Optimization of circuit solutions?

[WilliamAshley2019]

I just thought I would share some suggestions provided from a chat with AI. This post is included to start discussions on optimization of livespice, are these possible realistic, have they been considered, too complex etc..?. I am wondering if any of these optimization is implemented or possible in LiveSpice: The method for solving circuit equations in LiveSPICE is already quite advanced and efficient for its purpose of real-time audio signal simulation. It combines numerical integration with an optimized implementation of Newton's method to handle non-linearities, focusing on minimizing computational load during run-time by shifting as much work as possible to analysis-time. Here are some additional techniques and considerations that could further enhance the efficiency of solving these circuit equations:

Exploiting Sparsity:

Circuit matrices (like those from MNA) are often sparse. Using sparse matrix techniques can significantly reduce both memory usage and computation time. Libraries like Eigen or SuiteSparse can be integrated to handle sparse matrices more efficiently than dense matrix operations.
Adaptive Time Stepping:

Instead of using a fixed time step, adaptive time stepping can adjust the step size based on the solution's behavior. This allows larger steps when the solution is smooth and smaller steps when there are rapid changes, which can improve both accuracy and performance.
Parallelization and GPU Acceleration:

Modern processors and GPUs offer parallel computation capabilities. Using parallel algorithms or offloading suitable tasks to GPUs can drastically speed up simulations. Libraries like CUDA for NVIDIA GPUs or OpenCL for cross-platform parallel programming can be employed.
Model Order Reduction (MOR):

Techniques such as Proper Orthogonal Decomposition (POD) or Krylov subspace methods can reduce the dimensionality of the problem, especially for large-scale systems. MOR creates a reduced model that captures the essential dynamics with fewer equations.
Symbolic Computation:

Symbolic preprocessing can simplify expressions and reduce the number of operations required at run-time. Tools like SymPy can help with symbolic manipulation to derive more efficient forms of the equations to be solved.
Preconditioning Techniques:

Improving the conditioning of the system matrix can enhance the convergence rate of iterative solvers like Newton’s method. Preconditioners can transform the system into a form that is easier and faster to solve numerically.
Multi-Rate Simulation:

This approach can handle different parts of the circuit with different time steps. Critical components that require fine resolution are simulated with smaller time steps, while less critical parts use larger steps, balancing accuracy and performance.
Hybrid Analytical-Numerical Methods:

For certain circuit components or configurations, analytical solutions might be derivable. Combining these with numerical methods can reduce the computational burden.
Example of Enhanced Methodology
Let's illustrate the combination of sparse matrix techniques and GPU acceleration for solving the system:

Sparse Matrix Representation:

During the analysis phase, construct sparse representations of matrices
𝐴
A,
𝐵
B, and the Jacobian
𝐽
J. Utilize a sparse LU decomposition or iterative solvers like GMRES for solving these systems.
GPU-Accelerated Newton's Method:

Offload the computation of the Jacobian and its operations to the GPU. Libraries such as cuBLAS and cuSOLVER can perform these operations efficiently on the GPU.
Implementation Outline
python
Copy code
import numpy as np
from scipy.sparse import csc_matrix
from scipy.sparse.linalg import splu, gmres
import cupy as cp

Assume A, B, J, y(t), and c(t) are defined

A_sparse = csc_matrix(A)
B_sparse = csc_matrix(B)
J_sparse = csc_matrix(J)

Analysis phase: Precompute sparse LU decomposition

lu_A = splu(A_sparse)

Run-time phase: Use GPU for Newton's method iterations

def newton_iteration(y_prev, c_t, tol=1e-6, max_iter=50):
y = cp.array(y_prev) # Initial guess
for i in range(max_iter):
F_y = compute_F(y) # Compute non-linear part, possibly on GPU
r = J_sparse @ y + F_y - c_t # Residual
if cp.linalg.norm(r) < tol:
break
J_gpu = cp.array(J_sparse.todense()) # Transfer to GPU if not already
dy = cp.linalg.solve(J_gpu, -r) # Solve on GPU
y += dy # Update
return y

Example usage

y_next = newton_iteration(y_prev, c_t)
In this approach, we efficiently handle the sparse structure of the system using appropriate solvers and leverage GPU acceleration to speed up the iterative solution process.

By integrating these advanced techniques, you can further optimize the simulation of circuits for real-time performance, achieving higher speeds and maintaining numerical stability.