Conditions for the Qualified Convergence of Finite Difference Methods and the Quasi-Reversibility Method for Solving Linear Ill-Posed Cauchy Problems in a Hilbert Space

2019 ◽  
Vol 63 (10) ◽  
pp. 40-54 ◽  
Author(s):  
M. M. Kokurin
Keyword(s):  
Hilbert Space ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Cauchy Problems ◽  
Ill Posed ◽  
Difference Methods

A posteriori choice of time-discretization step in finite difference methods for solving ill-posed Cauchy problems in Hilbert space

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mikhail M. Kokurin
Keyword(s):  
Hilbert Space ◽  
Finite Difference ◽  
Finite Difference Methods ◽  
Time Discretization ◽  
Cauchy Problems ◽  
A Posteriori ◽  
Error Level ◽  
Selfadjoint Operators ◽  
Discretization Step ◽  
Ill Posed

Abstract Finite difference semidiscretization methods for solving an ill-posed Cauchy problem in a Hilbert space are investigated. The problems involve linear positively definite selfadjoint operators. We justify an a posteriori scheme for the choice of the time-discretization step and establish accuracy estimates in terms of the error level of input data.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

scholarly journals Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 206
Author(s):  
María Consuelo Casabán ◽  
Rafael Company ◽  
Lucas Jódar
Keyword(s):  
Stochastic Process ◽  
Finite Difference ◽  
Coming Out ◽  
Finite Difference Methods ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Finite Degree ◽  
Mean Square Stability ◽  
Reaction Models ◽  
Difference Methods

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

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