Determination and use of spectral embeddings of large-scale systems by substructuring

An apparatus for a technical analytical system, comprising:

• a memory;
• and
• a processor that is configured for:
• loading a matrix system into the memory;
• performing a domain decomposition into non-overlapping subdomains and a reordering of matrices of the matrix system;
• creating an interface variable projection subspace associated with interface variables of an adjacency graph of the matrix system, wherein the interface variables are related to nodes of the adjacency graph which are connected with nodes located in neighboring partitions;
• creating an internal variable projection subspace associated with internal variables of the adjacency graph of the matrix system, wherein the internal variables are related to nodes of the adjacency graph which are connected only to nodes located in the same partition;
• building a projection matrix based on the interface variable projection subspace and the internal variable projection subspace;
• determining eigenvalues that solve a Raleigh-Ritz eigenvalue problem utilizing the projection matrix;
• and
• utilizing the determined eigenvalues in a technical application system.

A computer program product for a technical analytical system, the computer program product comprising:

• one or more computer readable storage media, and program instructions collectively stored on the one or more computer readable storage media, the program instructions comprising program instructions for:
• loading a matrix system into the memory;
• performing a domain decomposition into non-overlapping subdomains and a reordering of matrices of the matrix system;
• creating an interface variable projection subspace associated with interface variables of an adjacency graph of the matrix system, wherein the interface variables are related to nodes of the adjacency graph which are connected with nodes located in neighboring partitions;
• creating an internal variable projection subspace associated with internal variables of the adjacency graph of the matrix system, wherein the internal variables are related to nodes of the adjacency graph which are connected only to nodes located in the same partition;
• building a projection matrix based on the interface variable projection subspace and the internal variable projection subspace;
• determining eigenvalues that solve a Raleigh-Ritz eigenvalue problem utilizing the projection matrix;
• and
• utilizing the determined eigenvalues in a technical application system.