Post on 24-Jan-2021
Gang Bao
Department of Mathematics
Zhejiang University
Inverse Problems for PDEs: Analysis,
Computation, and Applications
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Applications
stealth Geophysical inspection
Medical imaging pollution
cloakings
Application Example II: Seismic inversion
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1( , , , ) 0
Data : at 0; ( , ; )
( , , )
p x y z tt
z p
c
x y z
y
c
x t
To determine ?
Cloaking Other applications: Nondestructive testing CT, B-Scan,MRI,PAT submarine detection near-field, nano optics, super-
resolution
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反问题实例二:近场光学 - 科学前沿
突破衍射极限 1 = l/2
2014年诺贝尔化学奖Eric Betzig (92’, 93’) 考虑隐失场 1 = l/10,
(95’) 考虑 PSF
Stefan W. Hell (94’, 95’) STED
William E. Moerner 实验
2 0
1( ) as
0i r
u k u
uikr o r
r r
u u u
反障碍问题:类似Betzig 93’
1 2
1 2
( , , ), max
( , )
u x x d d f
f x x
l 通过近场测量数据 其中( ),
来反演重构障碍物形状
Application Example III:The secret of an ”Ermine”
Leonardo Da Vinci 'painted three Ermine portraits'
Layer Amplification Method
By Cotte Pascal
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Challenges
Nonlinear
Ill-posed
Incomplete data
Global versus local optimization
Large scale/multi scale computations
Uncertainty
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Opportunities
Diverse applications, high demands
industrial, medical, and military applications
Many basic questions are unanswered
USA: J. Keller, P. Lax, A. Calderon,A. Friedman,G. Uhlmann, V. Rokhlin…
Europe: J. Radon, R. Kress, H.Engl ,…Russia:Gelfand,Tikhonov,Lavrentiev…China: Feng Kang, Li Daqian,Zhang
Guanquan
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Calderon problem
ill-posed
Alessandrini’87, Mandache’01
hybridSeo, Nachman, …混合反问题方法
approach: data
multiple frequency!
stability
effective solution
super-resolution
Objective
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Stability for the IP of the Wave Eq.
Uniqueness:boundary control theory
Stability (partial results):Stefanov & Uhlmann (’98, ’05, 08, 12); Montalto(’12)
Uhlmann:CGO, No Caustics
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( )
: to find ( )
t uc x
uu c x
,
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Our New Stabilty Result
Gang Bao and H. Zhang: Sensitivity analysis of an inverse
problem for the wave equations in the presence of caustics,
J. of AMS, 27(2014), 953–981.
Results: Dynamic DtN too strong! Scattering relation Caustics, CGO not applicable Related:stability for lens rigidity problem
Methods:Gaussian beams/microlocal analysis
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Inverse Scattering Problems
Data:
boundary measurements, far-field or near-field
multiple frequency
single frequency
incomplete data or partial data
A reconstruction should be no more precise than that the data it represents
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Inverse Medium Scattering
Abstract Setting:
Previous work:
Dorn, Bertete-Aguirre, Berryman, Papanicolaou (1999),
Haddar and Monk (2002), Hohage (2003),
Colton and Kress, W. Chew et al (2003, 2004),
Sini and Thanh (2012), de Hoop, Scherzer, Qiu (2014)…
Optimization, initial guess/stability
( , ) Data( )M q k k
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Algorithm beyond Born
Born+ recursive linearization
0
1
0
* 1 1
1
Born
Do 1, 2, ...
Do ,...,
1 ( ) ( )
i i
k ki i
i i
k
k k
j j
j j j
k
j j
k k j
q
i
q q
j 1 m
q DM q R q
q q q
(wave number)
(incidence)
End
End
General Remarks
The UP based (data driven) recursive linearization methods are promising when the data sets are appropriate, such as a parameter family.
The methods can handle general media, structures, and sources.
Expected to be useful for inverse problems for other wave propagation models.
Perspectives/Opportunities
Inverse problems with incomplete data.
Standard theory fails: limited aperture,
phaseless data. Inverse problems involving
uncertainties;
Stability for the multi-frequency inverse
medium scattering
Near-field, nano imaging, nano
characterization; classical model ?
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Stability for Multi-frequency IMP 1-D
" 2( , ) (1 ( )) ( , ) 0, (0,1)x k k q x x k x
1-D Case:
IP: Given the reflection coefficientsto determine
0[0, ]k k( )q x
Theorem(Gang Bao, Triki ‘14)
0
100 0
0 0
0 ,
, 1( , )0
Let , (0,1), , , , 1.
Then for any
1 ( ) ( )
m
M q
M q mL L k k
q q C q q M q q q
k k
q q C d k d kk
,
Key:Trace formula
Nano Optics
Nordlander et al. Nano Letters 9, 887 (2009)
Classic
Quantum
Classic
Quantum
Savage et al. Nature 491, 574
(2012)
Multi-physics model is required.
Opportunities
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Multiphysics
Multiscale
Quantum: Cohen-Tannoudji et al. 89
Semi-classic: Stahl 87, Keller 96, Cho 03
high dimension
TDCDFT(Bao et al. 13)
Maxwell
TDCDFT
Stability of multiscale
),(),(
);,(,
Aj
jA
Nano Optics Modeling: Initial Attempt (FRG)
Multi-scale, multi-physics, wide open new frontier for
applied and computational mathematics!
Multiphysics modeling and multiscale computation of
nano-optical response, B., Liu, Luo, SIAP, 2013
A finite element solver for the Kohn-Sham equation
with a mesh redistribution technique, B., Hu, Liu,
JSC, 2012.
An h-adaptive FEM solver for the calculations of the
electronic structures, B., Hu, Liu, JCP, 2012
High harmonics, metal enhancements, 2014
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Related Work
Gang Bao, P. Li and H. Wu, A computational inverse diffraction
grating problem, JOSA A.
Gang Bao. J. Gao, J. Lin, and W. Zhang, Mode matching for the
electromagnetic scattering from three dimensional large cavities, IEEE
Trans. Antennas & Wave Propagation
Gang Bao, K. Yun, and Z. Zhou, Stability of the scattering from a large
electromagnetic cavity in two dimensions, SIAM J. Math. Anal.
Gang Bao, J. Lin, Imaging of reflective surfaces by near-field optics,
Optics Lett.
2012
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Related Work
Gang Bao, P. Li, J. Lv, Reconstruction of perfectly reflecting gratings
from intensity data, J.OSA A
Gang Bao, J. Qian, L. Ying, and H. Zhang, A convergent multiscale
Gaussian-beam parametrix for wave equations, Comm. in P.DE.
Gang Bao, J. Lin: Near-field imaging of the surface displacement on an
infinite ground plane, IPI,
Gang Bao, F. Triki: Reconstruction of a defect in an open waveguide,
Sci. China Math.
Gang Bao, P. Li: Near field imaging of infinite rough surfaces. SIAM
J. Appl. Math.
2013
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Related Work
Gang Bao, H. Zhang: Sensitivity analysis of an inverse problem for the wave
equations in the presence of caustics, J. of AMS
Gang Bao, S-N Chow, P. Li, and H-M Zhou, An inverse random source problem for
the Helmholtz equation in one dimension, Math. Comp.
Gang Bao, H. Zhang, and J. Zou, Unique determination of periodic polyhedral
structures by scattered electromagnetic fields Part II, Trans. Amer. Math. Soc.
Gang Bao, J. Lai and J. Qian: Fast multiscale Gaussian beam methods for wave
equations in bounded domains, J. Comput. Phys.
Gang Bao, J. Lin and S. Mefire: Numerical reconstruction of electromagnetic
inclusions in three dimensions, SIAM J. Imag. Sci,
Gang Bao, T. Cui and P. Li: Inverse diffraction grating of Maxwell's equations in
biperiodic structures, Optics Express
Gang Bao, H. Liu and J. Zou: On near-cloak in electromagnetic scattering, J. Math.
Pures Appl.
2014