ 7 ] J. Gray, Henri Poincaré: A Scientific Biography, Princeton Univ. Press, Princeton, NJ,
2013.
[ 8 ] C. Reid, Hilbert, George Allen & Unwin, London, Springer, Berlin, 1970. See also the
extensive publication of Hilbert’s lectures and algebraic number theory being published
by Springer.
[ 9 ] A. S. Troelstra and D. van Dalen, Constructivism in Mathematics: An Introduction, 2
volumes, Elsevier, Amsterdam, 1988.
[10] D. van Dalen, Mystic, Geometer, and Intuitionist: The Life of L. E. J. Brouwer, Volume
1: The Dawning Revolution, Oxford Univ. Press, Oxford, 1999; Volume 2: Hope and
Disillusion, Oxford Univ. Press, Oxford, 2005.
[11]
, The Selected Correspondence of L. E. J. Brouwer, Springer, London, 2011; in
English, with extra material available online at Springer.
[12]
, Logic and Structure, 5th edition, Springer, Berlin, 2013.
[13] W. van Stigt, The rejected parts of Brouwer’s dissertation on the foundations of mathematics, Historia Mathematica 6 (1979), 385–404.
[14] F. Verhulst, Henri Poincaré: Impatient Genius, Springer, New York, 2012.
[15] H. Weyl, Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis,
Berlin, Veit, 1918. English translation, The Continuum: A Critical Examination of the
Foundations of Analysis, Dover, New York, 1994.
[16]
, Gesammelte Abhandlungen, 4 volumes, Springer, Berlin, 1968.
UL7 #
(
9QB i)
· 15 ·