GMRES算法处理多个右端项的Block与PseudoBlock变体

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

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

实际实现考虑因素

1. 预处理：两种方法都可以结合预处理技术提高收敛速度
2. 并行化：Block GMRES更适合矩阵-块向量乘积的并行化
3. 内存管理：Block GMRES需要更多内存存储基向量
4. 重启策略：对于大型问题，两种方法都可能需要重启以避免内存爆炸

选择哪种方法取决于具体问题特性：当右端项高度相关时，Block GMRES通常更有效；当右端项独立或相关性低时，PseudoBlock GMRES可能更稳健。