Gang Bao

Department of Mathematics

Zhejiang University

Inverse Problems for PDEs: Analysis,

Computation, and Applications

1

2

Applications

stealth Geophysical inspection

Medical imaging pollution

cloakings

Application Example II: Seismic inversion

2

22

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

2

2

10

( )

: 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