Topics 1. Gas Flow ◮ Stationary gas flow in pipeline systems under additional assumptions: passive network, i.e., without components influencing gas flow actively (compressors, valves); horizontal pipes; constant friction coefficients. 2. Probabilistic Nomination Validation ◮ The probability of those balanced injection and random withdrawals, for which there exist arc flows and bounded node pressures fulfilling Kirchhoff’s Laws ?? 3. Explicit Feasibility Representation by Symbolic Computation ◮ Fundamental cycles of the network imply feasibility system of “privileged” multivariate polynomials of degree two; variable elimination; (reverse) lexicographic order; parametric system - comprehensive Gr¨ obner Base; Shape Lemma.
4. Spherical-Radial Decomposition of Gaussian Distribution ◮ Splits integration over Gaussian Laws into integration of uniform distributionn over unit sphere (sampling) and a χ 2 -distribution in dimension one;
4. Spherical-Radial Decomposition of Gaussian Distribution ◮ Splits integration over Gaussian Laws into integration of uniform distributionn over unit sphere (sampling) and a χ 2 -distribution in dimension one; 5. Performance Gains in Quasi-Monte-Carlo Sampling ◮ Sampling (of Probabilities, Function Values and Gradients) with implicit representation vs. Sampling with explicit formula vs. Comprehensive Gr¨ obner bases calculation within Sampling on the Unit Sphere: Initial experiments with probabilities → variance reduction by two decimals
Graph Theoretic Setting ◮ Gas network model = connected, directed (simple) graph G = ( V , E ) ,
Graph Theoretic Setting ◮ Gas network model = connected, directed (simple) graph G = ( V , E ) , ◮ with node-arc incidence matrix A + and in/out nomination vector b + .
Graph Theoretic Setting ◮ Gas network model = connected, directed (simple) graph G = ( V , E ) , ◮ with node-arc incidence matrix A + and in/out nomination vector b + . ◮ Kirchhoff 1 (mass preservation at nodes) A + q b + = with q ∈ R | E | denoting the gas flow in the pipes.
Graph Theoretic Setting ◮ Gas network model = connected, directed (simple) graph G = ( V , E ) , ◮ with node-arc incidence matrix A + and in/out nomination vector b + . ◮ Kirchhoff 1 (mass preservation at nodes) A + q b + = with q ∈ R | E | denoting the gas flow in the pipes. ◮ after deletion of the first row/component of A + , b + (slack or reference or root node) Aq = b gives the matrix A with full rank and the vector b . Now “simplex-like” b iff q B = ( A B ) − 1 b − ( A B ) − 1 A N q N Aq = b iff A B q B + A N q N =
Graph Theoretic Setting ◮ Gas network model = connected, directed (simple) graph G = ( V , E ) , ◮ with node-arc incidence matrix A + and in/out nomination vector b + . ◮ Kirchhoff 1 (mass preservation at nodes) A + q b + = with q ∈ R | E | denoting the gas flow in the pipes. ◮ after deletion of the first row/component of A + , b + (slack or reference or root node) Aq = b gives the matrix A with full rank and the vector b . Now “simplex-like” b iff q B = ( A B ) − 1 b − ( A B ) − 1 A N q N Aq = b iff A B q B + A N q N = The graph behind A B is a spanning tree T = ( V , E T ) of G , with edge flows q B in E T and edge flows q N in E \ T .
◮ Since T ⊆ G is a spanning tree, for every edge (chord) e ∈ E \ E T there exists a unique cycle in ( V , E T ∪ { e } ) , called fundamental cycle. → cardinality of | N | = minimum number of edges to be removed from G to obtain a tree.
◮ Since T ⊆ G is a spanning tree, for every edge (chord) e ∈ E \ E T there exists a unique cycle in ( V , E T ∪ { e } ) , called fundamental cycle. → cardinality of | N | = minimum number of edges to be removed from G to obtain a tree. ◮ Kirchhoff 2: (pressure drops along fundamental cycles sum up to zero) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q with p ∈ R | V | standing for the node pressures; squares as well as + moduli understood component-wise.
◮ Since T ⊆ G is a spanning tree, for every edge (chord) e ∈ E \ E T there exists a unique cycle in ( V , E T ∪ { e } ) , called fundamental cycle. → cardinality of | N | = minimum number of edges to be removed from G to obtain a tree. ◮ Kirchhoff 2: (pressure drops along fundamental cycles sum up to zero) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q with p ∈ R | V | standing for the node pressures; squares as well as + moduli understood component-wise.
Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b ( ω ) (1) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q (2) � p + min , p + max � p + ∈ (3) , Analytical as well as Algebraic Appeal ! Observations: ◮ System of multivariate polynomials of degree two with absolute values.
Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b ( ω ) (1) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q (2) � p + min , p + max � p + ∈ (3) , Analytical as well as Algebraic Appeal ! Observations: ◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator,
Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b ( ω ) (1) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q (2) � p + min , p + max � p + ∈ (3) , Analytical as well as Algebraic Appeal ! Observations: ◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients and Jacobian,
Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b ( ω ) (1) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q (2) � p + min , p + max � p + ∈ (3) , Analytical as well as Algebraic Appeal ! Observations: ◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients and Jacobian, Brouwer’s Fixed Point Theorem.
Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b ( ω ) (1) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q (2) � p + min , p + max � p + ∈ (3) , Analytical as well as Algebraic Appeal ! Observations: ◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients and Jacobian, Brouwer’s Fixed Point Theorem. ◮ Algebraic View: affine variety,
Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b ( ω ) (1) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q (2) � p + min , p + max � p + ∈ (3) , Analytical as well as Algebraic Appeal ! Observations: ◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients and Jacobian, Brouwer’s Fixed Point Theorem. ◮ Algebraic View: affine variety, polynomial ideal,
Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b ( ω ) (1) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q (2) � p + min , p + max � p + ∈ (3) , Analytical as well as Algebraic Appeal ! Observations: ◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients and Jacobian, Brouwer’s Fixed Point Theorem. ◮ Algebraic View: affine variety, polynomial ideal, (comprehensive) Gr¨ obner bases,
Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b ( ω ) (1) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q (2) � p + min , p + max � p + ∈ (3) , Analytical as well as Algebraic Appeal ! Observations: ◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients and Jacobian, Brouwer’s Fixed Point Theorem. ◮ Algebraic View: affine variety, polynomial ideal, (comprehensive) Gr¨ obner bases, elimination order,
Central Object of Study: Kirchhoff 1 + 2 plus pressure bounds Aq = b ( ω ) (1) ( A + ) ⊤ ( p + ) 2 = − Φ | q | q (2) � p + min , p + max � p + ∈ (3) , Analytical as well as Algebraic Appeal ! Observations: ◮ System of multivariate polynomials of degree two with absolute values. ◮ Analytical View: monotone, coercive operator, Lipschitzian gradients and Jacobian, Brouwer’s Fixed Point Theorem. ◮ Algebraic View: affine variety, polynomial ideal, (comprehensive) Gr¨ obner bases, elimination order, Shape Lemma.
0-dimensionality of affine variety
0-dimensionality of affine variety � � Aq = b and F · Φ · | q | · q = 0 . F circuit matrix , whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E
0-dimensionality of affine variety � � Aq = b and F · Φ · | q | · q = 0 . F circuit matrix , whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Φ · q 2 � � Aq = b , F · = 0 , q ≥ 0
0-dimensionality of affine variety � � Aq = b and F · Φ · | q | · q = 0 . F circuit matrix , whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Φ · q 2 � � Aq = b , F · = 0 , q ≥ 0 Denote the set of solutions of the above system by G = G b , Φ := { q ∈ C E | A q = b , F Φ q 2 = 0 } and Q = diag ( q ) Instead of checking the mere dimension of G , we will check when the tangent space at points q ∈ G b , Φ will be at least one-dimensional.
0-dimensionality of affine variety � � Aq = b and F · Φ · | q | · q = 0 . F circuit matrix , whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Φ · q 2 � � Aq = b , F · = 0 , q ≥ 0 Denote the set of solutions of the above system by G = G b , Φ := { q ∈ C E | A q = b , F Φ q 2 = 0 } and Q = diag ( q ) Instead of checking the mere dimension of G , we will check when the tangent space at points q ∈ G b , Φ will be at least one-dimensional.The tangent space T q ( G b , Φ ) in a point q ∈ G b , Φ is given by the linear equations A dq = 0 and F Φ Q dq = 0 .
0-dimensionality of affine variety � � Aq = b and F · Φ · | q | · q = 0 . F circuit matrix , whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Φ · q 2 � � Aq = b , F · = 0 , q ≥ 0 Denote the set of solutions of the above system by G = G b , Φ := { q ∈ C E | A q = b , F Φ q 2 = 0 } and Q = diag ( q ) Instead of checking the mere dimension of G , we will check when the tangent space at points q ∈ G b , Φ will be at least one-dimensional.The tangent space T q ( G b , Φ ) in a point q ∈ G b , Φ is given by the linear equations A dq = 0 and F Φ Q dq = 0 . A � � Thus, dim C T q ( G b , Φ ) ≥ 1 is equivalent to det = 0 . F Φ Q
0-dimensionality of affine variety � � Aq = b and F · Φ · | q | · q = 0 . F circuit matrix , whose rows correspond to the fundamental circuits and whose columns correspond to the edges in E Φ · q 2 � � Aq = b , F · = 0 , q ≥ 0 Denote the set of solutions of the above system by G = G b , Φ := { q ∈ C E | A q = b , F Φ q 2 = 0 } and Q = diag ( q ) Instead of checking the mere dimension of G , we will check when the tangent space at points q ∈ G b , Φ will be at least one-dimensional.The tangent space T q ( G b , Φ ) in a point q ∈ G b , Φ is given by the linear equations A dq = 0 and F Φ Q dq = 0 . A � � Thus, dim C T q ( G b , Φ ) ≥ 1 is equivalent to det = 0 . F Φ Q � ⊤ � ( A B ) − 1 A N Φ B q 2 B = Φ N q 2 and N �� � � ⊤ ( A B ) − 1 A N � ( A B ) − 1 A N � det Φ B Q B + Φ N Q N ) = 0 .
�� � � − 1 ⊤ b , b : ∃ ( q , p + ) fulfilling ( ?? ), ( ?? ), ( ?? ) M = .
�� � � − 1 ⊤ b , b : ∃ ( q , p + ) fulfilling ( ?? ), ( ?? ), ( ?? ) M = . Proposition: Let A = ( A B , A N ) be a partition into basis and nonbasis matrices. Let Φ B , Φ N and q B , q N be corresponding partitions of Φ and q . Denote g : R | V | × R | N | → R | V | – a multivariate, piece-wise quadratic mapping – such that � − 1 Φ B A ⊤ � A − 1 A − 1 � � � �� � g ( u , v ) := B ( u − A N v ) B ( u − A N v ) (4) . B
�� � � − 1 ⊤ b , b : ∃ ( q , p + ) fulfilling ( ?? ), ( ?? ), ( ?? ) M = . Proposition: Let A = ( A B , A N ) be a partition into basis and nonbasis matrices. Let Φ B , Φ N and q B , q N be corresponding partitions of Φ and q . Denote g : R | V | × R | N | → R | V | – a multivariate, piece-wise quadratic mapping – such that � − 1 Φ B A ⊤ � A − 1 A − 1 � � � �� � g ( u , v ) := B ( u − A N v ) B ( u − A N v ) (4) . B Then nomination b + is valid iff ( − 1 ⊤ b , b ) ∈ M
�� � � − 1 ⊤ b , b : ∃ ( q , p + ) fulfilling ( ?? ), ( ?? ), ( ?? ) M = . Proposition: Let A = ( A B , A N ) be a partition into basis and nonbasis matrices. Let Φ B , Φ N and q B , q N be corresponding partitions of Φ and q . Denote g : R | V | × R | N | → R | V | – a multivariate, piece-wise quadratic mapping – such that � − 1 Φ B A ⊤ � A − 1 A − 1 � � � �� � g ( u , v ) := B ( u − A N v ) B ( u − A N v ) (4) . B Then nomination b + is valid iff ( − 1 ⊤ b , b ) ∈ M IFF ∃ z such that ( b , z ) fulfils A ⊤ N g ( b , z ) = Φ N | z | z (5) and ) 2 + g i ( b , z ) � ) 2 + g i ( b , z ) � ( p max ( p min � � ≥ min max (6) i i i =1 ,..., n i =1 ,..., n ) 2 + g i ( b , z ) ( p min ) 2 ( p max ≤ � � min (7) 0 i i =1 ,..., n � ) 2 + g i ( b , z ) � ( p max ) 2 ( p min ≥ max (8) 0 i i =1 ,..., n
After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, | N | equations, and | N | variables.
After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, | N | equations, and | N | variables. � � ( A B ) − 1 b − ( A B ) − 1 A N q N � � ( A B ) − 1 b − ( A B ) − 1 A N q N � A ⊤ N ( A ⊤ B ) − 1 Φ B � � � − Φ N | q N | q N = 0 Solve this system in q N , and insert solution(s) below:
After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, | N | equations, and | N | variables. � ( A B ) − 1 b − ( A B ) − 1 A N q N � � � ( A B ) − 1 b − ( A B ) − 1 A N q N � A ⊤ N ( A ⊤ B ) − 1 Φ B � � � − Φ N | q N | q N = 0 Solve this system in q N , and insert solution(s) below: p 2 = 1 | V |− 1 p 2 � � ( A B ) − 1 b − ( A B ) − 1 A N q N � � ( A B ) − 1 b − ( A B ) − 1 A N q N � o − ( A ⊤ B ) − 1 Φ B � � �
After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, | N | equations, and | N | variables. � � ( A B ) − 1 b − ( A B ) − 1 A N q N � � ( A B ) − 1 b − ( A B ) − 1 A N q N � A ⊤ N ( A ⊤ B ) − 1 Φ B � � � − Φ N | q N | q N = 0 Solve this system in q N , and insert solution(s) below: p 2 = 1 | V |− 1 p 2 � � ( A B ) − 1 b − ( A B ) − 1 A N q N � � ( A B ) − 1 b − ( A B ) − 1 A N q N � o − ( A ⊤ B ) − 1 Φ B � � � Check p + ∈ � p + min , p + max �
After some formula manipulation, one arrives at the equivalent polynomial system with degree 2, | N | equations, and | N | variables. � ( A B ) − 1 b − ( A B ) − 1 A N q N � � � ( A B ) − 1 b − ( A B ) − 1 A N q N � A ⊤ N ( A ⊤ B ) − 1 Φ B � � � − Φ N | q N | q N = 0 Solve this system in q N , and insert solution(s) below: p 2 = 1 | V |− 1 p 2 � � ( A B ) − 1 b − ( A B ) − 1 A N q N � � ( A B ) − 1 b − ( A B ) − 1 A N q N � o − ( A ⊤ B ) − 1 Φ B � � � Check p + ∈ � p + min , p + max � meaning � �� � ) 2 + g i ( b , z ) � ) 2 + g i ( b , z ) � ) 2 � ( p min ( p max ( p min ) 2 , ( p max � ∩ � = ∅ max , min 0 0 i i i =1 ,..., n i =1 ,..., n
� b + : 1 ⊤ b + = 0 and ∃ ( q , p + ) : p + ∈ � p + min , p + max � � M = , Kirchhoff 1 and 2 .
� b + : 1 ⊤ b + = 0 and ∃ ( q , p + ) : p + ∈ � p + min , p + max � � M = , Kirchhoff 1 and 2 . Nomination Validation (passive network) – Decide b + ∈ M Given a balanced load vector b + . Do there exist arc flows q and node pressures p + within bounds p + min , p + max fulfilling the Kirchhoff Laws ??
“Regularity of Coefficients no Surprise” – Tailor F ( b , q N ) = 0 by properly directing G ∗ � � Φ Nl · | q Nl | ∗ = � � � � � Φ j b i − q Nh � � , � � � � j : e j ∈ E ( C l ) i ∈ I j h : C h ∈ H j or l = 1 , . . . , L , � � � � � � � � � Φ Nl ·| q Nl | q Nl = Φ j b i − q Nh b i − q Nh � � � � i ∈ I j h : C h ∈ H j i ∈ I j h : C h ∈ H j j : e j ∈ E ( C l ) with ◮ variables q N 1 , . . . , q NL , L = number of fundamental cycles. ◮ C l denotes the cycle containing q l , ◮ E ( C l ) the set of all edges, except for q Nl , with both ends in C l , ◮ I j = V ( T ( e he j )) the node set of the tree T which is rooted in the head e he of e j . j ◮ H j the set of all cycles C h , h ∈ { 1 , . . . , L } , the edge e j belongs to.
Φ 1 | β 1 − q N 1 | ∗ + Φ 2 | β 2 − q N 1 | ∗ + Φ 3 | β 3 − q N 1 − q N 4 − q N 5 | ∗ Φ N 1 | q N 1 | ∗ = Φ 4 | β 4 − q N 2 − q N 4 − q N 5 | ∗ + Φ 5 | β 5 − q N 2 | ∗ + Φ 6 | β 6 − q N 2 | ∗ Φ N 2 | q N 2 | ∗ = Φ 7 | β 7 − q N 3 − q N 4 − q N 5 | ∗ + Φ 8 | β 8 − q N 3 | ∗ + Φ 9 | β 9 − q N 3 | ∗ Φ N 3 | q N 3 | ∗ = Φ 3 | β 3 − q N 1 − q N 4 − q N 5 | ∗ + Φ 4 | β 4 − q N 2 − q N 4 − q N 5 | ∗ + Φ N 4 | q N 4 | ∗ = Φ 7 | β 7 − q N 3 − q N 4 − q N 5 | ∗ + Φ 10 | β 10 − q N 4 | ∗ + Φ 11 | β 11 − q N 4 | ∗ + Φ 3 | β 3 − q N 1 − q N 4 − q N 5 | ∗ + Φ 4 | β 4 − q N 2 − q N 4 − q N 5 | ∗ + Φ N 5 | q N 5 | ∗ = Φ 7 | β 7 − q N 3 − q N 4 − q N 5 | ∗ +
Two Cycles Sufficient ?? – Gallery of Real Gas Networks (I) Groningen voedingsstation(s) [entry-punten] compressor- en mengstation compressorstation N mengstation BBL exportstation installatie ondergrondse opslag L installatie voor vloeibaar aardgas N stikstofinjectie leiding – Groningen-gas N leiding – hoogcalorisch gas N leiding – laagcalorisch gas leiding – ontzwaveld gas leiding – stikstof L N
Two Cycles Sufficient ?? – Gallery of Real Gas Networks (II)
Two Cycles Sufficient ?? – Gallery of Real Gas Networks (III)
Two Cycles Sufficient ?? – Gallery of Real Gas Networks (III)
Two Cycles Sufficient ?? – Gallery of Real Gas Networks (IV)
Two Cycles Sufficient ?? – Gallery of Real Gas Networks (V)
OGE – High Caloric Grid Northern Germany Essentially 2 Circles !!
OGE - Low Caloric Grid: Always a Matter of Detail
Motivation: Probabilistic Setting Spheric-Radial Decomposition - Procedure for Approximating Gaussian Probabilities
Underlying Probability Distributions and their Decomposition Assume that ξ ∼ N ( µ, Σ) , i.e., the random vector ξ follows a multivariate Gaussian distribution with mean vector µ and positive definite covariance matrix Σ . Theorem (spheric-radial decomposition) Let ξ ∼ N (0 , R ) be some n -dimensional standard Gaussian distribution with zero mean and positive definite correlation matrix R .
Underlying Probability Distributions and their Decomposition Assume that ξ ∼ N ( µ, Σ) , i.e., the random vector ξ follows a multivariate Gaussian distribution with mean vector µ and positive definite covariance matrix Σ . Theorem (spheric-radial decomposition) Let ξ ∼ N (0 , R ) be some n -dimensional standard Gaussian distribution with zero mean and positive definite correlation matrix R . Then, for any Borel measurable subset M ⊆ R n it holds that � P ( ξ ∈ M ) = S n − 1 µ χ { r ≥ 0 | rLv ∈ M } d µ η ( v ) , where S n − 1 is the ( n − 1) -dimensional sphere in R n , µ η is the uniform distribution on S n − 1 , µ χ denotes the χ -distribution with n degrees of freedom and L is such that R = LL T (e.g., Cholesky decomposition).
Algorithm:
Algorithm: Let ξ ∼ N ( µ, Σ) and L such that LL T = Σ (e.g., Cholesky factorization).
Algorithm: Let ξ ∼ N ( µ, Σ) and L such that LL T = Σ (e.g., Cholesky factorization). 1. Sample N points { v 1 , . . . , v N } uniformly distributed on the sphere S n − 1 .
Algorithm: Let ξ ∼ N ( µ, Σ) and L such that LL T = Σ (e.g., Cholesky factorization). 1. Sample N points { v 1 , . . . , v N } uniformly distributed on the sphere S n − 1 . 2. Compute the one-dimensional sets M i := { r ≥ 0 | rLv i + µ ∈ M } for i = 1 , . . . , N .
Algorithm: Let ξ ∼ N ( µ, Σ) and L such that LL T = Σ (e.g., Cholesky factorization). 1. Sample N points { v 1 , . . . , v N } uniformly distributed on the sphere S n − 1 . 2. Compute the one-dimensional sets M i := { r ≥ 0 | rLv i + µ ∈ M } for i = 1 , . . . , N . 3. Set P ( ξ ∈ M ) ≈ N − 1 N � µ χ ( M i ) . i =1 Then nomination b + is valid iff ( − 1 ⊤ b , b ) ∈ M
Algorithm: Let ξ ∼ N ( µ, Σ) and L such that LL T = Σ (e.g., Cholesky factorization). 1. Sample N points { v 1 , . . . , v N } uniformly distributed on the sphere S n − 1 . 2. Compute the one-dimensional sets M i := { r ≥ 0 | rLv i + µ ∈ M } for i = 1 , . . . , N . 3. Set P ( ξ ∈ M ) ≈ N − 1 N � µ χ ( M i ) . i =1 Then nomination b + is valid iff ( − 1 ⊤ b , b ) ∈ M IFF ∃ z such that ( b , z ) fulfils A ⊤ N g ( b , z ) = Φ N | z | z (9) and ) 2 + g i ( b , z ) � ) 2 + g i ( b , z ) � ( p max ( p min � � ≥ min max (10) i i i =1 ,..., n i =1 ,..., n ) 2 + g i ( b , z ) ( p min ) 2 ( p max ≤ min � � (11) 0 i i =1 ,..., n � ) 2 + g i ( b , z ) � ) 2 ( p max ( p min ≥ max (12) 0 i i =1 ,..., n
1 1 Spheric-Radial Spheric-Radial Generic Sampling Generic Sampling 0.995 0.995 0.99 0.99 0.985 0.985 0.98 0.98 0.975 0.975 0.97 0.97 0.965 0.965 0.96 0.96 0 2000 4000 6000 8000 10000 0 2000 4000 6000 8000 10000 Figure : Moving average of computed probability with respect to number of iterations for Mersenne Twister (left) and QMC sampling (right)
Further Considerations:
Further Considerations: ◮ Networks with at most one fundamental cycle can be handled with the solution formula from highschool maths.
Further Considerations: ◮ Networks with at most one fundamental cycle can be handled with the solution formula from highschool maths. ◮ For mdels with two or more cycles computational algebra has something to offer, Gr¨ obner Bases .
With polynomials f 1 , . . . , f s in C [ x 1 , . . . , x n ] , the set � s � � � f 1 , . . . , f s � := h i f i : h 1 , . . . , h s ∈ C [ x 1 , . . . , x n ] i =1 (of “polynomial linear combinations”) is called the ideal generated by f 1 , . . . , f s . The polynomials f 1 , . . . , f s then are said to form a basis of I .
With polynomials f 1 , . . . , f s in C [ x 1 , . . . , x n ] , the set � s � � � f 1 , . . . , f s � := h i f i : h 1 , . . . , h s ∈ C [ x 1 , . . . , x n ] i =1 (of “polynomial linear combinations”) is called the ideal generated by f 1 , . . . , f s . The polynomials f 1 , . . . , f s then are said to form a basis of I . With polynomials f 1 , . . . , f s in C [ x 1 , . . . , x n ] , the set V ( f 1 , . . . , f s ) := { ( a 1 , . . . , a n ) ∈ C n : f i ( a 1 , . . . , a n ) = 0 , for all 1 ≤ i ≤ s } of common zeros in C n is called the affine variety defined by f 1 , . . . , f s .
With polynomials f 1 , . . . , f s in C [ x 1 , . . . , x n ] , the set � s � � � f 1 , . . . , f s � := h i f i : h 1 , . . . , h s ∈ C [ x 1 , . . . , x n ] i =1 (of “polynomial linear combinations”) is called the ideal generated by f 1 , . . . , f s . The polynomials f 1 , . . . , f s then are said to form a basis of I . With polynomials f 1 , . . . , f s in C [ x 1 , . . . , x n ] , the set V ( f 1 , . . . , f s ) := { ( a 1 , . . . , a n ) ∈ C n : f i ( a 1 , . . . , a n ) = 0 , for all 1 ≤ i ≤ s } of common zeros in C n is called the affine variety defined by f 1 , . . . , f s . Fact: If f 1 , . . . , f s and g 1 , . . . , g t are bases of the same ideal in C [ x 1 , . . . , x n ] , i.e., � f 1 , . . . , f s � = � g 1 , . . . , g t � , then their affine varieties coincide V ( f 1 , . . . , f s ) = V ( g 1 , . . . , g t ) . That means, the systems f 1 = 0 , . . . , f s = 0 and g 1 = 0 , . . . , g t = 0 have the same solution sets (in C n ).
Triangular Form of Polynomial Systems – Variables’ Elimination Fact: Given I = � f 1 , . . . , f s � ⊂ C [ x 1 , . . . , x n ] , under suitable assumptions, there exists a triangular basis G = { g 1 , . . . , g n } for I (Reduced Gr¨ obner Basis) , i.e., g 1 = g 1 ( x 1 , . . . , x n ) , g 2 = g 2 ( x 2 , . . . , x n ) . . . g n − 1 = g n − 1 ( x n − 1 , x n ) , g n = g n ( x n ) G can be computed by a finite algorithm (Buchberger’s Algorithm).
Shape Lemma Let I be a zero-dimensional radical ideal in C [ x 1 , . . . , x n ] such that all d complex roots of I have distinct x n coordinates. Then the reduced Gr¨ obner basis G of I in lexicographic monomial order has the shape − ϕ 1 ( x n ) x 1 − ϕ 2 ( x n ) x 2 . . . − ϕ n − 1 ( x n ) x n − 1 ϕ n ( x n ) where ϕ n is a univariate polynomial of degree d and the remaining ϕ i are polynomials of degree ≤ d − 1 .
b 1 1 q 01 q 13 0 3 q 12 b 0 b 3 q 02 q 23 2 b 2
Now the “red” system ( ?? ) of polynomial equations reads φ 02 | z 1 | z 1 = φ 01 | b 1 + b 2 + b 3 − z 1 | ( b 1 + b 2 + b 3 − z 1 ) + φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) φ 13 | z 2 | z 2 = φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) + φ 23 | b 3 − z 2 | ( b 3 − z 2 ) 0-dimensionality of variety: We obtain the equations q 2 1 + q 2 2 = q 2 q 2 2 + q 2 3 = q 2 and 4 5 (implying ( q 1 + q 3 )( q 1 − q 3 ) = ( q 4 + q 5 )( q 4 − q 5 ) ) and ( q 1 + q 2 + q 4 )( q 2 + q 3 + q 5 ) = q 2 2 . Elimination yields a single equation within the variables ( b 1 , b 2 , b 3 ) which decomposes into a product of three factors: b 3 · ( b 1 + b 2 + b 3 ) · ( b 2 1 + b 2 2 + b 2 3 − b 1 b 3 + 3 b 2 b 3 ) = 0 .
φ 02 | z 1 | z 1 = φ 01 | b 1 + b 2 + b 3 − z 1 | ( b 1 + b 2 + b 3 − z 1 ) � � � + φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � � φ 13 | z 2 | z 2 = φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � + φ 23 | b 3 − z 2 | ( b 3 − z 2 ) � � � y 1 ≥ y 2 + φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � � y 1 ≥ y 3 + φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � � � + φ 23 | b 3 − z 2 | ( b 3 − z 2 ) � � y 2 ≥ y 1 − φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � � y 2 ≥ y 3 + φ 23 | b 3 − z 2 | ( b 3 − z 2 ) � � � y 3 ≥ y 1 − φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � � − φ 23 | b 3 − z 2 | ( b 3 − z 2 ) � � � y 3 ≥ y 2 − φ 23 | b 3 − z 2 | ( b 3 − z 2 ) � � � b ∈ R 3 y 0 ≤ y 1 + φ 01 | b 1 + b 2 + b 3 − z 1 | ( b 1 + b 2 + b 3 − z 1 ) � M = ∃ z : . (13) + � � y 0 ≤ y 2 + φ 01 | b 1 + b 2 + b 3 − z 1 | ( b 1 + b 2 + b 3 − z 1 ) � � � � + φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � y 0 ≤ y 3 + φ 01 | b 1 + b 2 + b 3 − z 1 | ( b 1 + b 2 + b 3 − z 1 ) � � � � + φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � + φ 23 | b 3 − z 2 | ( b 3 − z 2 ) � � � y 0 ≥ y 1 + φ 01 | b 1 + b 2 + b 3 − z 1 | ( b 1 + b 2 + b 3 − z 1 ) � � � y 0 ≥ y 2 + φ 01 | b 1 + b 2 + b 3 − z 1 | ( b 1 + b 2 + b 3 − z 1 ) � � � + φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � y 0 ≥ y 2 + φ 01 | b 1 + b 2 + b 3 − z 1 | ( b 1 + b 2 + b 3 − z 1 ) � � � + φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � y 0 ≥ y 2 + φ 01 | b 1 + b 2 + b 3 − z 1 | ( b 1 + b 2 + b 3 − z 1 ) � � � � + φ 12 | b 2 + b 3 − z 1 − z 2 | ( b 2 + b 3 − z 1 − z 2 ) � � + φ 23 | b 3 − z 2 | ( b 3 − z 2 )
M i for a Single Sample Point | z 1 | z 1 = | r + 3 . 5 − z 1 | ( r + 3 . 5 − z 1 ) � � � + | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) � � � | z 2 | z 2 = | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) + | 0 . 5 − z 2 | (0 . 5 − z 2 ) � � � y 1 ≥ y 2 + | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) � � � y 1 ≥ y 3 + | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) � � � + | 0 . 5 − z 2 | (0 . 5 − z 2 ) � � � y 2 ≥ y 1 − | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) � � � y 2 ≥ y 3 + | 0 . 5 − z 2 | (0 . 5 − z 2 ) � � � � y 3 ≥ y 1 − | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) � � � − | 0 . 5 − z 2 | (0 . 5 − z 2 ) � � � y 3 ≥ y 2 − | 0 . 5 − z 2 | (0 . 5 − z 2 ) M 1 = r ≥ 0 ∃ z : . (14) � � � y 0 ≤ y 1 + | r + 3 . 5 − z 1 | ( r + 3 . 5 − z 1 ) � � � � y 0 ≤ y 2 + | r + 3 . 5 − z 1 | ( r + 3 . 5 − z 1 ) � � � + | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) � � � y 0 ≤ y 3 + | r + 3 . 5 − z 1 | ( r + 3 . 5 − z 1 ) � � � + | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) + | 0 . 5 − z 2 | (0 . 5 − z 2 ) � � � y 0 ≥ y 1 + | r + 3 . 5 − z 1 | ( r + 3 . 5 − z 1 ) � � � y 0 ≥ y 2 + | r + 3 . 5 − z 1 | ( r + 3 . 5 − z 1 ) � � � + | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) � � � y 0 ≥ y 2 + | r + 3 . 5 − z 1 | ( r + 3 . 5 − z 1 ) � � � + | 2 . 5 − z 1 − z 2 | (2 . 5 − z 1 − z 2 ) + | 0 . 5 − z 2 | (0 . 5 − z 2 ) �
Parametric Solution of F ( b , q N ) = 0 � � � � � � � � � Φ Nl · | q Nl | q Nl = Φ j b i − b i − q Nh q Nh � � � � j : e j ∈ E ( C l ) i ∈ I j h : C h ∈ H j i ∈ I j h : C h ∈ H j l = 1 , . . . , L , ◮ Case distinction of absolute values identifies validity regions for quadratic multivariate polynomials. ◮ In every region, reduced Gr¨ obner basis with lexicographic order yields triangular representation, at best, Shape Lemma applies. ◮ There is an extension of Buchberger’s Algorithm addressing parametric polynomial equations and computing a Comprehensive Gr¨ obner Basis . ◮ The idea is to handle parameters as additional variables and accompany the Buchberger iterations by a case distinction to separate parameter settings “leading to unwanted eliminations”. ◮ Successful experiments with package SINGULAR for instances with four pairwise interwoven cycles.
Three Cycles Let us now consider a network with three interconnected cycles given by the following node-arc incidence matrix: − 1 0 0 − 1 0 − 1 1 0 0 − 1 0 − 1 1 − 1 0 0 − 1 0 A + = = − 1 0 1 1 0 0 A B A N 0 0 1 0 1 1 b 1 1 q 01 q 13 q 12 0 3 q 02 q 23 b 0 2 b 3 b 2 q 03
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