Symbolic local Fourier Analysis Veronika Pillwein RISC Challenges - PowerPoint PPT Presentation

Symbolic local Fourier Analysis Veronika Pillwein RISC Challenges in 21st Century Experimental Mathematical Computation, July 21–25, 2014, ICERM, Brown University

Symbolic local Fourier Analysis (and some Special Functions Inequalities) Veronika Pillwein RISC Challenges in 21st Century Experimental Mathematical Computation, July 21–25, 2014, ICERM, Brown University

Cylindrical Algebraic Decomposition

Input: Polynomial Formulas ◮ Everything that can be formed from variables, rational (or real algebraic) numbers, + , − , · , /, = , � = , ≤ , <, ≥ , >, ¬ , ∧ , ∨ , ⇒ , ⇔ , True , False according to the usual syntactic rules.

Input: Polynomial Formulas ◮ Everything that can be formed from variables, rational (or real algebraic) numbers, + , − , · , /, = , � = , ≤ , <, ≥ , >, ¬ , ∧ , ∨ , ⇒ , ⇔ , True , False according to the usual syntactic rules. ◮ Examples: ◮ x 2 − ( x − 1)( y − 1) > 0 ∧ x ≥ y 2 ≥ 1 ⇒ y 2 − 3 x < 0 √ ◮ xz 2 − 2 y + z 3 > 1 ∧ x > y > z > 1 ⇒ xyz > 1 2

Input: Polynomial Formulas ◮ Everything that can be formed from variables, rational (or real algebraic) numbers, + , − , · , /, = , � = , ≤ , <, ≥ , >, ¬ , ∧ , ∨ , ⇒ , ⇔ , True , False according to the usual syntactic rules. ◮ Examples: ◮ x 2 − ( x − 1)( y − 1) > 0 ∧ x ≥ y 2 ≥ 1 ⇒ y 2 − 3 x < 0 √ ◮ xz 2 − 2 y + z 3 > 1 ∧ x > y > z > 1 ⇒ xyz > 1 2 ◮ Counterexamples: ◮ x cos( x ) − y sin( y ) > π 4 ⇒ cos( x ) sin( x ) < sin 2 ( x ) ◮ x exp( y ) − y 2 < xy ∨ log( y ) < x ⇒ x 2 + y 2 < 4

❘ ❘ ❘ ❘ Cylindrical Algebraic Decomposition (CAD)

Cylindrical Algebraic Decomposition (CAD) CAD is an algorithm which can ... ◮ decide whether a given polynomial formula is consistent ◮ decide whether a given polynomial formula is universal ◮ decide whether a given polynomial formula implies another one ◮ determine a certificate point for a given satisfiable formula ◮ determine the semialgebraic set of points ( x 1 , . . . , x n − 1 ) ∈ ❘ n − 1 such that there exists a number x n ∈ ❘ where a given formula is true at ( x 1 , . . . , x n − 1 , x n ) ◮ determine the semialgebraic set of all points ( x 1 , . . . , x n − 1 ) ∈ ❘ n − 1 such that for all numbers x n ∈ ❘ a given formula is true at ( x 1 . . . , x n − 1 , x n )

Quantifier Elimination In particular, given any polynomial formula Φ involving quantifiers ( ∀ , ∃ ) CAD can compute a polynomial formula Ψ without quantifiers that is equivalent (over ❘ ) to Φ.

Quantifier Elimination In particular, given any polynomial formula Φ involving quantifiers ( ∀ , ∃ ) CAD can compute a polynomial formula Ψ without quantifiers that is equivalent (over ❘ ) to Φ. Examples: y > − 1 ∧ B ≥ 2 − y ∀ x : 0 < x < 1 ⇒ x 2 − y + 1 ≤ B ( xy + 1) CAD − → y + 1

Quantifier Elimination In particular, given any polynomial formula Φ involving quantifiers ( ∀ , ∃ ) CAD can compute a polynomial formula Ψ without quantifiers that is equivalent (over ❘ ) to Φ. Examples: y > − 1 ∧ B ≥ 2 − y ∀ x : 0 < x < 1 ⇒ x 2 − y + 1 ≤ B ( xy + 1) CAD − → y + 1 ∀ x ∃ y : 0 < x < 1 ∧ 0 < y < x 2 ⇒ x 2 − y + 1 ≤ B ( xy + 1) CAD − → B ≥ 1

Quantifier Elimination In particular, given any polynomial formula Φ involving quantifiers ( ∀ , ∃ ) CAD can compute a polynomial formula Ψ without quantifiers that is equivalent (over ❘ ) to Φ. Examples: y > − 1 ∧ B ≥ 2 − y ∀ x : 0 < x < 1 ⇒ x 2 − y + 1 ≤ B ( xy + 1) CAD − → y + 1 ∀ x ∃ y : 0 < x < 1 ∧ 0 < y < x 2 ⇒ x 2 − y + 1 ≤ B ( xy + 1) CAD − → B ≥ 1 ∀ x ∃ y : x 2 + y 2 − 4 > 0 ⇔ ( x − 1)( y − 1) − 1 > 0 CAD − → True

Quantifier Elimination In particular, given any polynomial formula Φ involving quantifiers ( ∀ , ∃ ) CAD can compute a polynomial formula Ψ without quantifiers that is equivalent (over ❘ ) to Φ. Examples: y > − 1 ∧ B ≥ 2 − y ∀ x : 0 < x < 1 ⇒ x 2 − y + 1 ≤ B ( xy + 1) CAD − → y + 1 ∀ x ∃ y : 0 < x < 1 ∧ 0 < y < x 2 ⇒ x 2 − y + 1 ≤ B ( xy + 1) CAD − → B ≥ 1 ∀ x ∃ y : x 2 + y 2 − 4 > 0 ⇔ ( x − 1)( y − 1) − 1 > 0 CAD − → True Note: The execution of CAD may be computationally expensive!

Quantifier Elimination In particular, given any polynomial formula Φ involving quantifiers ( ∀ , ∃ ) CAD can compute a polynomial formula Ψ without quantifiers that is equivalent (over ❘ ) to Φ. Examples: y > − 1 ∧ B ≥ 2 − y ∀ x : 0 < x < 1 ⇒ x 2 − y + 1 ≤ B ( xy + 1) CAD − → y + 1 ∀ x ∃ y : 0 < x < 1 ∧ 0 < y < x 2 ⇒ x 2 − y + 1 ≤ B ( xy + 1) CAD − → B ≥ 1 ∀ x ∃ y : x 2 + y 2 − 4 > 0 ⇔ ( x − 1)( y − 1) − 1 > 0 CAD − → True Note: The execution of CAD may be computationally expensive! Implementations: Mathematica, Maple, QEPCAD, Redlog, etc.

Symbolic Local Fourier Analysis joint work with Stefan Takacs

Local Fourier analysis ◮ standard tool to analyze the convergence behaviour of a numerical method

Local Fourier analysis ◮ standard tool to analyze the convergence behaviour of a numerical method ◮ Model problem : optimization problem constrained to a partial differential equation Find a state y and a control u that minimize J ( y , u ) := 1 2 � y − y D � 2 2 � u � 2 L 2 (Ω) + α L 2 (Ω) , subject to − ∆ y = u in the domain Ω.

Local Fourier analysis ◮ standard tool to analyze the convergence behaviour of a numerical method ◮ Model problem : optimization problem constrained to a partial differential equation Find a state y and a control u that minimize J ( y , u ) := 1 2 � y − y D � 2 2 � u � 2 L 2 (Ω) + α L 2 (Ω) , subject to − ∆ y = u in the domain Ω. ◮ solution method: Finite Element Method (FEM) with a multigrid-solver (which ultimately means solving a large scale linear system)

Local Fourier analysis ◮ standard tool to analyze the convergence behaviour of a numerical method ◮ Model problem : optimization problem constrained to a partial differential equation Find a state y and a control u that minimize J ( y , u ) := 1 2 � y − y D � 2 2 � u � 2 L 2 (Ω) + α L 2 (Ω) , subject to − ∆ y = u in the domain Ω. ◮ solution method: Finite Element Method (FEM) with a multigrid-solver (which ultimately means solving a large scale linear system) ◮ robust with respect to parameters such as mesh-size and regularization parameters

Local Fourier Analysis ◮ Given: Iterative procedure of the form x ( k +1) = Ax ( k ) the convergence rate is related to the matrix norm of A which can be estimated by the spectral radius, i.e., the largest eigenvalue in absolute value

Local Fourier Analysis ◮ Given: Iterative procedure of the form x ( k +1) = Ax ( k ) the convergence rate is related to the matrix norm of A which can be estimated by the spectral radius, i.e., the largest eigenvalue in absolute value ◮ typically rates are obtained only by numerical interpolation

Local Fourier Analysis ◮ Given: Iterative procedure of the form x ( k +1) = Ax ( k ) the convergence rate is related to the matrix norm of A which can be estimated by the spectral radius, i.e., the largest eigenvalue in absolute value ◮ typically rates are obtained only by numerical interpolation ◮ symbolic local Fourier analysis: exact bounds

FEM and multigrid ◮ in FEM the given domain is subdivided in simple geometric objects (triangles, rectangles, tetrahedra, prisms, etc.)

FEM and multigrid ◮ in FEM the given domain is subdivided in simple geometric objects (triangles, rectangles, tetrahedra, prisms, etc.) ◮ Multigrid methods operate on two (or more) grids

FEM and multigrid ◮ in FEM the given domain is subdivided in simple geometric objects (triangles, rectangles, tetrahedra, prisms, etc.) ◮ Multigrid methods operate on two (or more) grids ◮ One step in a multigrid method consists of ◮ (pre)smoothing steps ↓ restriction ↓ ◮ coarse grid correction ↓ prolongation ↓ ◮ (post)smoothing steps

FEM and multigrid ◮ in FEM the given domain is subdivided in simple geometric objects (triangles, rectangles, tetrahedra, prisms, etc.) ◮ Multigrid methods operate on two (or more) grids ◮ One step in a multigrid method consists of ◮ (pre)smoothing steps ↓ restriction ↓ ◮ coarse grid correction ↓ prolongation ↓ ◮ (post)smoothing steps

FEM and multigrid ◮ in FEM the given domain is subdivided in simple geometric objects (triangles, rectangles, tetrahedra, prisms, etc.) ◮ Multigrid methods operate on two (or more) grids ◮ One step in a multigrid method consists of ◮ (pre)smoothing steps ↓ restriction ↓ ◮ coarse grid correction ↓ prolongation ↓ ◮ (post)smoothing steps