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Roadmap
The repository is primarily a numerical solution framework with:
__Core implementations in C++ (C++17 standard)__
__Multiple solver implementations including:__
__Krylov Subspace methods (GMRES, Conjugate Gradient)__
__Direct solvers (LU factorization)__
__Iterative solvers__
__Preconditioners (LU, MultiGrid)__
__Mixed precision capabilities__
__Python bindings using Pybind11__
__2. Optimization Guidelines__
a) Memory Management
Stack vs Heap: Use stack allocation for small matrices/vectors (< 1KB) Implement custom memory pools for frequent allocations Consider using memory alignment for SIMD operations b) SIMD Vectorization
C++
// Optimize for modern CPU architectures
#include <immintrin.h>
// Example vectorization for vector operations
void vectorAdd(double* a, double* b, double* c, size_t n) {
// Ensure proper alignment
#pragma omp simd align(a,b,c:32)
for (size_t i = 0; i < n; i++) {
c[i] = a[i] + b[i];
}
}
c) Cache Optimization
Implement cache-friendly data structures Use data layouts that minimize cache misses Consider matrix blocking techniques for large operations 2.2 Algorithm-Level Optimization
a) Linear Algebra Operations
Implement BLAS Level 3 operations where possible Use blocked algorithms for matrix operations Consider mixed-precision computing for appropriate problems b) Solver Optimization
C++
// Example of optimized Conjugate Gradient implementation
template<typename MatrixType, typename VectorType>
class OptimizedConjugateGradient : public IterativeSolver {
void solve(const MatrixType& A, VectorType& x, const VectorType& b) {
// Use blocked operations for matrix-vector products
// Implement pipelining for better cache utilization
// Consider mixed-precision operations where appropriate
}
};
2.3 Parallelization Strategy
a) Thread-Level Parallelism
Use OpenMP for shared-memory parallelization
Implement proper load balancing
Minimize synchronization points
C++
// Example of parallel implementation
#pragma omp parallel for schedule(dynamic)
for (size_t i = 0; i < n; i++) {
// Computational kernels
}
b) GPU Acceleration
Identify GPU-friendly algorithms Implement CUDA/OpenCL kernels for compute-intensive operations Use hybrid CPU-GPU execution where appropriate
Setup performance benchmarking framework
Implement comprehensive benchmarks Define performance metrics Setup CI/CD pipeline for performance testing Profile existing codebase
Identify bottlenecks Analyze memory access patterns Measure cache utilization
Memory optimization
Implement custom allocators Optimize data structures Improve cache utilization Algorithmic improvements
Optimize critical mathematical operations Implement blocked algorithms Add mixed-precision support
Thread-level parallelization
Implement OpenMP parallelization Optimize load balancing Reduce synchronization overhead GPU acceleration
Implement CUDA kernels for key operations Optimize memory transfers Implement hybrid CPU-GPU execution
SIMD optimization
Implement vectorized operations Optimize for different instruction sets Auto-vectorization improvements Specialized optimizations
Problem-specific optimizations Architecture-specific tuning Advanced preconditioners 4. Implementation Best Practices
C++
// Example of optimized class structure
template<typename T>
class OptimizedSolver {
private:
// Cache-aligned data structures
alignas(64) std::vector<T> data_;
`// Performance-critical methods`
`void computeIntensive() {`
`#pragma omp parallel`
`{`
`// Parallel implementation`
`}`
`}`
public:
// Public interface with performance documentation
void solve(const Problem& p) {
// Implementation
}
};
Implement performance regression tests Create benchmarking suite Maintain accuracy verification tests 5. Performance Monitoring and Maintenance
Setup continuous performance monitoring
Regular benchmarking Performance regression testing Resource utilization tracking Documentation and reporting
Performance characteristics documentation Optimization guidelines for contributors Regular performance reports