Chapter Summary

Chapter 4 Summary: Computational Tools for Inverse Problems

Key Points

• 1.

  The sensing operator $\mathbf{A}$ in RF imaging inherits Kronecker structure from the separable array geometry. The vec trick converts a dense $O(M^2N^2)$ matrix-vector product into two smaller matrix multiplications at cost $O(M_1 N_1 M_2 + M_2 N_2 N_1)$, reducing computation by orders of magnitude. When the factors are DFT matrices, the FFT provides an additional logarithmic speedup.

• 2.

  Matrix-free operators represent $\mathbf{A}$ and $\mathbf{A}^{H}$ as functions rather than stored matrices, enabling large-scale 3D reconstruction. GPU acceleration via CuPy or PyTorch provides massive parallelism for the underlying matrix multiplications and FFTs. Batched operations across frequencies amortize kernel launch overhead.

• 3.

  Automatic differentiation (AD) provides exact gradients through arbitrary computation graphs. Reverse mode (backpropagation) is efficient for scalar loss optimization (unrolled algorithms); forward mode is efficient for Jacobian-vector products (denoiser analysis). Gradient checkpointing trades computation for memory in deep unrollings.

• 4.

  Convergence diagnostics are integral to every reconstruction pipeline. The fixed-point residual is universal; primal and dual residuals provide ADMM-specific certificates; the discrepancy principle sets the noise-appropriate stopping level. Warm-starting from the matched-filter image saves 40--60% of iterations.