_-PhG dZddlmZddlZddlmZmZmZm Z ddl m Z ddl m Z ddlmZmZmZmZddlmZmZmZmZmZmZmZd d lmZmZmZmZd d l m!Z!d d l"m#Z#m$Z$m%Z%d dl&m'Z'ddddddddifdddifdddifddddfdfddddf dZ( ddZ)dS)z"Support for aggregation-based AMG.)warnN) csr_matrixisspmatrix_csrisspmatrix_bsrSparseEfficiencyWarning)MultilevelSolver)change_smoothers)eliminate_diag_dom_nodes get_blocksize levelize_strength_or_aggregation%levelize_smooth_or_improve_candidates) classical_strength_of_connection symmetric_strength_of_connection evolution_strength_of_connection#energy_based_strength_of_connectiondistance_strength_of_connectionalgebraic_distanceaffinity_distance)standard_aggregationnaive_aggregationlloyd_aggregationpairwise_aggregation)fit_candidates)jacobi_prolongation_smoother richardson_prolongation_smootherenergy_prolongation_smoother)relaxation_as_linear_operator hermitian symmetricstandardjacobiomegagUUUUUU?block_gauss_seidelsweep)r& iterations Fc Tt|sWt|sH t|}tdtn"#t $r}t d|d}~wwxYw|}|dvrtd||_ |j d|j dkrtd|tj tj t|j dt|z df|j tjt||j }ntj||j }t'|j dkr|d d}|j d|j dkrtd |j dt|krtd |j d kr||}ntj||j }t'|j dkr|d d}|j d|j dkrtd|j d|j dkrtd t-|| | \} } }t-|| | \} } }t/| | } t/|| }g}|t3j||d _||d _|j d kr ||d _t'|| krt|d jj dt|d jz | krlt=||||| | | t'|| krDt|d jj dt|d jz | klt3|fi|}t?||||S)aCreate a multilevel solver using classical-style Smoothed Aggregation (SA). Parameters ---------- A : csr_matrix, bsr_matrix Sparse NxN matrix in CSR or BSR format B : None, array_like Right near-nullspace candidates stored in the columns of an NxK array. The default value B=None is equivalent to B=ones((N,1)) BH : None, array_like Left near-nullspace candidates stored in the columns of an NxK array. BH is only used if symmetry='nonsymmetric'. The default value B=None is equivalent to BH=B.copy() symmetry : string 'symmetric' refers to both real and complex symmetric 'hermitian' refers to both complex Hermitian and real Hermitian 'nonsymmetric' i.e. nonsymmetric in a hermitian sense Note, in the strictly real case, symmetric and hermitian are the same. Note, this flag does not denote definiteness of the operator. strength : string or list Method used to determine the strength of connection between unknowns of the linear system. Method-specific parameters may be passed in using a tuple, e.g. strength=('symmetric',{'theta' : 0.25 }). If strength=None, all nonzero entries of the matrix are considered strong. Choose from 'symmetric', 'classical', 'evolution', 'algebraic_distance', 'affinity', ('predefined', {'C' : csr_matrix}), None aggregate : string or list Method used to aggregate nodes. Choose from 'standard', 'lloyd', 'naive', 'pairwise', ('predefined', {'AggOp' : csr_matrix}) smooth : list Method used to smooth the tentative prolongator. Method-specific parameters may be passed in using a tuple, e.g. smooth= ('jacobi',{'filter' : True }). Choose from 'jacobi', 'richardson', 'energy', None presmoother : tuple, string, list Defines the presmoother for the multilevel cycling. The default block Gauss-Seidel option defaults to point-wise Gauss-Seidel, if the matrix is CSR or is a BSR matrix with blocksize of 1. postsmoother : tuple, string, list Same as presmoother, except defines the postsmoother. improve_candidates : tuple, string, list The ith entry defines the method used to improve the candidates B on level i. If the list is shorter than max_levels, then the last entry will define the method for all levels lower. If tuple or string, then this single relaxation descriptor defines improve_candidates on all levels. The list elements are relaxation descriptors of the form used for presmoother and postsmoother. A value of None implies no action on B. max_levels : integer Maximum number of levels to be used in the multilevel solver. max_coarse : integer Maximum number of variables permitted on the coarse grid. diagonal_dominance : bool, tuple If True (or the first tuple entry is True), then avoid coarsening diagonally dominant rows. The second tuple entry requires a dictionary, where the key value 'theta' is used to tune the diagonal dominance threshold. keep : bool Flag to indicate keeping extra operators in the hierarchy for diagnostics. For example, if True, then strength of connection (C), tentative prolongation (T), and aggregation (AggOp) are kept. Other Parameters ---------------- cycle_type : ['V','W','F'] Structrure of multigrid cycle coarse_solver : ['splu', 'lu', 'cholesky, 'pinv', 'gauss_seidel', ... ] Solver used at the coarsest level of the MG hierarchy. Optionally, may be a tuple (fn, args), where fn is a string such as ['splu', 'lu', ...] or a callable function, and args is a dictionary of arguments to be passed to fn. Returns ------- ml : MultilevelSolver Multigrid hierarchy of matrices and prolongation operators See Also -------- MultilevelSolver, classical.ruge_stuben_solver, aggregation.smoothed_aggregation_solver Notes ----- - This method implements classical-style SA, not root-node style SA (see aggregation.rootnode_solver). - The additional parameters are passed through as arguments to MultilevelSolver. Refer to pyamg.MultilevelSolver for additional documentation. - At each level, four steps are executed in order to define the coarser level operator. 1. Matrix A is given and used to derive a strength matrix, C. 2. Based on the strength matrix, indices are grouped or aggregated. 3. The aggregates define coarse nodes and a tentative prolongation operator T is defined by injection 4. The tentative prolongation operator is smoothed by a relaxation scheme to improve the quality and extent of interpolation from the aggregates to fine nodes. - The parameters smooth, strength, aggregate, presmoother, postsmoother can be varied on a per level basis. For different methods on different levels, use a list as input so that the i-th entry defines the method at the i-th level. If there are more levels in the hierarchy than list entries, the last entry will define the method for all levels lower. Examples are: smooth=[('jacobi', {'omega':1.0}), None, 'jacobi'] presmoother=[('block_gauss_seidel', {'sweep':symmetric}), 'sor'] aggregate=['standard', 'naive'] strength=[('symmetric', {'theta':0.25}), ('symmetric', {'theta':0.08})] - Predefined strength of connection and aggregation schemes can be specified. These options are best used together, but aggregation can be predefined while strength of connection is not. For predefined strength of connection, use a list consisting of tuples of the form ('predefined', {'C' : C0}), where C0 is a csr_matrix and each degree-of-freedom in C0 represents a supernode. For instance to predefine a three-level hierarchy, use [('predefined', {'C' : C0}), ('predefined', {'C' : C1}) ]. Similarly for predefined aggregation, use a list of tuples. For instance to predefine a three-level hierarchy, use [('predefined', {'AggOp' : Agg0}), ('predefined', {'AggOp' : Agg1}) ], where the dimensions of A, Agg0 and Agg1 are compatible, i.e. Agg0.shape[1] == A.shape[0] and Agg1.shape[1] == Agg0.shape[0]. Each AggOp is a csr_matrix. Examples -------- >>> from pyamg import smoothed_aggregation_solver >>> from pyamg.gallery import poisson >>> from scipy.sparse.linalg import cg >>> import numpy as np >>> A = poisson((100,100), format='csr') # matrix >>> b = np.ones((A.shape[0])) # RHS >>> ml = smoothed_aggregation_solver(A) # AMG solver >>> M = ml.aspreconditioner(cycle='V') # preconditioner >>> x,info = cg(A, b, tol=1e-8, maxiter=30, M=M) # solve with CG References ---------- .. [1996VaMaBr] Vanek, P. and Mandel, J. and Brezina, M., "Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems", Computing, vol. 56, no. 3, pp. 179--196, 1996. http://citeseer.ist.psu.edu/vanek96algebraic.html zImplicit conversion of A to CSRzSArgument A must have type csr_matrix or bsr_matrix, or be convertible to csr_matrixN)r!r nonsymmetriczOExpected "symmetric", "nonsymmetric" or "hermitian" for the symmetry parameter rrzexpected square matrixdtypez1The shape of near null-space modes B is incorrectz\Having less target vectors, B.shape[1], than blocksize of A can degrade convergence factors.r+zIThe number of left and right near null-space modes B and BH must be equal) rrrrr Exception TypeErrorasfptype ValueErrorsymmetryshapenpkrononesintr r-eyeasarraylenreshapecopyr r appendrLevelABBH_extend_hierarchyr )r@rArBr3strength aggregatesmooth presmoother postsmootherimprove_candidates max_levels max_coarsediagonal_dominancekeepkwargselevelsmls ]/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/pyamg/aggregation/aggregation.pysmoothed_aggregation_solverrSs+t 1  F!2!2F F1 A 24K L L L L F F F>??DE F F AAAA788 8AJwqzQWQZ1222 y GBGSM!,<,|j|j} |dj} ||t|dz \} }| dkrt|fi|}n| dkrt|fi|}n| dkrt|fi|}n| d vr!d |vrt|fi|}nt|| fi|}n| d krt|fi|}n| d kr|d }n^| dkrt|fi|}nJ| dkrt!|fi|}n6| |}nt#dt%| ||\}}|rt'||fi|}||t|dz \} }| dkrt)|fi|d}n| dkrt+|fi|d}nt| dkrt-|fi|d}nZ| dkrt/|fi|d}n@| d kr|d}nt#dt%| ||t|dz \} }| yt1j|jddf|j}t9| |f||| z} | |d_|jdkr#t9| |f| || z} | |d_t;|| \}} |jdkrt;|| \} } ||t|dz \} }| dkrt=|||| fi|}nT| dkrt?||fi|}n?| dkrtA|||| ddiffi|}n$| |}nt#dt%| |j}|dkr|j}n|dkr |j}n|dkr||t|dz \} }| dkr(t=| | || fi|j}n| dkr&t?| | fi|j}n~| dkr.tA| | || ddiffi|}|j}nJ| |j}n.t#dt%| t#d|r'||d_!||d_"||d_||d_#||d_$|%tMj'||z|z}||_||d_| |d_|jdkr| |d_dSdS) zExtend the multigrid hierarchy. Service routine to implement the strength of connection, aggregation, tentative prolongation construction, and prolongation smoothing. 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