GMRES算法处理多个右端项的Block与PseudoBlock变体

Block与PseudoBlock GMRES简介

在处理多个右端项的线性方程组时，Block GMRES和PseudoBlock GMRES是两种常用的变体算法：

Block GMRES：同时处理所有右端项，构建一个大的Krylov子空间
PseudoBlock GMRES：独立处理每个右端项，但共享Arnoldi过程的信息

主要区别

特性	Block GMRES	PseudoBlock GMRES
处理方式	同时处理所有右端项	独立处理但共享信息
子空间	单个大子空间	多个子空间共享基
内存需求	较高	相对较低
收敛性	可能更快(当右端项相关)	更稳健(当右端项独立)
实现复杂度	较高	中等

Block GMRES伪代码

def block_GMRES(A, B, max_iter, tol, restart=None):"""A: 系数矩阵(n×n)B: 右端项矩阵(n×s), s为右端项数量max_iter: 最大迭代次数tol: 容忍误差restart: 重启周期(可选)"""n, s = B.shapeX = initial_guess(n, s)  # 初始解矩阵for cycle in range(max_cycles):# 计算初始残差块R = B - A @ Xβ = norm(R, 'fro')Q = [R / β]  # 初始正交基H = []  # Hessenberg矩阵for j in range(1, max_iter+1):# Arnoldi过程W = A @ Q[-1]for i in range(j):H[i,j-1] = dot(Q[i], W)W = W - Q[i] * H[i,j-1]H[j,j-1] = norm(W, 'fro')if H[j,j-1] < eps:breakQ.append(W / H[j,j-1])# 解最小二乘问题e1 = zeros(j*s+1, s)e1[:s] = β * eye(s)y = lstsq(H[:j+1,:j], e1)# 更新解X = X + Q[:j] @ y# 检查收敛if norm(R, 'fro') < tol:return Xif restart:# 重启逻辑X = current_solutioncontinuereturn X

PseudoBlock GMRES伪代码

def pseudoBlock_GMRES(A, B, max_iter, tol):"""A: 系数矩阵(n×n)B: 右端项矩阵(n×s), s为右端项数量max_iter: 最大迭代次数tol: 容忍误差"""n, s = B.shapeX = zeros(n, s)  # 初始解矩阵residuals = [norm(B[:,i] - A @ X[:,i]) for i in range(s)]active_systems = list(range(s))  # 未收敛的系统索引# 共享的Krylov子空间Q_shared = []H_shared = []while active_systems:# 选择最不收敛的系统作为种子seed_idx = argmax([residuals[i] for i in active_systems])b_seed = B[:, seed_idx]x_seed = X[:, seed_idx]r_seed = b_seed - A @ x_seedβ = norm(r_seed)q1 = r_seed / β# 标准GMRES Arnoldi过程Q = [q1]H = zeros(max_iter+1, max_iter)for j in range(max_iter):# Arnoldi步骤v = A @ Q[j]for i in range(j+1):H[i,j] = dot(Q[i], v)v = v - H[i,j] * Q[i]H[j+1,j] = norm(v)if H[j+1,j] < eps:breakQ.append(v / H[j+1,j])# 对种子系统求解e1 = zeros(j+2, 1)e1[0] = βy = lstsq(H[:j+2,:j+1], e1)x_seed = x_seed + Q[:j+1] @ y# 对其他系统使用相同子空间for i in active_systems:if i == seed_idx:continuer_i = B[:,i] - A @ X[:,i]β_i = norm(r_i)e1_i = zeros(j+2, 1)e1_i[0] = β_iy_i = lstsq(H[:j+2,:j+1], e1_i)X[:,i] = X[:,i] + Q[:j+1] @ y_iresiduals[i] = norm(B[:,i] - A @ X[:,i])# 检查收敛active_systems = [i for i in active_systems if residuals[i] > tol]if not active_systems:break# 更新共享子空间信息Q_shared = QH_shared = H[:j+2,:j+1]return X