Modeling Physics with Differential-Algebraic Equations Lecture 2 - PowerPoint PPT Presentation

Modeling Physics with Differential-Algebraic Equations Lecture 2 Structural Analysis: Index Reduction COMASIC (M2) Khalil Ghorbal k halil.ghorbal@inria.fr K. Ghorbal (INRIA) 1 COMASIC M2 1 / 23

Summary lecture 1 1 Ordinary Differential Equations: • Cauchy-Lipschitz theorem: existence and uniqueness of solutions • Liouville theorem: no closed form solutions in general • Numerical integration: convergence and stability • Qualitative analysis: invariant regions 2 Differential-Algebraic Equations • Informal Introduction • Examples • Different Forms K. Ghorbal (INRIA) 2 COMASIC M2 2 / 23

Outline 1 Matching Problem 2 BLT Decomposition 3 Pantelides Algorithm K. Ghorbal (INRIA) 2 COMASIC M2 2 / 23

Structural Analysis of Systems of Equations: Example Consider the following system of equations:  f 1 ( x 1 ) = 0  f 2 ( x 1 , x 2 , x 3 ) = 0 f 3 ( x 3 ) = 0  K. Ghorbal (INRIA) 3 COMASIC M2 3 / 23

Structural Analysis of Systems of Equations: Example Consider the following system of equations:  f 1 ( x 1 ) = 0  f 2 ( x 1 , x 2 , x 3 ) = 0 f 3 ( x 3 ) = 0  Incidence Matrix x 1 x 2 x 3   1 0 0 f 1 1 1 1 f 2   0 0 1 f 3 K. Ghorbal (INRIA) 3 COMASIC M2 3 / 23

Structural Analysis of Systems of Equations: Example Consider the following system of equations:  f 1 ( x 1 ) = 0  f 2 ( x 1 , x 2 , x 3 ) = 0 f 3 ( x 3 ) = 0  Incidence Matrix Bipartite Graph (Bigraph) x 1 f 1 x 1 x 2 x 3   1 0 0 f 1 1 1 1 x 2 f 2 f 2   0 0 1 f 3 x 3 f 3 K. Ghorbal (INRIA) 3 COMASIC M2 3 / 23

Bigraphs and Matching Bipartite graph ( F , V , E ) • F : set of equations • V : set of variables (disjoint with F ) • E : subset of the cartesian product F × V Example: a triangle is not a bipartite graph. Matching Problem Given a bipartite graph ( F , V , E ), assign one and only one equation f ∈ F to each variable v ∈ V such that ( f , v ) ∈ E . K. Ghorbal (INRIA) 4 COMASIC M2 4 / 23

Matching Problem: Intuitions Algorithm For each equation, pick up its first unmatched variable. The procedure succeeds whenever all variables are matched. Matching x 1 f 1 x 2 f 2 x 3 f 3 K. Ghorbal (INRIA) 5 COMASIC M2 5 / 23

Matching Problem: Intuitions Algorithm (incomplete/wrong) For each equation, pick up its first unmatched variable. The procedure succeeds whenever all variables are matched. Matching Backtrack ( f 2 , x 3 ) x 1 f 1 x 1 f 1 x 2 f 2 x 3 f 2 x 3 f 3 x 2 f 3 K. Ghorbal (INRIA) 5 COMASIC M2 5 / 23

Matching Problem: Algorithm Let G : ( V , E ) denote a graph • Matching : A matching is a set of pairwise non-adjacent edges. • Maximum Matching : A matching having the maximum cardinality, denoted ν ( G ). • Perfect Matching : A matching such that every vertex of the graph is incident to exactly one edge of the matching. K. Ghorbal (INRIA) 6 COMASIC M2 6 / 23

Matching Problem: Algorithm Let G : ( V , E ) denote a graph • Matching : A matching is a set of pairwise non-adjacent edges. • Maximum Matching : A matching having the maximum cardinality, denoted ν ( G ). • Perfect Matching : A matching such that every vertex of the graph is incident to exactly one edge of the matching. K. Ghorbal (INRIA) 6 COMASIC M2 6 / 23

Matching Problem: Algorithm Let G : ( V , E ) denote a graph • Matching : A matching is a set of pairwise non-adjacent edges. • Maximum Matching : A matching having the maximum cardinality, denoted ν ( G ). • Perfect Matching : A matching such that every vertex of the graph is incident to exactly one edge of the matching. K. Ghorbal (INRIA) 6 COMASIC M2 6 / 23

Matching Problem: Algorithm Let G : ( V , E ) denote a graph • Matching : A matching is a set of pairwise non-adjacent edges. • Maximum Matching : A matching having the maximum cardinality, denoted ν ( G ). • Perfect Matching : A matching such that every vertex of the graph is incident to exactly one edge of the matching. Proposition G has a perfect matching if and only if | G | = 2 ν ( G ). ➻ For systems of n variables and n equations: (1) compute ν ( G ), (2) check if ν ( G ) = n . K. Ghorbal (INRIA) 6 COMASIC M2 6 / 23

Hopcroft-Karp Algorithm (1973) • Input: bipartite graph ( U , V , E ) • Output: maximum cardinality matching � • Complexity: (worst case) O ( | E | | V | ) Augmented Shortest Path At each phase: • BFS: alternate between U and V where one starts from an unmatched variable in U and reaches an unmatched variable in V while following a matched edge from V to U (this gives an augmented shortest path). • DFS: selects one shortest path out of the many selected ones by the BFS. • update the matching set K. Ghorbal (INRIA) 7 COMASIC M2 7 / 23

Example Source: Wikipedia K. Ghorbal (INRIA) 8 COMASIC M2 8 / 23

Maximum Transversal Problem Matching Incidence Matrix x 1 f 1 x 1 x 2 x 3   1 0 0 f 1 x 3 f 2 1 1 1   f 2   0 0 1 f 3 x 2 f 3 Maximum Transversal Problem Finding a permutation that places a maximum number of non-zero on the diagonal of a sparse matrix. K. Ghorbal (INRIA) 9 COMASIC M2 9 / 23

Maximum Transversal Problem Matching Incidence Matrix x 1 f 1 x 1 x 2 x 3   1 0 0 f 1 x 3 f 2 1 1 1   f 2   0 0 1 f 3 x 2 f 3 x 1 x 3 x 2   1 − − Maximum Transversal Problem f 1 Finding a permutation that 0 1 −   f 3   places a maximum number of 1 1 1 f 2 non-zero on the diagonal of a sparse matrix. ❥ Very useful prior to decomposing the matrix using Gaussian elimination. K. Ghorbal (INRIA) 9 COMASIC M2 9 / 23

Outline 1 Matching Problem 2 BLT Decomposition 3 Pantelides Algorithm K. Ghorbal (INRIA) 9 COMASIC M2 9 / 23

Block-Lower-Triangular (BLT) Form Goal: Given a square matrix M ( m ij ∈ { 0 , 1 } ), find a permutation to put M in a block lower triangular form. x 1 x 2 x 3 x 4  1 0 1 0  f 1 1 1 0 1 f 2     0 1 0 0 f 3   1 1 1 0 f 4 x 2 x 1 x 3 x 4  1 0 0 0  f 3 0 1 1 0 f 1     1 1 1 0 f 4   1 1 0 1 f 2 K. Ghorbal (INRIA) 10 COMASIC M2 10 / 23

BLT Decomposition Blocks of dimension > 1 (called algebraic loops ) are (numerically) solved by Gaussian elimination if linear or Newton methods if nonlinear. K. Ghorbal (INRIA) 11 COMASIC M2 11 / 23

BLT Decomposition Blocks of dimension > 1 (called algebraic loops ) are (numerically) solved by Gaussian elimination if linear or Newton methods if nonlinear. Three main steps 1 Find a (perfect) matching 2 Construct a dependency graph 3 Find strongly connected components (Tarjan’s algorithm) K. Ghorbal (INRIA) 11 COMASIC M2 11 / 23

BLT Decomposition: Example Perfect Matching Bipartite Graph (Bigraph) Incidence Matrix x 1 f 1 x 1 x 2 x 3 x 4   1 0 1 0 f 1 x 2 1 1 0 1 f 2   f 2    0 1 0 0  f 3   1 1 1 0 x 3 f 3 f 4 x 4 f 4 K. Ghorbal (INRIA) 12 COMASIC M2 12 / 23

BLT Decomposition: Example Dependency Graph Incidence Matrix Dependency Graph f 1 x 1 x 2 x 3 x 4   1 0 1 0 f 1 1 1 0 1   f 2   f 4 f 2  0 1 0 0  f 3   1 1 1 0 f 4 f 3 K. Ghorbal (INRIA) 13 COMASIC M2 13 / 23

BLT Decomposition: Example Strongly Connected Components Dependency Graph BLT Decomposition f 1 x 2 x 1 x 3 x 4  1 0 0 0  f 3 2 0 1 1 0 f 1   f 4 f 2 3   1 1 1 0 f 4   1 1 0 1 f 2 f 3 1 K. Ghorbal (INRIA) 14 COMASIC M2 14 / 23

Strongly Connected Components (SCC) Definitions • A directed graph is strongly connected if every vertex is reachable from every other vertex. • Strongly connected components of a directed graph form a partition into subgraphs which are themselves strongly connected. • By contracting each strongly connected component into one vertex, one obtains a condensation of the original graph into a directed acyclic graph . K. Ghorbal (INRIA) 15 COMASIC M2 15 / 23

Strongly Connected Components (SCC) Definitions • A directed graph is strongly connected if every vertex is reachable from every other vertex. • Strongly connected components of a directed graph form a partition into subgraphs which are themselves strongly connected. • By contracting each strongly connected component into one vertex, one obtains a condensation of the original graph into a directed acyclic graph . K. Ghorbal (INRIA) 15 COMASIC M2 15 / 23

Strongly Connected Components (SCC) Definitions • A directed graph is strongly connected if every vertex is reachable from every other vertex. • Strongly connected components of a directed graph form a partition into subgraphs which are themselves strongly connected. • By contracting each strongly connected component into one vertex, one obtains a condensation of the original graph into a directed acyclic graph . K. Ghorbal (INRIA) 15 COMASIC M2 15 / 23