Numerical-Asymptotic Integral Equation Methods for High Frequency Scattering Simon Chandler-Wilde University of Reading, UK www.reading.ac.uk/~sms03snc With: Steve Langdon , Timo Betcke , Ivan Graham , Marko Lindner , Markus Melenk , Peter Monk , Valery Smyshlyaev Postdocs: Dave Hewett , Joel Philips , Euan Spence Plus 8 PhDs Funding: • EPSRC project(s) across Bath/Reading/UCL with BAE Systems , Institute of Cancer Research , Met Office , Schlumberger Cambridge Research as project partners. • 3 NERC/EPSRC CASE Studentships at Bath & Reading • EPSRC Fellowships for Euan, Timo Linz, November 2011 1
Aim of Our High Frequency Wave Projects Develop numerical methods which use oscillatory basis functions to represent solutions with hugely reduced numbers of degrees of freedom. Domain-based formulations (Plane wave DG, UWVF. etc.). Timo Betcke, Joel Phillips, Ivan Graham, Steve Langdon, SNCW + 5 PhDs at Bath/Reading/UCL. BEM-based methods. Timo Betcke, SNCW, Ivan Graham, Dave Hewett, Tatiana Kim, Steve Langdon, Euan Spence, Ashley Twigger. 2
Aim of Our High Frequency Wave Projects Develop numerical methods which use oscillatory basis functions to represent solutions with hugely reduced numbers of degrees of freedom. Domain-based formulations (Plane wave DG, UWVF. etc.). Timo Betcke, Joel Phillips, Ivan Graham, Steve Langdon, SNCW + 5 PhDs at Bath/Reading/UCL. BEM-based methods. Timo Betcke, SNCW, Ivan Graham, Dave Hewett, Tatiana Kim, Steve Langdon, Euan Spence, Ashley Twigger. 3
A Simple Generic Time Harmonic Scattering Problem � � ❅ � ∆ u + k 2 u = 0 � � � � ❅ � � � � ❅ � � � Ω + ❅ ❘ u i , incident wave ❅ � u = 0 Γ Obstacle k = 2 π λ > 0 is the wave number and λ the corresponding wavelength . 4
Typical Multiscale Solution: many length scales as k → ∞ : k − 1 , k − 1 / 3 , k − 1 / 2 , ... 5
The Computational Challenge Can we achieve, with integral equation methods, ‘prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency’ to quote from the title of Bruno, Geuzaine, Monro, and Reitich, Phil Trans R Soc Lond A (2004) . 6
The Computational Challenge Can we achieve, with integral equation methods, ‘prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency’ to quote from the title of Bruno, Geuzaine, Monro, and Reitich, Phil Trans R Soc Lond A (2004) Answer: 1. YES for some classes of 2D and 3D problems. 2. For more general classes significant improvements possible and promising research area. 7
The Associated Mathematical Challenge ... PROVING EVERYTHING! • Best approximation results using novel approximation spaces • Stability • Convergence • Error estimates for fully discrete schemes . 8
The Associated Mathematical Challenge ... PROVING EVERYTHING! • Best approximation results using novel approximation spaces • Stability • Convergence • Error estimates for fully discrete schemes THE (large) NOVELTY IS THAT WE NEED TO DO THIS IN THE LIMIT AS k → ∞ with N approximately fixed (not the classical N → ∞ with k fixed). 9
I will only scrape the surface today. For more details: • Read survey article by C-W & Graham (and related articles by Huybrechs & Olver, Monk, Motamed & Runborg) in Highly Oscillatory Problems , CUP, 2009. • C-W, Graham, Langdon, Spence, to appear in Acta Numerica 2012. 10
The Scattering Problem � � ❅ � ∆ u + k 2 u = 0 � � � ❅ � � � � � ❅ � � � Ω + ❅ ❅ ❘ u i , incident wave � u = 0 Γ Lipschitz obstacle 11
� � ❅ � ∆ u + k 2 u = 0 � � � ❅ � � � � � ❅ � � � Ω + ❅ ❅ ❘ u i , incident wave � u = 0 Γ Lipschitz obstacle Green’s representation theorem: G ( x, y ) ∂u � u ( x ) = u i ( x ) − x ∈ Ω + , ∂n ( y ) ds ( y ) , Γ where e i k | x − y | := 1 4 H (1) G ( x, y ) := i 0 ( k | x − y | ) (2D), | x − y | (3D). 4 π 12
� � ❅ � ∆ u + k 2 u = 0 � � � � ❅ � � � � ❅ � � � Ω + ❅ ❅ ❘ u i , incident wave � u = 0 Γ Lipschitz obstacle Taking a linear combination of Dirichlet and Neumann traces of the previous equation, we get the boundary integral equation � ∂u 1 ∂u � ∂G ( x, y ) � ∂n ( x ) + + i ηG ( x, y ) ∂n ( y ) ds ( y ) = f ( x ) , x ∈ Γ , 2 ∂n ( x ) Γ where f ( x ) := ∂u i ∂n ( x ) + i ηu i ( x ) . 13
� � ❅ � ∆ u + k 2 u = 0 � � � ❅ � � � � � ❅ � � � Ω + ❅ ❘ u i , incident wave ❅ � u = 0 Γ Lipschitz obstacle � ∂u 1 ∂u � ∂G ( x, y ) � ∂n ( x ) + + i ηG ( x, y ) ∂n ( y ) ds ( y ) = f ( x ) , x ∈ Γ . 2 ∂n ( x ) Γ Theorem (Mitrea 1996, C-W & Langdon 2007) If η ∈ R , η � = 0 , then this integral equation is uniquely solvable in L 2 (Γ) . 14
� ∂u � ∂G ( x, y ) 1 ∂u � ∂n ( x ) + + i ηG ( x, y ) ∂n ( y ) ds ( y ) = f ( x ) , x ∈ Γ . 2 ∂n ( x ) Γ in operator form A ∂u ∂n = f. Theorem If η ∈ R , η � = 0 , then this integral equation is uniquely solvable in L 2 (Γ) . In fact (C-W & Monk 2008, C-W, Graham, Langdon, Lindner 2009), if scatterer is starlike and η = 1 + k then (in 3D) � A − 1 � ≤ C, � A � ≤ C (1 + k ) , cond A ≤ C (1 + k ) . 15
The Subtlety of Behaviour of � A � and � A − 1 � ∼ k 1 / 3 , ∼ 1 ∼ k 1 / 2 , ∼ 1 ∼ k 1 / 2 m , ∼ k 7 / 5 ∼ k 1 / 2 m , ∼ e γk m m Details : see C-W, Graham et al (2009), Betcke, C-W, Graham et al (2011). 16
Mechanism for Exponential Growth: Exponential Localization of Eigenmodes 17
� ∂u � ∂G ( x, y ) 1 ∂u � ∂n ( x ) + + i ηG ( x, y ) ∂n ( y ) ds ( y ) = f ( x ) , x ∈ Γ . 2 ∂n ( x ) Γ Conventional BEM: Approximate ∂u/∂n by a piecewise polynomial, i.e. N ∂u � ∂n ( x ) ≈ a j b j ( x ) , j =1 where b 1 ( x ) , . . . , b N ( x ) are the piecewise polynomial basis functions. . 18
� ∂u 1 ∂u � ∂G ( x, y ) � ∂n ( x ) + + i ηG ( x, y ) ∂n ( y ) ds ( y ) = f ( x ) , x ∈ Γ . 2 ∂n ( x ) Γ Conventional BEM: Approximate ∂u/∂n by a piecewise polynomial, i.e. N ∂u � ∂n ( x ) ≈ a j b j ( x ) , j =1 where b 1 ( x ) , . . . , b N ( x ) are the piecewise polynomial basis functions. Applying a Galerkin method or a collocation method we get a linear system to solve with N degrees of freedom, namely the unknown values of a 1 , . . . , a N . 19
� ∂u � ∂G ( x, y ) 1 ∂u � ∂n ( x ) + + i ηG ( x, y ) ∂n ( y ) ds ( y ) = f ( x ) , x ∈ Γ . 2 ∂n ( x ) Γ Conventional BEM: Apply a Galerkin method, approximating ∂u/∂n by a piecewise polynomial of degree p , leading to a linear system to solve with N degrees of freedom. Problem: N of order of k d − 1 and cost is ... close to O ( N ) if a fast multipole method is used. This is fantastic , but still infeasible as k → ∞ . . 20
Alternative: Reduce N by using new oscillatory basis functions which can represent the solution well. Specifically, let’s try N e M ∂u � � ∂n ( x ) ≈ a ij exp(i kg i ( x )) b ij ( x ) , i =1 j =1 with a ij ∈ C the unknown coefficients, g 1 ( x ) , . . . , g M ( x ) known phase functions , b ij ( x ) conventional BEM basis functions . . 21
Alternative: Reduce N by using new oscillatory basis functions which can represent the solution well. Specifically, let’s try M N e ∂u � � ∂n ( x ) ≈ a ij exp(i kg i ( x )) b ij ( x ) , i =1 j =1 with a ij ∈ C the unknown coefficients, g 1 ( x ) , . . . , g M ( x ) known phase functions , b ij ( x ) conventional BEM basis functions . Moreover, let’s have #dof N = MN e much less than conventional BEM, ideally N = O (1) as k → ∞ , the ‘high frequency O (1) algorithm’ holy grail. . 22
Alternative: Reduce N by using new oscillatory basis functions which can represent the solution well. Specifically, let’s try N e M ∂u � � ∂n ( x ) ≈ a ij exp(i kg i ( x )) b ij ( x ) , i =1 j =1 with a ij ∈ C the unknown coefficients, g 1 ( x ) , . . . , g M ( x ) known phase functions , b ij ( x ) conventional BEM basis functions . The Plan: let’s have #dof N = MN e which is N = O (1) as k → ∞ , and then we will achieve the ‘high frequency O (1) CPU time algorithm’ holy grail. . 23
Alternative: Reduce N by using new oscillatory basis functions which can represent the solution well. Specifically, let’s try M N e ∂u � � ∂n ( x ) ≈ a ij exp(i kg i ( x )) b ij ( x ) , i =1 j =1 with a ij ∈ C the unknown coefficients, g 1 ( x ) , . . . , g M ( x ) known phase functions , b ij ( x ) conventional BEM basis functions . The Plan: let’s have #dof N = MN e which is N = O (1) as k → ∞ , and then we will achieve the ‘high frequency O (1) CPU time algorithm’ holy grail. No! Unfortunately, N = O (1) �⇒ CPU time = O (1) . 24
The Snag: our N 2 matrix entries are highly oscillatory integrals When we use the Galerkin method , typical matrix entries in 3D are 1 � � 4 π | x − y | exp[i k ( | x − y | + y · d i − x · d m )] b ij ( y ) b mn ( x ) ds ( y ) ds ( x ) . Γ ij Γ mn Each entry is a 4-dimensional, increasingly oscillatory integral as k → ∞ . . 25
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