## CPM/cz148

Closest Point Method. Chen, Wathen and Zhu
Name cz148 CPM 2559 148 148 1,527 1,527 2D/3D Problem No 2012 Y. Chen, A. Wathen, S. Zhu T. Davis
Structural Rank 148 true 2 2 0 42.6% 23.1% no no real
SVD Statistics
Matrix Norm 1.929639e+02
Minimum Singular Value 7.804709e-02
Condition Number 2.472404e+03
Rank 148
sprank(A)-rank(A) 0
Null Space Dimension 0
Full Numerical Rank? yes
Download Singular Values
Download ```Closest Point Method, Yujia Chen, Andy Wathen, Shengxin Zhu A method for computing on surfaces. For more information see: http://people.maths.ox.ac.uk/~macdonald/closestpoint/ One of the matrices in this collection (cz10228) is solved poorly by x=A\b in MATLAB R2012a. The solution is to use this setting: spparms ('piv_tol', 0.5) x = A\b which changes the pivot tolerance. MATLAB has a test which checks the accuracy of x=A\b when using UMFPACK, and if the accuracy is poor, it refactorizes the matrix with a piv_tol of 1.0 (standard partial pivoting). For the cz10228 matrix, the test is nearly, but not quite, triggered. The other matrices do not cause this problem. Further details from Shengxin Zhu: Generally speaking, the matrix comes from the numerical solution of Poisson equation on a unit circle by solving an embedding PDE posed in a narrow band around the circle. Of course the easiest way to solve PDE on a unit circle is to parametrize it and then solve a 1D problem; but here we just want to test the effectiveness of the embedding method for solving PDEs on general curves or surfaces. The method extends the original surface PDE to a band around the surface, and then solve the extended PDE on a Cartesian grid by a finite difference scheme. In order to define such a PDE, one has to both define proper embedding differential operator and extend the solution from the surface to the embedding space. One natural way is to enforce the solution being constant along the normal direction of the surface so that the surface differential operator could be replaced by standard Cartesian differential operator. Here in the case of Poisson equation on a unit circle, we solve a standard Poisson equation on a 2D cartesian grid around the circle with suitable right hand side and Neumann boundary condition on the boundary of the band. We enforce the Neumann boundary condition by taking the value of each boundary point to be the value of its closest point on the circle; and since the closest point is usually not a grid point, its value is obtained by interpolation of values of the surrounding points. Putting the process into a sparse matrix, and change one row of the matrix to fix the value of one point of the solution and ensure the matrix is nonsingular. I tried different bandwidth, different interpolation order for the value of the closest points, 2nd order and 4-th order finite difference scheme to the Laplace operator. In most cases, MATLAB backslash works pretty well in solving the resulting linear system. The size of the matrix which makes MATLAB backslash not work is not the largest among all, and its condition number is not largest among all. By the way, the AMG solver by Notay and a geometric multigrid solver written by myself works pretty well for this case.```