The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Fast direct solvers for elliptic partial differential equations on locally-perturbed geometries Yabin Zhang (Joint work with Adrianna Gillman ) 1
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Definition of “fast” A numerical linear algebraic method is fast if its execution time scales asymptotically less than the cost of classic linear algebra techniques. 2
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Definition of “fast” A numerical linear algebraic method is fast if its execution time scales asymptotically less than the cost of classic linear algebra techniques. What is a “direct solver”? Given a pre-set tolerance ǫ and a linear system A A Ax = b , a direct A − 1 − T solver constructs an operator T T T so that � A A T T � ≤ ǫ . 2
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Definition of “fast” A numerical linear algebraic method is fast if its execution time scales asymptotically less than the cost of classic linear algebra techniques. What is a “direct solver”? Given a pre-set tolerance ǫ and a linear system A A Ax = b , a direct A − 1 − T solver constructs an operator T T T so that � A A T T � ≤ ǫ . For a direct solver to be fast, the cost of constructing T T T and applying T T T to a vector needs to be low. 2
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Motivation https://altairhyperworks.com/product/FEKO/Applications-Antenna-Placement 3
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Motivation https://altairhyperworks.com/product/FEKO/Applications-Antenna-Placement G Marple, A. Barnett, A. Gillman, and A. Veerapaneni, A Fast Algorithm for Simulating Multiphase Flows Through Periodic Geometries of Arbitrary Shape. 3
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Model problem ν x Consider the Laplace BVP x Ω − ∆ u ( x ) = 0 for x ∈ Ω , Γ u ( x ) = f ( x ) for x ∈ Γ = ∂ Ω . 4
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Model problem ν x Consider the Laplace BVP x Ω − ∆ u ( x ) = 0 for x ∈ Ω , Γ u ( x ) = f ( x ) for x ∈ Γ = ∂ Ω . The solution to the BVP can be represented as a double-layer potential � ∂G ( x, y ) u ( x ) = σ ( y ) dl ( y ) , x ∈ Ω ∂ν y Γ where σ ( x ) is an unknown boundary charge density and � � G ( x, y ) = − 1 1 2 π log is the Green’s function. | x − y | 4
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Model problem ν x Consider the Laplace BVP x Ω − ∆ u ( x ) = 0 for x ∈ Ω , Γ u ( x ) = f ( x ) for x ∈ Γ = ∂ Ω . The solution to the BVP can be represented as a double-layer potential � ∂G ( x, y ) u ( x ) = σ ( y ) dl ( y ) , x ∈ Ω ∂ν y Γ where σ ( x ) is an unknown boundary charge density and � � G ( x, y ) = − 1 1 2 π log is the Green’s function. | x − y | Enforcing the boundary condition yields the boundary integral equation (BIE) � − 1 ∂G ( x, y ) 2 σ ( x ) + σ ( y ) dl ( y ) = f ( x ) , for x ∈ Γ . ∂ν y Γ 4
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry The discretized linear system σ = ( σ ( x 1 ) , . . . , σ ( x n )) T , � f = ( f ( x 1 ) , . . . , f ( x n )) T , I Let � I I be the D ij = ∂G ( x i ,x j ) identity matrix, and D D D be a matrix with entries D D w j , ∂ν xj then the discretized BIE can be written as σ = ( − 1 σ = � A I I + D D D ) � A A� 2 I f 5
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry The discretized linear system σ = ( σ ( x 1 ) , . . . , σ ( x n )) T , � f = ( f ( x 1 ) , . . . , f ( x n )) T , I Let � I I be the D ij = ∂G ( x i ,x j ) identity matrix, and D D D be a matrix with entries D D w j , ∂ν xj then the discretized BIE can be written as σ = ( − 1 σ = � A I I + D D D ) � A A� 2 I f A is called the coefficient matrix . A A 5
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry The discretized linear system σ = ( σ ( x 1 ) , . . . , σ ( x n )) T , � f = ( f ( x 1 ) , . . . , f ( x n )) T , I Let � I I be the D ij = ∂G ( x i ,x j ) identity matrix, and D D D be a matrix with entries D D w j , ∂ν xj then the discretized BIE can be written as σ = ( − 1 σ = � A I I + D D D ) � A A� 2 I f A is called the coefficient matrix . A A Properties of the coefficient matrix A A : A ◮ A A A is a dense matrix. ◮ The size of A A A depends on the number of discretization points N on the boundary Γ . ◮ A A A is data-sparse. ◮ Particularly, the off-diagonal blocks of A A A are low-rank. 5
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Data-sparse property of the coefficient matrix S ∈ R m × n is ǫ -rank if it has exactly k = k ( ǫ ) Definition: A matrix S S singular values that are greater than ǫ . S S S is called a low-rank matrix if k << m . 6
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Data-sparse property of the coefficient matrix S ∈ R m × n is ǫ -rank if it has exactly k = k ( ǫ ) Definition: A matrix S S singular values that are greater than ǫ . S S S is called a low-rank matrix if k << m . Let’s verify that the off-diagonal blocks of the coefficient matrix A A A are indeed low-rank by an example: 6
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Data-sparse property of the coefficient matrix S ∈ R m × n is ǫ -rank if it has exactly k = k ( ǫ ) Definition: A matrix S S singular values that are greater than ǫ . S S S is called a low-rank matrix if k << m . Let’s verify that the off-diagonal blocks of the coefficient matrix A A A are indeed low-rank by an example: Γ c Γ τ τ Boundary: Γ = Γ τ ∪ Γ c τ τ ) ∈ R 100 × 900 A (Γ τ , Γ c Matrix block: A A 6
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Data-sparse property of the coefficient matrix S ∈ R m × n is ǫ -rank if it has exactly k = k ( ǫ ) Definition: A matrix S S singular values that are greater than ǫ . S S S is called a low-rank matrix if k << m . Let’s verify that the off-diagonal blocks of the coefficient matrix A A A are indeed low-rank by an example: 10 0 10 -5 ǫ = 10 − 10 , k = 10 10 -10 Γ c Γ τ τ ǫ = 10 − 16 , k = 19 10 -15 Boundary: Γ = Γ τ ∪ Γ c 10 -20 τ 0 20 40 60 80 100 τ ) ∈ R 100 × 900 A (Γ τ , Γ c A (Γ τ , Γ c Matrix block: A A The singular values of A A τ ) 6
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry Block-separable matrix A matrix A A A of dimension ( np ) × ( np ) is block-separable if it consists p × p blocks each of size n × n : e.g. for p = 4 , D D D 11 A A 12 A A A A 13 A A A 14 A D A A A A 21 D D 22 A A 23 A A 24 A A = A . A A A 31 A A A 32 D D D 33 A A A 34 A A A D A A 41 A A 42 A A 43 D 44 D And each of the off-diagonal block admits the factorization ˜ V ∗ A = U A V A ij A U U i A A ij V j n × n n × k k × k k × n where the rank k is significantly smaller than the block size n . A. Gillman, P. Young, and P.G. Martinsson, A direct solver with O(N) complexity for integral equations on one-dimensional domains 7
The Basics Boundary Integral Equation The HBS Representation and Inversion Locally Perturbed Geometry U 1 ˜ U 1 ˜ U 1 ˜ D D D 11 U U A A 12 V A V V ∗ U U A A 13 V A V V ∗ U U A A 14 V A V ∗ V 2 3 4 U 2 ˜ U 2 ˜ U 2 ˜ V ∗ V ∗ V ∗ U U A A A 21 V V D D 22 D U U A A A 23 V V U U A A A 24 V V Then we have A 1 3 4 A = A , U 3 ˜ U 3 ˜ U 3 ˜ V ∗ V ∗ V ∗ U U A 31 V A A V U U A 32 V A A V D D D 33 U U A A A 34 V V 1 2 4 U 4 ˜ U 4 ˜ U 4 ˜ U A V V ∗ U A V V ∗ U A V ∗ V D U A 41 V A U A 42 V A U A A 43 V D D 44 1 2 3 and it can be factored as A A A = ˜ ˜ ˜ 0 0 0 A A A 12 A A A 13 A 14 A A U U U 1 V V ∗ V 1 ˜ ˜ ˜ U U U 2 A 21 A A 0 0 0 A A A 23 A A A 24 V ∗ V V 2 + ˜ ˜ ˜ U U U 3 0 V V ∗ V A A A 31 A A 32 A 0 0 A A A 34 3 ˜ ˜ ˜ U 4 U U V V ∗ V 0 A 41 A A A A 42 A A A A 43 0 0 4 � �� � � �� � � �� � = U U = V V U V ∗ = ˜ A A A D D D 11 D D D 22 , D D D 33 D D D 44 � �� � = D D D 8
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