Multiscale Cholesky preconditioning for ill-conditioned problems

Many computer graphics applications boil down to solving sparse systems of linear equations. While the current arsenal numerical solvers available in various specialized libraries and for different architectures often allow efficient scalable solutions image processing, modeling simulation applications, an increasing number problems face large-scale ill-conditioned --- a challenge which typically chokes both direct factorizations (due high memory requirements) iterative (because slow convergence). We propose novel approach preconditioning such emerge from discretization over unstructured meshes partial differential equations with heterogeneous anisotropic coefficients. Our consists simply performing fine-to-coarse ordering multiscale sparsity pattern degrees freedom, using we apply incomplete Cholesky factorization. By further leveraging supernodes cache coherence, graph coloring improve parallelism diagonal shifting remedy negative pivots, obtain preconditioner which, combined conjugate gradient solver, far exceeds performance existing carefully-engineered involving bad mesh elements and/or contrast also back core concepts behind our simple solver theoretical foundations linking recent method operator-adapted wavelets used homogenization traditional factorization matrix, providing us clear bridge between analysis that leverage numerically.