33,467 research outputs found
The Lie-group shooting method for boundary-layer problems with suction/injection/reverse flow conditions for power-law fluids
No full textThe concept of best vector used to solve ill-posed linear inverse problems
No full textThe iterative algorithms based on the concept of best vector are proposed to solve an ill-conditioned linear system: Bx-b=0, which might be a discretization of linear inverse problem. In terms of r:=Bx-b and a monotonically increasing positive function Q(t) of a time-like variable t, we define a future cone in the Minkowski space, wherein the discrete dynamics of the proposed algorithm is evolved. We propose two methods to approximate the best vector B-1r, and obtain three iterative algorithms for solving x, which we label them as the steepest-descent and optimal vectors iterative algorithm (SOVIA), the mixed optimal iterative algorithm (MOIA), as well as the optimal vector iterative algorithm (OVIA). These algorithms are compared with the relaxed steepest descent method (RSDM), the conjugate gradient method (CGM) and an optimal iterative algorithm with an optimal descent vector (OIA/ODV) by testing several ill-posed linear inverse problems
A Lie-Group Adaptive Method to Identify Spatial-Dependence Heat Conductivity Coefficients
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The Lie-Group Shooting Method for Solving Multi-dimensional Nonlinear Boundary Value Problems
No full textThe Method of Fundamental Solutions for Solving the Backward Heat Conduction Problem with Conditioning by a New Post-Conditioner
No full textA revision of relaxed steepest descent method from the dynamics on an invariant manifold
No full textBased-on the ordinary differential equations defined on an invariant manifold, we propose a theoretical procedure to derive a Relaxed Steepest Descent Method (RSDM) for numerically solving an ill-posed system of linear equations when the data are polluted by random noise. The invariant manifold is defined in terms of a squared-residual-norm and a fictitious time-like variable, and in the final stage we can derive an iterative algorithm including a parameter, which is known as the relaxation parameter. Through a Hopf bifurcation, this parameter indeed plays a major role to switch the situation of slow convergence to a new situation with faster convergence. Several numerical examples, including the first-kind Fredholm integral equation and backward heat conduction problem, are examined and compared with exact solutions, revealing that the RSDM has superior computational efficiency and accuracy even for the highly ill-conditioned linear equations with a large noise imposed on the given data
A Globally Optimal Iterative Algorithm to Solve an Ill-Posed Linear System
No full textAn iterative algorithm based on the critical descent vector is proposed to solve an ill-posed linear system: Bx = b. We define a future cone in the Minkowski space as an invariant manifold, wherein the discrete dynamics evolves. A critical value αc in the critical descent vector u = αcr + BTr is derived, which renders the largest convergence rate as to be the globally optimal iterative algorithm (GOIA) among all the numerically iterative algorithms with the descent vector having the form u = αr + BTr to solve the ill-posed linear problems. Some numerical examples are used to reveal the superior performance of the GOIA
An inverse problem for computing a leading coefficient in the Sturm–Liouville operator by using the boundary data
No full textUsing a Lie-group adaptive method for the identification of a nonhomogeneous conductivity function and unknown boundary data
Full text linkOnly the left-boundary data of temperature and heat flux are used to estimate an unknown parameter function α(x) in Tt(x,t) = ∂(α(x)Tx)/∂x + h(x,t), as well as to recover the right-boundary data. When α(x) is given the above problem is a well-known inverse heat conduction problem (IHCP). This paper solves a mixed-type inverse problem as a combination of the IHCP and the problem of parameter identification, without needing to assume a function form of α(x) a priori, and without measuring extra data as those used by other methods. We use the one-step Lie-Group Adaptive Method (LGAM) for the semi-discretizations of heat conduction equation, respectively, in time domain and spatial domain to derive algebraic equations, which are used to solve α(x) through a few iterations. To test the stability of the present LGAM we also add a random noise in the initial data. When α(x) is identified, a sideways approach is employed to recover the unknown boundary data. The convergence speed and accuracy are examined by numerical examples
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