_-Ph2zdZddlZddlmZmZddlmZddlm Z ddl m Z ddZ d Z dd Zdd Zd ZddZddZdS)a Functions to compute C/F splittings for use in Classical AMG. Overview -------- A C/F splitting is a partitioning of the nodes in the graph of as connection matrix (denoted S for strength) into sets of C (coarse) and F (fine) nodes. The C-nodes are promoted to the coarser grid while the F-nodes are retained on the finer grid. Ideally, the C-nodes, which represent the coarse-level unknowns, should be far fewer in number than the F-nodes. Furthermore, algebraically smooth error must be well-approximated by the coarse level degrees of freedom. Representation -------------- C/F splitting is represented by an array with ones for all the C-nodes and zeros for the F-nodes. C/F Splitting Methods --------------------- RS : Original Ruge-Stuben method - Produces good C/F splittings. - May produce AMG hierarchies with relatively high operator complexities. - See References [1] and [4] PMIS: Parallel Modified Independent Set - Very fast construction with low operator complexity. - Convergence can deteriorate with increasing problem size on structured meshes. - Uses method similar to Luby's Maximal Independent Set algorithm. - See References [1] and [3] PMISc: Parallel Modified Independent Set in Color - Fast construction with low operator complexity. - Better scalability than PMIS on structured meshes. - Augments random weights with a (graph) vertex coloring - See References [1] CLJP: Cleary-Luby-Jones-Plassmann - Parallel method with cost and complexity comparable to Ruge-Stuben. - Convergence can deteriorate with increasing problem size on structured meshes. - See References [1] and [2] CLJP-c: Cleary-Luby-Jones-Plassmann in Color - Parallel method with cost and complexity comparable to Ruge-Stuben. - Better scalability than CLJP on structured meshes. - See References [1] Summary ------- In general, methods that use a graph coloring perform better on structured meshes [1]. Unstructured meshes do not appear to benefit substantially from coloring. ======== ======== ======== ========== method parallel in color cost ======== ======== ======== ========== RS no no moderate PMIS yes no very low PMISc yes yes low CLJP yes no moderate CLJPc yes yes moderate ======== ======== ======== ========== References ---------- .. [1] Cleary AJ, Falgout RD, Henson VE, Jones JE. "Coarse-grid selection for parallel algebraic multigrid" Proceedings of the 5th International Symposium on Solving Irregularly Structured Problems in Parallel. Springer: Berlin, 1998; 104-115. .. [2] David M. Alber and Luke N. Olson "Parallel coarse-grid selection" Numerical Linear Algebra with Applications 2007; 14:611-643. .. [3] Hans De Sterck, Ulrike M Yang, and Jeffrey J Heys "Reducing complexity in parallel algebraic multigrid preconditioners" SIAM Journal on Matrix Analysis and Applications 2006; 27:1019-1039. .. [4] Ruge JW, Stuben K. "Algebraic multigrid (AMG)" In Multigrid Methods, McCormick SF (ed.), Frontiers in Applied Mathematics, vol. 3. SIAM: Philadelphia, PA, 1987; 73-130. N) csr_matrixisspmatrix_csr)vertex_coloring)amg_core)remove_diagonalFc t|stdt|}|j}t j|jdd}t j|jdfd}tj |jd|j |j |j |j |||r,tj |jd|j |j ||S)aCompute a C/F splitting using Ruge-Stuben coarsening. Parameters ---------- S : csr_matrix Strength of connection matrix indicating the strength between nodes i and j (S_ij) second_pass : bool, default False Perform second pass of classical AMG coarsening. Can be important for classical AMG interpolation. Typically not done in parallel (e.g. Hypre). Returns ------- splitting : ndarray Array of length of S of ones (coarse) and zeros (fine) Examples -------- >>> from pyamg.gallery import poisson >>> from pyamg.classical.split import RS >>> S = poisson((7,), format='csr') # 1D mesh with 7 vertices >>> splitting = RS(S) See Also -------- amg_core.rs_cf_splitting References ---------- .. [1] Ruge JW, Stuben K. "Algebraic multigrid (AMG)" In Multigrid Methods, McCormick SF (ed.), Frontiers in Applied Mathematics, vol. 3. SIAM: Philadelphia, PA, 1987; 73-130. expected csr_matrixrintcdtype)r TypeErrorrTtocsrnpemptyshapezerosrrs_cf_splittingindptrindicesrs_cf_splitting_pass2)S second_passr splitting influences U/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/classical/split.pyRSrcsH !  /-...A  A6222I!'!*f555I QWQZXqyXqy&& ((( =&qwqz18'(y) = = = clt|}t|\}}}}~~t||S)a6C/F splitting using the Parallel Modified Independent Set method. Parameters ---------- S : csr_matrix Strength of connection matrix indicating the strength between nodes i and j (S_ij) Returns ------- splitting : ndarray Array of length of S of ones (coarse) and zeros (fine) Examples -------- >>> from pyamg.gallery import poisson >>> from pyamg.classical.split import PMIS >>> S = poisson((7,), format='csr') # 1D mesh with 7 vertices >>> splitting = PMIS(S) See Also -------- MIS References ---------- .. [6] Hans De Sterck, Ulrike M Yang, and Jeffrey J Heys "Reducing complexity in parallel algebraic multigrid preconditioners" SIAM Journal on Matrix Analysis and Applications 2006; 27:1019-1039. r _preprocessMIS)rweightsGrs rPMISr%s9@ A"1~~GQ1 1 q'??rJPcpt|}t||\}}}}~~t||S)ahC/F splitting using Parallel Modified Independent Set (in color). PMIS-c, or PMIS in color, improves PMIS by perturbing the initial random weights with weights determined by a vertex coloring. Parameters ---------- S : csr_matrix Strength of connection matrix indicating the strength between nodes i and j (S_ij) method : string Algorithm used to compute the initial vertex coloring: * 'MIS' - Maximal Independent Set * 'JP' - Jones-Plassmann (parallel) * 'LDF' - Largest-Degree-First (parallel) Returns ------- splitting : array Array of length of S of ones (coarse) and zeros (fine) Examples -------- >>> from pyamg.gallery import poisson >>> from pyamg.classical.split import PMISc >>> S = poisson((7,), format='csr') # 1D mesh with 7 vertices >>> splitting = PMISc(S) See Also -------- MIS References ---------- .. [7] David M. Alber and Luke N. Olson "Parallel coarse-grid selection" Numerical Linear Algebra with Applications 2007; 14:611-643. )coloring_methodr )rmethodr#r$rs rPMIScr*s@P A"1f===GQ1 1 q'??rc Rt|stdt|}d}|rd}|j}t j|jdd}tj |jd|j |j |j |j |||S)a4Compute a C/F splitting using the parallel CLJP algorithm. Parameters ---------- S : csr_matrix Strength of connection matrix indicating the strength between nodes i and j (S_ij) color : bool use the CLJP coloring approach Returns ------- splitting : array Array of length of S of ones (coarse) and zeros (fine) Examples -------- >>> from pyamg.gallery import poisson >>> from pyamg.classical.split import CLJP >>> S = poisson((7,), format='csr') # 1D mesh with 7 vertices >>> splitting = CLJP(S) See Also -------- MIS, PMIS, CLJPc References ---------- .. [8] David M. Alber and Luke N. Olson "Parallel coarse-grid selection" Numerical Linear Algebra with Applications 2007; 14:611-643. r rr r ) rr rrrrrrrcljp_naive_splittingrr)rcolorcoloridrrs rCLJPr0sD !  /-...AG   A6222I !!'!*"#(AI"#(AI"+") +++ rcBt|}t|dS)aCompute a C/F splitting using the parallel CLJP-c algorithm. CLJP-c, or CLJP in color, improves CLJP by perturbing the initial random weights with weights determined by a vertex coloring. Parameters ---------- S : csr_matrix Strength of connection matrix indicating the strength between nodes i and j (S_ij) Returns ------- splitting : array Array of length of S of ones (coarse) and zeros (fine) Examples -------- >>> from pyamg.gallery import poisson >>> from pyamg.classical.split import CLJPc >>> S = poisson((7,), format='csr') # 1D mesh with 7 vertices >>> splitting = CLJPc(S) See Also -------- MIS, PMIS, CLJP References ---------- .. [1] David M. Alber and Luke N. Olson "Parallel coarse-grid selection" Numerical Linear Algebra with Applications 2007; 14:611-643. T)r.)rr0)rs rCLJPcr2%s%F A    rc t|stdt|}tj|jdd}d|dd<t j}|)||jd|j|j ddd||d n=|dkrtd||jd|j|j ddd||| |S) aCompute a maximal independent set of a graph in parallel. Parameters ---------- G : csr_matrix Matrix graph, G[i,j] != 0 indicates an edge weights : ndarray Array of weights for each vertex in the graph G maxiter : int Maximum number of iterations (default: None) Returns ------- mis : array Array of length of G of zeros/ones indicating the independent set Examples -------- >>> from pyamg.gallery import poisson >>> from pyamg.classical.split import MIS >>> import numpy as np >>> G = poisson((7,), format='csr') # 1D mesh with 7 vertices >>> w = np.ones((G.shape[0],1)).ravel() >>> mis = MIS(G,w) See Also -------- fn = amg_core.maximal_independent_set_parallel r rr r Nr,zmaxiter must be >= 0) rr rrrrr maximal_independent_set_parallelrr ValueError)r$r#maxitermisfns rr"r"Ls> !  /-...A (171:V , , ,C CF  2B 171:qxB1c7BGGGG Q;;344 4 171:qxB1c7GLLL Jrct|std|jd|jdkrtd|j|jd}t t j|jd|j|j f||f}|j }||z}d|j dd<t j |d }|0|t jt#|z}n\t%||}|dz}|t jt#||z|z z}||||fS) aPreprocess splitting functions. Parameters ---------- S : csr_matrix Strength of connection matrix method : string Algorithm used to compute the vertex coloring: * 'MIS' - Maximal Independent Set * 'JP' - Jones-Plassmann (parallel) * 'LDF' - Largest-Degree-First (parallel) Returns ------- weights: ndarray Weights from a graph coloring of G S : csr_matrix Strength matrix with ones T : csr_matrix transpose of S G : csr_matrix union of S and T Notes ----- Performs the following operations: - Checks input strength of connection matrix S - Replaces S.data with ones - Creates T = S.T in CSR format - Creates G = S union T in CSR format - Creates random weights - Augments weights with graph coloring (if use_color == True) r rr,zexpected square matrix, shape=int8r )rN)axis)rr rr6rronesnnzrrrrdataravelsumrandomrandlenrmax)rr(Nrr$r#coloring num_colorss rr!r!saF !  /-...wqzQWQZC!'CCDDD  ABGAE000!)QXFQ ! ! !A  A AAAF111Ihquu!u}}%%GBINN3w<<888"1o66\\^^a' binnS\\::XE   Q1 r)F)r&)N)__doc__numpyr scipy.sparserr pyamg.graphrpyamgrpyamg.util.utilsrrr%r*r0r2r"r!rrrPsYYt33333333'''''',,,,,,5555p###L++++\3333l$$$N0000h<<<<<<r