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| Yuusuke ISO |
| Complex Systems Research Group Leader |
| Department of Applied Analysis and Complex Dynamical Systems |
| Graduated School of Informatics |
The Mathematical Analysis of Complex Systems research group is studying a novel field called the "science of complexity" from the viewpoints of science and engineering, and aims to truly integrate these two disciplines. In the science of complexity, phenomena that have proven intractable to conventional simplified models are addressed through various applications of computers and analysis based on new concepts such as fractal analysis. The Mathematical Analysis of Complex Systems research group consists of scientists and engineers and endeavors to complement the shortcomings of each group while respecting the emphasis of the scientists on principles and the emphasis of the engineers on manufacturing. Our goal is a new integration of science and engineering both in research and graduate education.
More specifically, in addition to furthering the studies of individual researchers, this COE program addresses research subjects across the group, and aims toward a graduate education based on this concept. In particular, our research and educational activities (beyond individual research) in FY 2003 include: fractal analysis and probability theory, mathematical analysis of brain models, mathematical analysis of chaos, new developments in computational engineering, and the social contribution of sample value control theory. Of these five subjects, the first three place greater emphasis on theory, while the last subject targets the social contribution of academic activities.
For the heat kernel associated with a Dirichlet form on a general measure metric space, we have shown that the local Nash inequality combined with the exit time estimate is equivalent to a kind of upper estimate of the heat kernel under the volume doubling property. This result can be applied to the heat kernel of the diffusion process of Brownian motion on a line segment that is time-changed with respect to a self-similar measure to obtain a detailed evaluation of the heat kernel under a volume doubling measure. Moreover, for a general resistance form, we have provided a definition of the Green function when the boundary consists of a finite number of points, and have shown that the Green function is uniformly Lipschitz continuous with respect to the resistance metric. Furthermore, we used the Green function to provide a definition of a measure-valued Laplacian, and studied the Lipschitz continuity of functions with respect to the resistance metric in the domain of the Dirichlet form or Laplacian on the self-similar set.
The following research results have been achieved in the fields of fractal sets and infinite-dimensional spaces, which have structures that are essentially different from the Euclidean space:
In the field of applied probability theory, our research has focused on developing more efficient simulation schemes for behavior analysis of probabilistic systems constructed from the Ito equation. We envisioned developing techniques for analysis of probabilistic systems and reliability analysis of time-varying systems based on probability measure transformation, and extending such techniques to risk-analysis applications in a broad sense. In particular, we targeted a scheme that may contribute to estimating very small first arrival probabilities, which is highly desired in risk-analysis. We omitted the restriction that the driving noise of the system is a Wiener process, and extended the scope of the scheme to systems that may receive composite Poisson processes and their modified processes as inputs of driving noise. We then provided a formulation of the first arrival probability as the solution process of this system to a given set. On the basis of a method of probability measure transformation, we constructed an efficient simulation scheme that can accurately estimate even a very small first arrival probability. This method of probability measure transformation, which relies on the Girsanov-Meyer theorem, generalizes conventional approaches that have been developed for Wiener processes. Assuming several specific systems, we conducted numerical experiments to confirm that our proposed scheme works effectively for estimating extremely small first arrival probabilities, which cannot be practically estimated using normal simulations. We have also proposed the outline of a method of selecting an optimal measure transformation, and have achieved preliminary results by applying it to several specific subjects in risk-analysis.
| KIGAMI, Jun (Professor, Applied Analysis) |
| HINO, Masanori (Associate Professor, Applied Analysis) |
| TANAKA, Hiroaki (Associate Professor, Complex Dynamics) |
Recent physiological experiments suggest the possibility that firing correlations in the neural network may play an important role in higher functions such as integration of visual stimuli and attention switching. One topic of interest of the Mathematical Brain Models group is to theoretically verify the following mechanisms in both directions:
Ultimately, we hope to explore the mechanism by which high-level functions are realized by cooperation of these two mechanisms in an autonomous and distributed manner. With respect to I, we have proposed a theoretical explanation of the significance of gap junctions between a homogeneous pair of inhibitory neurons that are ubiquitous in the cerebral cortex from the viewpoint of the realization of various firing patterns. Our analysis has demonstrated that a variety of synchronous and asynchronous states can be realized by adjusting the coupling strength ratio between normal chemical synaptic coupling and the gap junction, which coexist between inhibitory neurons, within a physiologically reasonable range.
With respect to II, in a competitive neural network, which is important in column formation, we have theoretically demonstrated the possibility that the correlation of input firing timings can control function. Further, we built an associative memory model based on the circuit structure of the cerebral cortex, and demonstrated that the switching of recalled patterns can be controlled by synchronous firing inputs. This interesting result suggests the possibility that synchronous firings can be a signal for switching of neural activities (e.g., switching of behaviors).
Another interest of the Mathematical Brain Models group is to reproduce millimeter-scale phenomena of the information processing system in the cerebral cortex from nanometer-scale events of neural ion channels through theoretical frameworks and large-scale computer simulations, thereby elucidating the information processing mechanism. We have constructed a new model of the neuron based on the stochastic behavior of ion channels spatially distributed on the plasma membrane, and have mathematically formulated this model as a multivariable stochastic partial differential equation. Numerical calculations for this model showed that random fluctuation (noise) is derived from the stochastic behavior of ion channels. It was also demonstrated that the random noise added to weak external input signals generates stochastic resonance, which enhances the signal detection performance of neurons. The reliability of input detection is maximized when the fluctuation of membrane potential falls within an appropriate range. Such a collection of neurons can be used to estimate the magnitude of the input current and the input timing with high reliability. In order to analyze millimeter-scale phenomena, we constructed a large-scale neural network using our new neuron model. Since the model has a local coupling structure, its neural activity has not only temporal, but also spatial dynamics, which generate a variety of very complex spatial-temporal structures. In spite of this complex activity, the temporal average of the spatial-temporal dynamics obtained from one coupling structure indicated a high correlation with another coupling structure having completely different random external inputs. On the other hand, for a different coupling structure, the average activities showed no correlation even for the same external inputs. Therefore, a fine coupling structure greatly affects the global dynamics of the system. Furthermore, we compared our large-scale neural network model with experiments of optical recording in the early visual cortex. Physiological experiments show that neuron firings occur with high probability when the surrounding global activity approaches a certain state. Similar results were obtained using our model, which confirms its validity.
| KUBO, Masayoshi (Lecturer, Applied Analysis) |
| AOYAGI, Toshio (Lecturer, Complex Systems Synthesis) |
With an emphasis on cooperation with society in our research in FY 2003, we pursued basic studies primarily directed towards digital signal processing through the application of sample value control theory. In particular, to contribute technological benefits to society, we set the goal of achieving practical noise elimination in deteriorated acoustic signals. This research was conducted in cooperation with the Kyoto Digital Archive Research Center to support its cultural project of archiving SP (78 rpm) records and conserving their sound quality in optimal conditions.
Existing SP records comprise numerous sources of cultural and historical value dating from before World War II, including jiuta (traditional music handed down around Kyoto) and prewar campaign speeches. It is of great social significance to conserve and pass these materials on to subsequent generations. As the owners of SP records have aged, the concern about the loss of these important materials has grown.
In this project, we have begun by digitizing SP record sources offered by a collector in Kyoto. More specifically, these SP records were digitized in CD format with a sampling frequency of 44.1 kHz, and analyzed. The results of this analysis have demonstrated the following points about the archiving of SP records:
In light of this preliminary analysis, we applied digital signal processing theory based on sample value control theory to our project as follows:
This strategy resulted in adequate recovery of high frequencies and reduction of noise. The results are presented on the website of the Kyoto Digital Archive Research Center with evaluation samples. Note, however, that the site is now (as of April 2004) under renewal in association with an organizational change. This project was also published in the newspaper Kyoto Shimbun, and broadcast by KBS (Kyoto Broadcasting System) on November 5, 2003.
| YAMAMOTO, Yutaka (Professor, Complex Systems Synthesis) |
| FUJIOKA, Hisaya (Associate Professor, Complex Systems Synthesis) |
| NAGAHARA, Masaaki (Research Associate, Complex Systems Synthesis) |
We study chaotic motions that appear in nonlinear physical phenomena from the viewpoints of fluid dynamics and statistical physics.
In our fluid dynamics approach, the exact velocity field of flow in the presence of an axial pressure gradient was numerically determined in a partitioned-pipe mixer (PPM), which is composed of two kinds of plates crossed at a fixed angle and alternately placed in a rotating cylindrical duct along its axis. Furthermore, we investigated the mixing efficiency and mixing process due to the chaotic motion of fluid particles under this velocity field. More specifically, we found that the regular region, which prevents uniform mixing, can be reduced by the ratio of the lengths of the two kinds of plates; this reduction leads to a higher mixing efficiency. Moreover, we calculated the separation curve U(n), which is the set of initial positions of fluid particles that reach the leading edge of any of the plates within n periods. We showed that it is possible to quantify the mixing efficiency based on the distribution density of U(n). We also demonstrated that the mixing in n periods is efficient when the region with densely distributed U(n) is large. These studies are not only of theoretical interest, but are also interesting for basic research applications to fluid mixing.
Our statistical physics approach has focused on the following subjects: (1) nonlinear dynamics of anisotropic XY spin systems in a strong oscillating magnetic field, (2) construction of stochastic models for coarsened enstrophy dynamics in two-dimensional turbulence, and (3) anomalous scaling rules in deterministic diffusion.
In (1), a magnetic material in a strong oscillating magnetic field exhibits a complex and diverse motion depending on the strength and frequency of the magnetic field. In particular, we have studied anisotropic XY spin systems in a strong oscillating magnetic field from the viewpoint of nonlinear physics. The models we have studied can be placed into four categories of motion according to the strength of anisotropy. Among these models, we found that Ising-type and XY-type motions with broken time symmetry allow motions with Neel and Bloch walls. In these cases, we theoretically and experimentally examined the linear stability of the Neel walls to determine the existence region of the Neel and Bloch walls. We found that the stable existence regions of the two kinds of walls may significantly change in cases of different anisotropies. In agreement with experimental data, we also theoretically determined that the strength of the Bloch walls increases with the 1/2 power of the deviation from the transition point at which the Neel walls vanish and the Bloch walls occur. Furthermore, the types of motion of Neel and Bloch walls were clarified by computer experiments.
With respect to (2), in a two-dimensional developed Navier-Stokes turbulence, the nonlinear term causes energy cascades to occur toward the long-wavelength region and enstrophy cascades to occur toward the short-wavelength region. The fluctuation of the coarsened enstrophy dissipation rate is an important quantity that characterizes two-dimensional turbulence. In this project, we first qualitatively derived the probability density of the coarsened enstrophy dissipation rate fluctuation in light of the self-similarity of enstrophy cascades in developed turbulence. In order to consider the spatial variation of dissipation rate fluctuation, we then phenomenologically constructed the probability density of the coarsened enstrophy dissipation rate field, and used the projection operator method to derive the Langevin dynamics that reproduce the static probability density. Comparing the numerical calculation of the NS equation with the result of the Langevin dynamics, we found not only a qualitative agreement, but also a good quantitative agreement of the spectral intensity for the time series of the coarsened enstrophy dissipation rate.
Even without thermal motion, the trajectory of a dynamical system may diffuse due to the mixing properties of self-generated chaos. This is called deterministic diffusion. If the control variables of the chaotic dynamical system are varied, the characteristic time for attenuation of velocity correlation becomes extremely long around the bifurcation point. In (3), continuous-time random walk theory was applied to the crossover between the shorter time region below the characteristic time, where strongly correlated motions such as ballistic motion and anomalous diffusion are observed, and the longer time region above the characteristic time, where normal diffusion is observed. We thus analytically determined the mean square displacement and higher-order moments of position as a function of time, and demonstrated the occurrence of a curved time scale and a scaling rule reminiscent of extended self-similarity. We compared this result with numerical calculations for several dynamical systems exhibiting deterministic diffusion and found good agreement.
| FUNAKOSHI, Mitsuaki (Professor, Complex Dynamics) |
| FUJISAKA, Hirokazu (Professor, Complex Dynamics) |
| MIYAZAKI, Syuji (Lecturer, Complex Dynamics) |
| KANEKO, Yutaka (Research Associate, Complex Dynamics) |
| TUTU, Hiroki (Research Associate, Complex Dynamics) |
We have established numerical calculation techniques, including a new design for the computational environment, and have applied these techniques in applications such as inverse problem analysis, fracture mechanics, and global environmental simulations.
To engineer a new computational environment, we designed and implemented a fast multiple-precision computational environment that can cope with several thousands of decimal digits and run on the most common current computer architecture. Our research interests include the establishment of fast numerical solvers for applied inverse problems, such as nondestructive testing and computer tomography (CT), which are of increasing importance to the fields of engineering and medicine. Past theoretical studies based on numerical experiments have found that effective utilization of multiple-precision numerical calculation contributes to high-accuracy numerical calculations for ill-posed problems, such as inverse problems. Ill-posed problems (in the sense of Hadamard) are extremely sensitive to rounding error during calculation, as well as observation error and discretization error in numerical calculation. Rapid increases in rounding error may lead to failure of the numerical calculation even if a mathematically established formula is used. In multiple-precision calculation, a sufficient number of digits are allocated to perform the numerical calculation virtually without rounding error; this may allow the realization of high-accuracy numerical calculation for problems sensitive to rounding error. Our research has also demonstrated that the balance among observation error, discretization error, and rounding error is important in reliable numerical calculations for applied inverse problems. By using multiple-precision numerical calculation, we have made new discoveries in regard to the application of so-called regularization methods.
Our new environment also has the capacity to perform interval operations. It enables reliability evaluation of numerical solutions for rounding error and observation error in ill-posed problems through a posteriori error evaluation. Such evaluation has been difficult when using conventional numerical-analytic approaches.
We further developed high-accuracy discretization methods intended for the multiple-precision numerical calculation environment based on spectral methods in view of the balance among observation error, discretization error, and rounding error.
In the fields of computational engineering and computational physics, we have achieved successful results in studies of global environment simulators, fracture mechanics, and the quantum Monte Carlo method. More specifically, in relation to environmental problems, we conducted numerical analysis focused on mathematical models of weather and oceanic phenomena and the solvers of such problems. In particular, we studied data assimilation numerical solvers (so-called inverse methods), which enable oceanic dynamics simulators to be consistent with observational data and to predict real oceanic conditions as precisely as possible. We have proposed a simplified model to reduce the computational load of large-scale problems, and have addressed the quantitative relationship between high and medium resolutions and the systematic treatment of phenomena with different scales.
In the field of fracture mechanics, our research has focused on numerical simulation of crack development phenomena. We have developed software programs for numerical simulation of the bifurcation phenomenon in crack configuration that occurs in fractures when crack development is suppressed at low speed. Since it is difficult to treat regions including cracks with current general-purpose programs, we designed and implemented a dedicated solver using the boundary element method. As a mathematical model of crack development phenomena, we adopted the mathematical model proposed by Avner Friedman and Yong Liu in 1996 that uses the J-integral. For this model, the existence and uniqueness of a solution to the equation has already been proven under certain conditions, and we also consider it to be a satisfactory model from the viewpoint of mathematical analysis. In the numerical simulation, we used the Galerkin boundary element method with singular elements introduced in view of the singularity of solution at the tip of the crack. We confirmed the convergence of a solution for a steady crack.
In the field computational physics, we continued to collaborate with Dr. Naoki Kawashima (Tokyo Metropolitan University) on algorithms of the quantum Monte Carlo method. We have proposed a coarsening algorithm for quantum spin systems with easy-magnetization-plane anisotropy.
| ISO, Yuusuke (Professor, Applied Analysis) |
| NOGI, Tatsuo (Professor, Complex Systems Synthesis) |
| WAKANO, Isao (Lecturer, Applied Analysis) |
| FUJIWARA, Hiroshi (Research Associate, Applied Analysis) |
| HARADA, Kenji (Research Associate, Complex Systems Synthesis) |
| HIGASHIMORI, Nobuyuki (COE Fellow) |
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