研究集会「異常拡散の数理とシミュレーション手法ならびに関連する課題」(Conference on Diffusion in Heterogeneous Media and Related Topics)

数理科学研究科棟Room 002

6 March (Thursday) 2014
10:00-10:10: Opening
10:10-11:00:
J. Cheng (Fudan Univ.)"The Mathematical Model for the Contamination Problems and Related
Inverse Problems"
11:10-12:00
W. Rundell (Texas A&M Univ.)"Inverse problems with fractional derivatives in the space variable".
13:30-14:20
T. Kumagai (Kyoto Univ.)"Simple random walk on the two -dimensional uniform spanning tree a ndits scaling limits"
14:30-15:20
Y. Luchko (Beuth Technical University of Applied Sciences):"Neutral-fractional diffusion-wave equation and some properties of its
fundamental solution"
15:20-16:10
S. Hirano (Univ. Tsukuba)“An integral representation and numerical simulation of dynamic earthquake faulting occurring in a
two-layered medium”
16:20-17:10
M. Cristofol (Aix-Marseille Univ.)“New kind of observations in an inverse parabolic problem”
18:00-: Group discussions

7 March (Friday)
10:00-10:50
L. Ling (Hong Kong Baptist Univ.)"Numerical Caputo differentiation by radial basis functions"
11:00-11:50
Benny Hon (City Univ. of Hong Kong)"Kernel-based approximation method for backward time -fractional
diffusion problem"13:30-14:20
Yasumasa Miyazawa (JAMSTEC)"Dispersion of materials in ocean: observation and modeling"
14:20-15:10
Y． Hatano (Tsukuba Univ.)
TBA
15:10-16:00
T. Aoyama (Tsukuba Univ.)"Diffusion of radioactive Suspended Particulate Matter (SPM) inthe atmosphere"
16:10-16:40
Z. Li (The Univ. of Tokyo)“Initial-boundary value problem for multi-time-fractional diffusion equation and its inverseproblems”
16:40-17:10
Y. Liu (The Univ. of Tokyo)“Well-posedness and numerical simulation for multi-term time-fractional diffusion equations with positive constantcoefficients”
17:10-17:50
T. Li (Southeast Univ.)“ On the efficient computations for transmission eigenvalue problems arising in wave
scatterings”

8 March (Saturday)
10:00-10:50
R. Guglielmi (Bayreuth University)"Null controllability of degenerate/singular parabolic equation"
10:50-11:40
J. Liu (Southeast Univ.)“Total variation regularization for backward time-fractional diffusion problem”
11:40-12:30
Daijun Jiang (Central China Normal University)“Convergence Rates of Tikhonov Regularizations for Parameter Identification ina Parabolic-Elliptic System”

主 催 ・ 共 催 ： 数 物 フ ロ ン テ ィ ア ･リ ー デ ィ ン グ 大 学 院 プ ロ グ ラ ム 、
卓越した大学院拠点形成支援補助金、
数学協働プログラム

研究代表者：
坪井俊、山本昌宏

ABSTRACTJ. Cheng (Fudan Univ.)"The Mathematical Model for the Contamination Problems and Related
Inverse Problems"
In this talk, we discuss the motivation for the study of the abnormal
diffusion models for the contamination problems. From the practical point ofview, several inverse problems proposed and studied Theoretic results, forexample, the uniqueness and stability are shown. The po ssibility of theapplication of these studies is mentioned.

W. Rundell (Texas A&M Univ.)"Inverse problems with fractional derivatives in the space variable".
This talk will look at some classical inverse problems where thehighest order term in the sp atial direction is replaced by a fractional
derivative. There are again some surprising results over what is known in
the usual case of integer order derivatives but also quite different fromthe case of fractional diffusion in the time variable.
The talk will give some answers, but pose many more open problems.

T. Kumagai (Kyoto Univ.)"Simple random walk on the two -dimensional uniform spanning tree andits scaling limits"
In this talk, we will first discuss anomalous asymptotic behavior of
simple random walk on the two-dimensional uniform spanning tree. We then show
the existence of subsequential scaling limits for the random walks, anddescribe the limits as diffusions on the limiting random real trees. Anomalous
heat transfer on the random real trees will be derived by estimating heat
kernels of the diffusions. This is an on -going joint project with M.T. Barlow(UBC) and D. Croydon (Warwick).

Y. Luchko (Beuth Technical University of Applied Sciences)"Neutral-fractional diffusion-wave equation and some properties of its
fundamental solution"
Recently, the so called neutral -fractional diffusion-wave equation was
introduced and analysed in the case of one spatial variable. In contrast to
the fractional diffusion of diffusion -wave equations, the neutral-fractional
diffusion-wave equation contains fractional derivatives of the same order bothin space and in time. The fundamental solution of the neutral -fractional
diffusion-wave equation was shown to exhibit properties of both the solutionsof the diffusion equation and those of the wave equation.
In the one-dimensional case, the fundamental solution of the neutral -fractional
diffusion-wave equation can be interpreted as a spatial probability density
function evolving in time. At the same time, it can be treated as a damped
wave whose amplitude maximum and the gravity and mass centres propagate withthe constant velocities that depend just on the equation order.
In this talk, the problems mentioned above are considered for the multi dimensional neutral-fractional diffusion-wave equation. To illustrate
analytical findings, some results of numerical calculations and plots arepresented.

Shiro Hirano（Univ. of Tsukuba）“An integral representation and numerical simulation of dynamic earthquake faulting occurring in a two-layered
medium”
Earthquakes are dynamic rupture and slip along cracks embedded in rocks.
Wave field due to dynamic rupture is represented as an integral transform of slip along an expanding crack, so thatbounded integration kernels for the transform are required depending on surrounding media to simulate earthquakes
numerically.
Although the kernel for an infinite homogeneous medium have been obtained, actual rocks surrounding faults are
inhomogeneous and often modeled as a layered medium.
Hence we here derive the kernels for a two-layered medium and simulate dynamic rupture propagation intersectingwith a material interface.

M. Cristofol (Aix-Marseille Univ.)“New Kind of Observations in an Inverse Parabolic Problem”
In this talk, I analyze the inverse problem of determining the reaction termf(x,u) in reaction-diffusion equations of the form ¥partial_t u-D¥partial_{xx}u= f(x,u), where f is assumed to be periodic with respect to x in R. Startingfrom a family of exponentially decaying initial conditions u_{0, ¥lambda}, I willshow that the solutions u_ ¥lambda of this equation propagate with constant
asymptotic spreading speeds w_ ¥lambda. The main result shows that the linearizationof faround the steady state 0, ¥partial_u f(x,0), is uniquely determined (up to a symmetry) am
ong a subset of piecewise linear functions, by the observation of
the asymptotic spreading speeds w_ ¥lambda.

L. Ling (Hong Kong Baptist Univ.)"Numerical Caputo differentiation by radial basis functions"
Previously, based on the method of (radial powers) radial basis functions,
we proposed a procedure for approximating derivative values fromone-dimensional scattered noisy data. In this talk, we show that the same
approach also allows us to approximate the values of (Caputo) fractional
derivatives (for orders between 0 and 1). With either an a priori or aposteriori strategy of choosing the regularization paramete r, our convergence
analysis shows that the approximated fractional derivative values converge at
the same rate as in the case of integer order 1.

Yasumasa Miyazawa (JAMSTEC)"Dispersion of materials in ocean: observation and modeling"
This talk provides a brief introduction to a view of materials dispersion
in physical oceanography. A classical theory has been developed partly
motivated by necessity of tracking artificial radionuclides emitted into theocean. The turbulence viscosity/diffusion coeffi cients involved in the theory,
which could be basically isotropic and stationary, have been frequently
adopted for the analyses in the literature. Recent severe marine accidents
including the Deep Water Horizon oil rig in Mexico Bay (2010) and Fukushima
Dai-ichi Nuclear Plant (2011) have revealed quite un -isotropic andun-stationary features of materials dispersion in ocean through utilization of
the advanced measurement and simulation techniques. A Lagrangian -type concept
originating from the classi cal continuum mechanics has been recently proposed
for understanding of the complex features of materials dispersion in ocean.

T. Aoyama (Tsukuba Univ.)"Diffusion of radioactive Suspended Particulate Matter (SPM) inthe atmosphere"
Meteorological approach of non-reactive matters in the atmosphere Practical
puff model of Japan Atomic Energy Agency Gaussian extensions to include
chemical reactions Equation of motion of Gaussian puff Decay and gamma -rayflux of fission products; Interaction with human body Simulation forradioactive SPM in the Fukushima accident; comparison with observations .

Z. Li (The Univ. of Tokyo)“Initial-boundary value problem for multi-time-fractional diffusion equation and its inverse problems”
We investigate the initial-boundary value problem for multi-term time-fractional diffusion equation. By means of theMittag-Leffler function and the eigenfunction expansion, the problem is reduced to an integral equation for asolution. The fixed-point theorem is then applied to prove the unique existence. For the case of the homogeneous
equation, the solution can be analytically extended to a sector in the complex plane. Moreover, the decay rate of thesolution is shown to be determined by the lowest order of fractional time-derivative.
For the inverse problem, we focus on the determination of the fractional orders in the diffusion model by interior
observation or Dirichlet-to-Neumann map, which is important for experimentally evaluating the anomaly of thediffusion in heterogeneous medium.

Y. Liu (The Univ. of Tokyo)“Well-posedness and numerical simulation for multi-term time-fractional diffusion equations with positive constantcoefficients”
We investigate the initial-boundary value problem for time-fractional diffusion equations, where the timedifferentiation consists of a finite summation of Caputo derivatives with decreasing orders and positive constantcoefficients. Based on several new properties of multinomial Mittag-Leffler functions, we establish the uniquenessand stability of the solution with respect to initial value and source term, from which the continuous dependency ofLipschitz type upon various coefficients is also verified. Concerning the numerical simulation, we develop astandard Galerkin finite element method and a fully discrete scheme, and give error estimates. This is a joint workwith Dr. Bangti Jin (UC Riverside), Prof. Raytcho Lazarov and Mr. Zhi Zhou (Texas A&M Univ.).

T. Li (Southeast Univ.)“ On the efficient computations for transmission eigenvalue problems arising in wave
scatterings”
Abstract: Transmission eigenvalue problems arise in many applied areas such as wave
scatterings. It was found in resent years that such problems are related to somereconstruction schemes for inverse scattering problems. However, the efficientcomputations for the transmission states are very difficult due to some extra complex
eigenvalues for the discrete system which are not physical meaningful. In this talk, wewill introduce our recent works for the efficient computations of transmission
eigenvalues for our discrete system which is of the size of 100 thousands. Moreover,out scheme can yield the real eigenvalues automatically.

R. Guglielmi (Bayreuth University)"Null controllability of degenerate/s ingular parabolic equation"
We study the controllability properties of a class of generalized Grushin
operator of parabolic type, when the degeneracy occurs in the interior of
the space domain. We show that the null controllability holds true only when
the degeneracy of the operator is weak, and possibly under minimum time
assumptions. The positive results lies on suitable Carleman estimates.
Moreover, in connection with the study of the Laplace -Beltrami operator on
manifold with almost Riemannian metrics, we analyze the case of a Grushin
operator with a inverse square singular potential, and we show partial
positive result on the null controllability of this operator.

J. Liu (Southeast Univ.)“Total variation regularization for backward time-fractional diffusion problem”
Consider a 2-dimensional backward problem for time-fractional diffusion
process, which can be considered as image de-blurring where the blurring process is
assumed to be slow diffusion. In order to avoid the over-smoothing effect for object
image with edges and to construct a fast reconstruction scheme, the total variation
regularizing term and the data residual error in the frequency domain are coupled toconstruct the cost functional. The well-posedness of this optimization problem isstudied. The minimizer is sought approximately using the iteration process for a seriesof optimization problems with Bregman distance as penalty term.
This iteration reconstruction scheme is essentially a new regularizing scheme withcoupling parameter in the cost functional and the iteration stopping times as two
regularizing parameters. We give the choice strategy for the regularizing parameters
in terms of the noise level of measurement data, which yields the optimal error estimate
on the iterative solution. The series optimization problems are solved by alternative
iteration with explicit exact solution and therefore the amount of computation is much
weakened. Numerical implementations are given to support our theoretical analysis on
the convergence rate and to show the significant reconstruction improvements.

Daijun Jiang (Central China Normal University)“Convergence Rates of Tikhonov Regularizations for Parameter Identification ina Parabolic-Elliptic System”
We shall study the convergence rates of the Tikhonov regularizations for the identification of the diffusivity$q(¥x)$ in a parabolic-elliptic system. The $H^1$ regularization and a mixed $L^p$-$H^1$ regularization areconsidered. For the $H^1$ regularization, we present a simple and easily interpretable source condition, underwhich the regularized solutions will be shown to converge in the standard rate in terms of the noise level of the data.
The convergence is analyzed
in three different approaches, which result in the same convergence rate but require quite different conditions on themeasurement time and the identifying parameters. For the mixed $L^p$-$H^1$ regularization, we will achieve somedesired convergence rate by using the Bregman distance and some new source condition and nonlinearity condition.