Dimensionality Reduction Of Flow Based On Weighted Residue

Dimensionality reduction has kept the attention of scientists, it is the transformation or projection of.high-dimensional data or dynamical system into a meaningful representation of reduced dimensionality. The analysis of high-dimensional data or dynamical system in various disciplines is faced with three puzzles. The fist one is the blessing of dimensionality which shows that the abundance information of the high-dimensional data or dynamical system enables scientist to make a new discovery of physical phenomenon, means a revolution in methodology. The second is the curse of dimensionality which shows incredible geometric properties in high-dimensional,,space and has challenged visualization. The studies showed that usual mathematical objects like spheres and cubes behave strangely and do not share the same nice properties as in the two-or three-dimensional case. In the other word, the perception of low-dimensional space is nearly useless in high-dimensional one. Moreover, most high-dimensional nonlinear systems of partial differential equations (PDEs) are impossible to solve analytically. In fact,,basic properties such as the existence and uniqueness of solutions, which was so obvious in the linear case, no longer hold for nonlinear dynamical systems. Even if we do find a solution of such a system with a specific initial condition, high-dimensional system may exhibit chaotic behavior that a property makes a particular solution essentially worthless of understanding the behavior of the system. Therefore, it is critical to develop dimensionality reduction method for overcoming the challenge of high dimensionality.During the 1900s, dimensionality reduction went through several eras. The fist era mainly relied on spectral methods like principal components analysis and the classical metric multidimensional scaling. Next, the second era consisted of the generalization of multidimensional scaling into nonlinear variants many of them being based on distance preservation and among which Sammon’s nonlinear mapping is probably the most emblematic representative. At the end of the century, the field of nonlinear dimensionality reduction was deeply influenced by neural network approaches; the Kohonen’s self-organizing map is the most prominent examples of this stream. The beginning of the new century witnessed the rebirth of spectral approaches, starting with the discovery of kernel principal component analysis. Furthermor, although the formulation of approximate inertial manifold is different, the process of dimensionality reduction still depend on traditional Galerkin projection method. These methods mentioned above are highly sophisticated, but many of them give poor results when the intrinsic dimensionality of the underlying manifold exceeds four or five. In such case, the reduced dimensional space becomes high enough to observe undesired effects related to the curse of dimensionality. Consequently, a new techniques is required to take up this ultimate challenge.The objective of this dissertation is to study the dimensionality reduction method of optimal truncated low-dimensional dynamical systems based on weighted residue (POT-WR) which developed by use the theory of constructing optimal low-dimensional dynamical system, either based on known databases or partial differential equations. A conjugate gradient algorithm by FORTRAN code is developed to solve the POT-WR equations, and three numerical cases are considered to illustrate the result of dimensionality reduction. Two optimal processes and two convergent criterions are considered for one-dimensional linear and nonlinear partial differential equations:The local optimal result of POT-WR equations; the global optimal result of POT-WR equations with carrying out the enumeration algorithm; the global convergent criterion of error and weighted residue. For two-dimensional nonlinear partial equation, the global optimal result with carrying coarse-grained algorithm and the comparison between POT-WR base and proper orthogonal decomposition (POD) base are considered. The dissertation is generally outlined into 5 chapters. The main contributions are stated as following.(1) The result of dimensionality reduction of linear POT-WR is investigated based on one-dimensional heat transfer equation with three different initial iteration bases. The result showed that the POT-WR is good agreement with the analytical solutions. The global optimal bases were independent of initial iteration bases. For both convergent criterion mentioned above, POT-WR converged on analytical solution.(2) The result of dimensionality reduction of nonlinear POT-WR is investigated based on Burgers’equation with three different initial iteration bases. The result of local optimal process showed that local optimal bases depend on initial iteration bases for nonlinear partial differential equation. However, the global optimal bases are independent of initial iteration bases; and they are coupled nonlinear bases. The dominant characteristics of the dynamics are well captured in case of few bases used only. For the small viscosity coefficient, the shock wave of Burgers’equation was captured by the last order of global optimal bases. The investigation of the convergent criterion of the global optimal searches illustrates that the application of the weighted residue of the equation under consideration as the convergent criterion is an effectivepvay to bound the error of the solution.(3) The result of dimensionality reduction of two-dimensional nonlinear POT-WR is investigated based on unsteady Navier-Stokes equations for Newtonian incompressible fluids with coarse-grained global optimal algorithm, which firstly filters the noise of system by choosing initial iterative bases,hen projects in a low-dimensional space by POT-WR. The results show that the new method characterizes the detail of flow field by a few of bases efficiently. The boundary formulation was discussed in details with Dirichlet, Nuemann and Robin boundary condition of Navier-Stokes equation. The lid-driven cavity case was investigated for illustrated the approximate result of three and five POT-WR bases. The result indicated that with the lager order number of the truncation, the higher approximate accuracy can obtain. There is no bijection relationship between POT-WR base and POD base. Moreover, the cavity flow can be characterized only by fluctuant POT-WR bases. The POT-WR bases are a set of nonlinear coupled bases; they do not have the composition properties as POD bases or other linear bases, but work as a whole and automatically adjust themselves to represent the nonlinear feature of the Navier-Stokes equations. The same as Burgers’case, the main characteristic of system was captured by the last order of POT-WR bases.