Tensorial Model for Photolithography Aerial Image Simulation

Algorithm 1

Lower rank-(𝐾₁,𝐾₂,𝐾₃,𝐾₄) tensor approximation (LRTA-(𝐾₁,𝐾₂,𝐾₃,𝐾₄))

(1) Input: data tensor 𝒯𝒞𝒞, and dimensions 𝐾₁,𝐾₂,𝐾₃,𝐾₄ of all 𝑛-mode signal subspaces.

(2) Initialization  𝑘=0: For 𝑛=1 to 4, calculate the projectors 𝐏₀^(𝑛) given by HOSVD-(𝐾₁,𝐾₂,𝐾₃,𝐾₄):

 (a) 𝑛-mode unfold 𝒯𝒞𝒞 into matrix 𝐓𝐂𝐂_𝑛;

 (b) Compute the SVD of 𝐓𝐂𝐂_𝑛;

 (c) Compute matrix 𝐔₀^(𝑛) formed by the 𝐾_𝑛 eigenvectors associated with the 𝐾_𝑛 largest singular values of

    𝐓𝐂𝐂_𝑛. 𝐔₀^(𝑛) is the initial matrix of the 𝑛-mode signal subspace orthogonal basis vectors;

 (d) Form the initial orthogonal projector 𝐏₀^(𝑛)=𝐔₀^(𝑛)𝐔^(𝑛)𝑇₀ on the 𝑛-mode signal subspace;

 (e) Compute the HOSVD-(𝐾₁,𝐾₂,𝐾₃,𝐾₄) of tensor 𝒯𝒞𝒞 given by

  ℬ₀=𝒯𝒞𝒞×₁𝐏₀⁽¹⁾×₂⋯×₄𝐏₀⁽⁴⁾;

(3) ALS loop: Repeat until convergence, that is, for example, while ‖ℬ_𝑘+1−ℬ_𝑘‖²>𝜀, 𝜀>0 being a prior

   fixed threshold,

 (a) For 𝑛=1 to 4:

  (i) Form ℬ^(𝑛),𝑘: ℬ^(𝑛),𝑘=𝒯𝒞𝒞×₁𝐏⁽¹⁾_𝑘+1×₂𝐏⁽²⁾_𝑘+1×₃𝐏⁽³⁾_𝑘+1×₄𝐏⁽⁴⁾_𝑘+1;

  (ii) 𝑛-mode unfold tensor ℬ^(𝑛),𝑘 into matrix 𝐁_𝑛^(𝑛),𝑘;

  (iii) Compute matrix 𝐂^(𝑛),𝑘=𝐁_𝑛^(𝑛),𝑘𝐓𝐂𝐂^𝑇_𝑛;

  (iv) Compute matrix 𝐔^(𝑛)_𝑘+1 composed of the 𝐾_𝑛 eigenvectors associated with the 𝐾_𝑛 largest eigenvalues

     of 𝐂^(𝑛),𝑘. 𝐔_𝑘^(𝑛) is the matrix of the 𝑛-mode signal subspace orthogonal basis vectors at the 𝑘th

     iteration;

  (v) Compute 𝐏^(𝑛)_𝑘+1=𝐔^(𝑛)_𝑘+1𝐔^(𝑛)𝑇_𝑘+1;

 (b) Compute ℬ_𝑘+1=𝒯𝒞𝒞×₁𝐏⁽¹⁾_𝑘+1×₂⋯×₄𝐏⁽⁴⁾_𝑘+1;

 (c) Increment 𝑘.

(4) Output: the estimated signal tensor is obtained through 𝒯𝒞𝒞=𝒯𝒞𝒞×₁𝐏_𝑘⁽¹⁾_stop×₂⋯×₄𝐏_𝑘⁽⁴⁾_stop. 𝒯𝒞𝒞 is

   the rank-(𝐾₁,𝐾₂,𝐾₃,𝐾₄) approximation of 𝒯𝒞𝒞, where 𝑘_stop is the index of the last iteration after the

   convergence of TUCKALS3 algorithm.