RESEARCH SUMMARY
Kui Ren
Department of Mathematics, University of Texas at Austin
ren@math.utexas.edu
http://www.math.utexas.edu/users/ren
My research interests include mainly: i) Inverse Problems and Mathematical Imaging and
ii) Numerical Analysis and Scientific Computation. My research has been supported by National Science Foundation, through grants DMS-0914825 and DMS-1321018, and University
of Texas at Austin.
Highlights of Results. Here are some of the main results of my research in recent years:
(i) Together with my collaborators, we have performed constructive mathematical analysis on elliptic inverse problems in quantitative photoacoustic tomography (QPAT)[8]. The
analysis led to not only uniqueness and stability results on the inverse problems but also a
direct reconstruction method that is non-iterative. This work has been selected by the journal “Inverse Problems” as an “Insights” article 1 ; (ii) Together with my collaborators, we
have proposed an imaging algorithm for QPAT using multi-spectral data [10]. We showed
that spectral data allow us to reconstruct all three interesting coefficients uniquely and
stably. This is the first work in the field to achieve simultaneous reconstruction of three
optical coefficients. This work have been selected as a feature article and a “Highlight of
2012” article by the journal “Inverse Problems” 2 ; (iii) Together with my collaborators, we
have recently proposed a new hybrid imaging scheme, called fluorescence photoacoustic tomography (fPAT), aiming at stabilizing classical fluorescence tomography with ultrasound
data [37, 38]. We constructed a mathematical model, a system of elliptic PDE, for the
new imaging modality and a complete mathematical theory on related inverse problems;
(iv) Together with my collaborators, we proposed the first inverse transport algorithm in
the field for optical tomography with frequency-domain data [31, 32]. Mathematical and
numerical analysis later showed that the algorithm can reduce significantly the so-called
cross-talk between the scattering and absorption coefficients, especially in high-frequency
regime. The reconstruction algorithm has been used to image small animals (mainly to
monitor tumor growth in small animals) in real-world experiments. The two initial publications on the reconstruction algorithm have combined about 25 citations each year since
their publication dates according to Google Scholar; (v) With my collaborators, we have recently designed an imaging scheme in PAT to reconstruct acoustic and optical properties of
heterogeneous media using differential acoustic measurement [24, 29]. This scheme provides
a solution to the problem of unknown ultrasound speed in photoacoustic tomography.
1
See the insight news article on the journal’s webpage: http://iopscience.iop.org/0266-5611/labtalkarticle/46460
2
See the links to the collection of “Highlight of 2012” articles on the journal’s webpage:
http://iopscience.iop.org/0266-5611/page/Highlights of 2012
1
1
Inverse Boundary Value Problems in Applications
Inverse boundary value problems aim at reconstructing coefficients in partial differential
equations (PDEs) from the knowledge of over-determined Cauchy boundary data. These
inverse problems find applications in various modern imaging techniques. In particular,
diffuse optical tomography (DOT) is a non-invasive medical imaging modality that uses
near infra-red (NIR) light sources to probe physical properties of biological tissues. In a
DOT experiment, one sends NIR photons into the tissue to be probed, say Ω ⊂ Rd . One
then measures the photon currents that exit the tissue through its surface ∂Ω. From the
measurements, one attempts to recover the absorption and scattering properties of the
tissue which provide useful diagnostic information.
The distribution of NIR photon density u(x, v) in tissues is accurately modeled by the
following phase-space radiative transport equation (RTE):
Z
v · ∇u + σ(x)u = σs (x)
K(v, v0 )u(x, v0 )dv0 , in Ω × Sd−1
(1)
d−1
S
u = f (x, v),
on Γ− .
Here Sd−1 is the unit sphere in Rd , dv is the normalized surface measure on Sd−1 , and
K(v, v0 ) is the scattering phase function that describes the way that photons traveling
in direction v getting scattered into direction v0 . The boundary sets Γ± are defined as
Γ± = {(x, v) ∈ ∂Ω × Sd−1 s.t. ± v · n(x) > 0} with n(x) the unit outer normal vector at
x ∈ ∂Ω. The light source is modeled by the function f (x, v). The positive functions σa (x) =
σ(x) − σs (x) and σs (x) are the absorption and the scattering coefficients, respectively.
Mathematically, diffuse optical tomography can be formulated as an inverse boundary value
problem for the transport equation that aims at reconstructing the absorption σa and
scattering σs coefficients from boundary data given by the albedo operator:
Z
v · n(x) u(x, v) dv
(2)
Λ : f (x, v) 7→ J(x) =
Sd−1
+ (x)
d−1
with Sd−1
: v · n(x) > 0}.
+ (x) ≡ {v ∈ S
This inverse boundary value problem is mathematically and computationally challenging.
There are two practically important issues to be addressed. First, the inverse problem is in
general severely ill-conditioned, as we showed in [4] in a simplified setting. What we mean
by “severely ill-conditioned” is that even though the inverse problem may admit a unique
solution, when no regularization is applied, noise contained in the measurements will be
amplified more than what would result from an arbitrary number of differentiations during
the numerical inversion procedure. It is thus true that to obtain stable reconstructions,
we have to provide more data than these given in (2). Second, this inverse boundary
value problem is computationally very expensive to solve due to the fact that the transport
equation (1) is posed in a high-dimensional phase space. Model-based iterative schemes are
usually very slow.
Frequency-Domain Inverse RTE. To stabilize the inverse problem, we modulate the
NIR illumination source to collect frequency-domain boundary data. In [31, 32], we de2
veloped an inverse transport algorithm based on frequency-domain data. This is the first
frequency-domain reconstruction algorithm that is based on the radiative transport equation. Our mathematical and numerical analysis later [16, 26, 33, 35] showed that frequencydomain data stabilize the inversion significantly in certain situations, enabling us to simultaneously reconstruct the absorption and scattering coefficients in the transport equation.
Fast Reconstruction Algorithms. To improve the efficiency of numerical reconstruction algorithms, we developed several strategies that utilize the phase-space structure of the
transport equation [1, 14, 26, 27, 28]. This structure allows us to combine efficient preconditioning methods (based on the tensor structure in the discretized scattering operator) with
efficient numerical optimization methods to derive reconstruction algorithms that can be
effectively parallelized. We also developed algorithms that utilize the ill-conditioningness
nature of the inverse problems to efficiently parameterize the unknown coefficients with
least amount of generalized Fourier coefficients [17]. This strategy allows us to significantly
reduce the number of the unknowns in optimization-based reconstruction algorithms.
2
Hybrid Multi-Physics Inverse Problems
Hybrid inverse problems refer to inverse problems of coupled systems of two or more PDEs of
different types. This type of inverse problems are the mathematical foundations for emerging multi-modality coupled-physics imaging techniques where one combines two imaging
modalities based on different physics to obtain the advantages of both. Photoacoustic tomography (PAT) is one of such modalities that combine the high-contrast diffuse optical
tomography with the high-resolution ultrasound imaging. In a PAT experiment, one sends
a pulse of NIR light into a biological tissue. The tissue absorbs part of the incoming light
and heats up due to the absorbed energy. The heating then results in the expansion of the
tissue and the expansion generates compressive (acoustic) waves. One then measures the
time-dependent acoustic signal that arrives on the surface of the tissue. From the knowledge of the acoustic measurements, one reconstructs the optical properties as well as the
photoacoustic efficiency of the tissue.
The model for the propagation of NIR light in biological tissues is again the radiative
transport equation (1). The photon energy that is absorbed at location x ∈ Ω per unit
volume is the product of the absorption coefficient and the photon density. The heating
due to this absorbed energy generates an initial pressure field, denoted by H, that is of the
form:
Z
H(x) = γ(x)
σa (x)u(x, v)dv ≡ γ(x)σa (x)hui(x)
(3)
Sd−1
where the positive function γ(x) is the nondimensional Grüneisen coefficient which in the
current formulation measures the photoacoustic efficiency of the tissue. The initial pressure
field H then propagates according to the acoustic wave equation, with the wave speed c(x),
3
1 ∂ 2p
− ∆p = 0,
in R+ × Rd
c2 (x) ∂t2
(4)
∂p
d
p(0, x) = H(x),
(0, x) = 0 in R .
∂t
The time-dependent pressure signal p(t, x) is then measured on the surface of the tissue
for long enough time and the objective is to reconstruct the coefficients σa , σs and the
Grüneisen coefficient γ from this measurement.
The inverse problem in PAT is a two-step problem. It can be split into two inverse problems:
the problem of reconstructing interior data H from boundary pressure data
Problem 2.1. To reconstruct the initial pressure field H from data
ΛI : H(x) 7→ ΛI H = p|R+ ×∂Ω ;
(5)
and the problem of reconstructing optical coefficients from interior data H
Problem 2.2. To reconstruct (γ,σa ,σs ) from the data
ΛII : f (x, v) 7→ ΛII f = H(x).
(6)
The second step of PAT, Problem 2.2, is often called quantitative PAT (QPAT). It is a
nonlinear inverse coefficient for the transport equation with interior data.
Quantitative Photoacoustic Tomography. For the QPAT problem, we showed in [8,
9, 34], in diffusive regime, that if one of the coefficients in (γ,σa ,σs ) is known, then the
other two can be reconstructed uniquely with Lipschitz type of stability using only data
from two illuminations f1 and f2 . Not only that, we can construct an explicit reconstruction
procedure, which only requires solutions of a transport equation with known vector field
(given by the data) and a diffusion equation with known coefficients, to solve the nonlinear
inverse problem. We showed that one can reconstruct all three coefficients (γ,σa ,σs ) simultaneously, again in a unique and stable way, with multi-spectral data, that is data collected
from illuminations of different wavelengths [10]. These results are recently generalized
in [24] to the transport model (1) in both the non-scattering regime with two illuminations,
for applications in sectional photoacoustic tomography, and the full scattering regime with
the full albedo data ΛII .
Quantitative Thermoacoustic Tomography. Similar to what happens in PAT, We
can combine microwave imaging with ultrasound imaging to obtain a hybrid imaging modalities called thermoacoustic tomography (TAT). The working principle of TAT is identical
to PAT except that one replaces the NIR light source in PAT with a microwave source in
TAT. The propagation of microwave in tissue is described by the Maxwell’s equation:
−∇× ∇× E + k 2 E + ikµ(x)E = 0, Ω
n(x) × E = g(x),
∂Ω.
(7)
The initial pressure field generated in this case is of the form
H(x) = γ(x)µ(x)|E|2 (x).
4
(8)
The pressure field propagates following the same acoustic wave equation (4).
In [11] we proved the uniqueness of reconstructing the conductivity µ (assuming γ = 1)
in quantitative TAT (QTAT) and showed that the inverse problem is stable in Lipschitz
sense. We proposed a fixed point algorithm for the reconstruction of µ. In [23] we generalized the numerical result to reconstruct both γ and µ using data collected from multiple
illuminations.
Quantitative Fluorescence Photoacoustic Tomography. We propose recently to
combine the classical fluorescence tomography with ultrasound tomography to obtain another hybrid imaging modality: fluorescence photoacoustic tomography (fPAT). The mathematical model for light propagation in fPAT is a system of coupled radiative transport
equations:
Z
v · ∇ux + σx ux = σs,x
K(v, v0 )ux (x, v0 )dv0 ,
in X
d−1
SZ
(9)
v · ∇um + σm um = σs,m
K(v, v0 )um (x, v0 )dv0 + ησa,xf (x)hux i(x), in X
ux (x, v) = gx (x, v),
Sd−1
um (x, v) = 0,
The datum obtained from acoustic reconstruction in fPAT is of the form
H(x) = γ(x) σa,x (x)hux i(x) + σa,m (x)hum i(x) .
on Γ−
(10)
where the absorption coefficients σa,x = σx − σs,x and σa,m = σm − σs,m . The objective is
to reconstruct the coefficients η and σa,xf from data H. We studied this inverse problem
numerically in [37] and provided a complete mathematical theory on the uniqueness and
stability of the hybrid inverse problem in the diffusive regime in [38].
Photoacoustic Tomography with Unknown Sound Speed. In all the existing mathematical results on PAT, the ultrasound speed c(x) are always assumed known. In practical
applications, however, c(x) is not known, or at least not known exactly. This motivates the
work in [36] where we characterized numerically the impact of the uncertainty in the sound
speed on the quality of the reconstructions of the optical coefficients, and showed that a
small error in the sound speed can result in a large reconstruction error in the coefficients.
We then proposed in [29] a systematic method based on differential measurement that
would allow us to reconstruct simultaneously the sound speed and the optical properties in
a unique and stable way with minor additional assumptions.
3
Transport in Complex Media
In the analysis and simulation of the inverse problems we just described, we have to model
the propagation of particles and waves in heterogeneous (random) media. We have studied
this modeling problem in several different settings.
In [3, 5] we studied the problem of propagation of NIR photons in highly scattering media
with non-scattering regions. Classical macroscopic description of the propagation fails in
5
this case with extended void regions. Instead we derived a generalized macroscopic diffusion
model in which we model the extended void regions as an interface on which a tangential
diffusion process is supported. The model we derived is computationally as cheap as the
classical diffusion model but is significantly more accurate. We showed later in [5] that we
can actually locate these void regions from boundary Cauchy data using this generalized
diffusion model by solving an inverse problem of shape reconstruction.
In [2, 6] we studied the problem of imaging extended targets in discrete random media using
high-frequency acoustic/electromagnetic waves. We considered the regime where there are
significant amount of multiple scattering between waves and the underlying medium. In
this case, the phase of the measured wave field contains mainly information of the multiple
scattering due to the random media, and thus does not provide much information on the
targets. We suggested that in this case, wave energy density is a more robust quantity
to consider. We showed that the wave energy density actually solves a kinetic equation
whose coefficient are determined by low-order statistics of the random media. Thus the
imaging problem can be reformulated as an inverse transport problem of reconstructing
the macroscopic property of the media and the main characteristics of the targets. We
proposed several methods to reconstruct the shape and location parameters of the targets
using synthetic energy density data [6] and real-world experimental measurements [2].
We have also performed numerical studies on the propagation of photon density waves in
heterogeneous media [30, 31] for applications in frequency-domain optical tomography. We
proposed efficient computational algorithms for solving the transport and diffusion models
of such wave propagation problem.
In [12, 15], we studied the transport of electrons and holes in semiconductor devices for
applications in device optimization and design. We studied the inverse problem of doping
profiling where the aim is to reconstruct doping profiles of devices from current-voltage
characteristics of the devices. We proposed several numerical reconstruction strategies to
solve the Boltzmann-Poisson transport equation and the related inverse problem. We later
generalized the ideas in [18] to study the transport of charged particles in semiconductorelectrolyte solar cells, aiming at improving the efficiency of the cell through computational
optimization of the charge transport process.
4
Randomness in Inverse Problems
Randomness is ubiquitous in study of inverse and imaging problems. The most well-known
example is random noises presented in measured data used in solving inverse problems. In
the past, significant efforts have been spent on dealing with this type of random noises.
Linear and nonlinear regularization theory were developed.
Our incomplete knowledge about the underlying mathematical models can also induce
randomness in the study of inverse problems. For instance, in inverse problems in imaging,
we are often in the situation that the underlying PDE models have multiple coefficients,
while we aim at reconstructing only some of these coefficients assuming that the others
are known. However if we make an error in the coefficients that we assumed known, the
error could propagate into the reconstructed coefficients (often after being significantly
6
amplified). We thus have to estimate the stability of the inverse solutions with respect to
these coefficients that assumed known. These stability estimates determine whether or not
the corresponding imaging method is physically feasible.
In [36], we studied QPAT problem with sound speed that contains random perturbations.
We formulate the problem as an uncertain quantification problem and presented a systematic way to numerically investigate the impact of random perturbations on the reconstruction of the optical coefficients. We showed that a small error in the sound speed can result
in a large reconstruction error in the optical coefficients. It is thus important to model the
sound speed accurately.
In many inverse problems, the forward maps are smoothing (regularizing) operators. The inverse maps are usually unbounded. Thus only the low frequency component of the object of
interest can be faithfully reconstructed from noisy measurements. The high-frequency components are treated as random and neglected. In many cases however, the neglected high
frequency component may significantly affect the measured data. In [7], using simple scaling arguments, we characterized the influence of the random high-frequency components.
We showed how to eliminate the effect of the random high-frequency components in an
inverse spectral problem to obtain better reconstructions of the low-frequency components
of the unknown, i.e., to reduce the uncertainty in the reconstruction of the low-frequency
components. Using the same idea, we were able to study inverse transport problems in random environments and obtained more stable reconstruction of absorption and scattering
coefficients from noisy measurements [13].
5
Multiscale Analysis and Fluid Dynamics
Before shifting my research interests to inverse problems and imaging, I have done some
research on multiscale analysis of nonlinear dynamical systems and fluid dynamics. Results
of these analysis can be found in [19, 20, 21, 22, 39, 40, 41].
Let s(t) be a given sequence indexed by t coming out of a stochastic dynamical system. We
define the so-called structure function dl (t) = s(t + l) − s(t) at scale l. We are interested
in how the probability distributions and statistical behavior of dl (t) scale as a function of
l. Intuitively, if s(t) is a Brownian motion, then dl (t) follows the Gaussian distribution
with variance scale linearly with l. For general s(t), the behavior of dl (t) characterize the
deviation of s(t) from Brownian motion. In most cases, this deviation happens at the tails of
the distribution function, which means that events of large amplitude but small probability
actually play central roles in the statistical properties [20] of the system. We observe very
similar structures in velocity fields of turbulent fluids in Taylor-Couette flow and wake
flows [39, 41], nucleotides density profiles of DNA sequence [40] as well as solutions of other
nonlinear dynamical systems [25]. These multiscale analysis in fact provided additional
information for some phase space reconstruction problems in nonlinear dynamics [21]. They
also enabled us to establish a relationship between wavelet transform and invariant measure
of iterated function systems (IFS) [22] and to find homoclinic solutions (basically solitons)
to many functional equations [19, 21].
7
References
[1] G. S. Abdoulaev, K. Ren, and A. H. Hielscher. Optical tomography as a PDE-constrained
optimization problem. Inverse Problems, 21:1507–1530, 2005.
[2] G. Bal, L. Carin, D. Liu, and K. Ren. Experimental validation of a transport-based imaging
method in highly scattering environments. Inverse Problems, 23:2527–2539, 2007.
[3] G. Bal and K. Ren. Generalized diffusion model in optical tomography with clear layers. J.
Opt. Soc. Am. A, 20:2355–2364, 2003.
[4] G. Bal and K. Ren. Atmospheric concentration profile reconstructions from radiation measurements. Inverse Problems, 21:153–168, 2005.
[5] G. Bal and K. Ren. Reconstruction of singular surfaces by shape sensitivity analysis and
level set method. Math. Models Methods Appl. Sci., 8:1347–1373, 2006.
[6] G. Bal and K. Ren. Transport-based imaging in random media. SIAM J. Appl. Math.,
68:1738–1762, 2008.
[7] G. Bal and K. Ren. Physics-based models for measurement correlations. application to an
inverse Sturm-Liouville problem. Inverse Problems, 25, 2009. 055006.
[8] G. Bal and K. Ren. Multi-source quantitative PAT in diffusive regime. Inverse Problems,
27, 2011. 075003.
[9] G. Bal and K. Ren. Non-uniqueness results for a hybrid inverse problem. In G. Bal, D. Finch,
P. Kuchment, J. Schotland, P. Stefanov, and G. Uhlmann, editors, Tomography and Inverse
Transport Theory, volume 559 of Contemporary Mathematics, pages 29–38. Amer. Math.
Soc., Providence, RI, 2011.
[10] G. Bal and K. Ren. On multi-spectral quantitative photoacoustic tomography in diffusive
regime. Inverse Problems, 28, 2012. 025010.
[11] G. Bal, K. Ren, G. Uhlmann, and T. Zhou. Quantitative thermo-acoustics and related
problems. Inverse Problems, 27, 2011. 055007.
[12] Y. Cheng, I. Gamba, and K. Ren. Recovering doping profiles in semiconductor devices with
the Boltzmann-Poisson model. J. Comput. Phys., 230:3391–3412, 2011.
[13] T. Ding and K. Ren. Inverse coefficient problem for transport in stochastic environment.
Preprint, 2013.
[14] T. Ding and K. Ren. Inverse transport calculations in optical imaging with subspace optimization algorithms. Submitted to J. Comput. Phys., 2013.
[15] I. Gamba and K. Ren. Identifying localized anomalies in doping profiles of semiconductor
devices. Preprint, 2013.
[16] X. Gu, K. Ren, and A. H. Hielscher. Frequency-domain sensitivity analysis for small imaging
domains using the equation of radiative transfer. Applied Optics, 46:6669–6679, 2007.
[17] X. Gu, K. Ren, J. Masciotti, and A. H. Hielscher. Parametric image reconstruction using
the discrete cosine transform for optical tomography. J. Biomed. Opt., 14, 2010. Art. No.
064003.
[18] Y. He, I. M. Gamba, H. C. Lee, and K. Ren. On the modeling and simulation of
semiconductor-electrolyte solar cells. Submitted to SIAM J. Appl. Math., 2013.
[19] S.-D. Liu, Z.-T. Fu, S.-K. Liu, and K. Ren. The homoclinic orbit solution for functional
equation. Commun. Theor. Phys., 38:553–554, 2002.
[20] S.-D. Liu, S.-K. Liu, Z.-T. Fu, K. Ren, and Y. Guo. The most intensive fluctuation in chaotic
time series and relativity principle. Chaos Solitons Fractals, 15:227–230, 2003.
8
[21] S.-D. Liu, S.-K. Liu, S. Liang, K. Ren, and Z.-T. Fu. Several problems in studying of nonlinear
dynamics. Prog. Nat. Sci., 12:1–7, 2002.
[22] S.-K. Liu, Z.-T. Fu, S.-D. Liu, and K. Ren. Scaling equation for invariant measure. Commun.
Theor. Phys., 39:295–296, 2003.
[23] Y. Ma and K. Ren. Quantitative thermoacoustic tomography with multiple illuminations.
Submitted, 2013.
[24] A. V. Mamonov and K. Ren. Quantitative photoacoustic imaging in radiative transport
regime. Comm. Math. Sci., 12, 2014.
[25] K. Ren. Hierarchical structure analysis of multi-scale fluctuations from highly nonlinear
physical systems. Master’s thesis, Peking University, Beijing, China, 2001.
[26] K. Ren. Inverse Problems in Transport and Diffusion Theory with Applications in Optical
Tomography. PhD thesis, Columbia University, New York, 2006.
[27] K. Ren. Recent developments in numerical techniques for transport-based medical imaging
methods. Commun. Comput. Phys., 8:1–50, 2010.
[28] K. Ren. Fast numerical methods for optical tomography with large data set. Submitted, 2013.
[29] K. Ren. Simultaneous reconstruction of ultrasound speed and absorption coefficient in QPAT
with differential measurements. Submitted, 2013.
[30] K. Ren, G. Abdoulaev, G. Bal, and A. H. Hielscher. Modeling photon density wave propagation in turbigaod media with the equation of radiative transfer. In Proceedings of the
Biomedical Topical Meeting on Advances in Optical Imaging and Photon Migration. Optical
Society of America, 2004.
[31] K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher. Algorithm for solving the equation
of radiative transfer in the frequency domain. Optics Lett., 29:578–580, 2004.
[32] K. Ren, G. Bal, and A. H. Hielscher. Frequency domain optical tomography based on the
equation of radiative transfer. SIAM J. Sci. Comput., 28:1463–1489, 2006.
[33] K. Ren, G. Bal, and A. H. Hielscher. Transport- and diffusion-based optical tomography in
small domains: A comparative study. Applied Optics, 46:6669–6679, 2007.
[34] K. Ren, H. Gao, and H. Zhao. A hybrid reconstruction method for quantitative photoacoustic
imaging. SIAM J. Imag. Sci., 6:32–55, 2013.
[35] K. Ren, B. Moa-Anderson, G. Bal, X. Gu, and A.H. Hielscher. Frequency-domain tomography in small animals with the equation of radiative transfer. In B. Chance, R. R. Alfano,
B. J. Tromberg, M. Tamura, and E. M. Sevick-Muraca, editors, Optical Tomography and
Spectroscopy of Tissue VI, pages 111–120. SPIE, 2005.
[36] K. Ren and S. Vallélian. Characterizing uncertainties in quantitative photoacoustic tomography. Preprint, 2013.
[37] K. Ren and H. Zhao. Imaging heterogeneous media with fluorescence photoacoustic tomography. Submitted, 2013.
[38] K. Ren and H. Zhao. Quantitative fluorescence photoacoustic tomography. SIAM J. Imag.
Sci., 2013. In Press.
[39] Z. S. She, K. Ren, G. S. Lewis, and H. L. Swinney. Scalings and structures in turbulent
Couette-Taylor flow. Phys. Rev. E, 64, 2001. 016308.
[40] J. Wang, Q. Zhang, K. Ren, and Z. S. She. Multi-scaling hierarchical structure analysis on
the sequence of E.-coli complete genome. Chinese Sci. Bull., 43:1988–1992, 2001.
[41] Z. Zou, K. Ren, W. Su, Z. Gu, and Z. S. She. Hierarchical similarity in an inhomogeneous
turbulent wake flow sbehind multi-rectangular pillars. Acta Mechanica Sinica, 35:519–523,
2003.
9