Differential Algebra (DA) based Fast Multipole Method (FMM) He - PowerPoint PPT Presentation

Differential Algebra (DA) based Fast Multipole Method (FMM) He Zhang, Martin Berz, Kyoko Makino Department of Physics and Astronomy Michigan State University June 27, 2010 He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

Space charge effect Space charge effect : the fields of the charged particles in a bunch on each other affect the motion of themselves. Pair-to-pair method, time consuming, O ( N 2 ) Tree code, O ( N log N ). (The field of the electrons far away from the observer point, can be represented by the field of a multipole.) Fast multipole method, much faster, O ( N ). (Multipole expansion and local expansion, recursive progress.) He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

Space charge effect � L � � M M O ( N log N ) O ( N ) He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

History of FMM 1 J.Barnes, and P.Hut. A Hierarchical O ( N log N ) Force-Calculation Algorithm. Nature Vol. 324, pp. 446-449, Dec. 4, 1986 2 L.Greengard, and V.Rokhlin. A Fast Algorithm for Particle Simulations. J. Comput. Phys. 73, pp. 325-348, 1987 3 J.Carrier, L.Greengard, and V.Rokhlin. A Fast Adaptive Multipole Algorithm for Particle Simulations. SIAM J. Sci. Stat. Comput. Vol. 9, No. 4, pp. 669-686, July 1988. 4 R.Beatson and L.Greengard. A Short Course on Fast Multipole Methods. Numerical Methematics and Scientific Computation, Wavelets, Multilevel Methods and Elliptic PDEs. Oxford University Press, pp. 1-37, 1997 5 B.Shanker and H.Huang. Accelerated Cartesian Expansions - A Fast Method for Computing of Potentials of the Form R − ν for All Real v. J. Comput. Phys. 226, pp. 732-753, 2007 He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

Some concepts Cut box, near region, and far region. First Level Second Level Near region (neighbors) Far region He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

Some concepts Interaction list and how it works. Interaction list Interaction list Already calculated He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

Brief introduction of FMM The process of FMM can be described as follows. Select the level of boxes according to the accuracy needed. From the finest level to the coarest level, calculate the multipole expansion of the charges inside each box. For each box in each level, convert the multipole expansions of the boxes in its interaction list into its local expansion. From the coarest to the finest level, translate the local expansion of each box into its child boxes and add it to the local expansion of the child box In the finest level, in each box calculate the potential or field by the local expansion on each particle inside. The potentials or fields of the particles inside the box or in its near region are calculated directly. He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame Automatic Taylor expansion of a function f ( x + δ x ) = f ( x ) + f ′ ( x ) δ x + 1 2! f ′′ ( x ) δ x 2 + 1 3! f ′′′ ( x ) δ x 3 + ... In Cosy, f ( x + da (1)) = f ( x )+ f ′ ( x ) da (1)+ 1 2! f ′′ ( x ) da (1) 2 + 1 3! f ′′′ ( x ) da (1) 3 + ... Composition of two maps G ( x ) = G ( F ) ◦ F ( x ) , or G ( x ) = G ( F ( x )) In COSY, it can be done by the command POLVAL L P NP A NA R NR He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame Multipole expansion from charges � � R ( x, y, z ) R ( x, y, z ) � R 1 � R ( � R ( � � φ � R i ) = φ � M ) R 2 M S (0 , 0 , 0) S (0 , 0 , 0) � R i He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame Multipole expansion from charges (for boxes in the finest level) N Q i � φ c 2 m = ( x i − x ) 2 + ( y i − y ) 2 + ( z i − z ) 2 � i =1 x 2 + y 2 + z 2 � Q i / = � 1 + x 2 i + y 2 i + z 2 2 x i x 2 y i y 2 z i z i x 2 + y 2 + z 2 − x 2 + y 2 + z 2 − x 2 + y 2 + z 2 − x 2 + y 2 + z 2 Q i · d 1 = , � 1 + ( x 2 i + y 2 i + z 2 i ) d 2 1 − 2 x i d 2 − 2 y i d 3 − 2 z i d 4 with x 2 + y 2 + z 2 = 1 1 x 2 + y 2 + z 2 = x x d 1 = r , d 2 = r 2 , � x 2 + y 2 + z 2 = y y x 2 + y 2 + z 2 = z z d 3 = r 2 , d 4 = r 2 , He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame Multipole expansion in the higher level boxes � � � M M M � � M M He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame Translation of a Multipole Expansion � � R ( x ′ , y ′ , z ′ ) R ( x, y, z ) � M ′ S ′ ( x ′ s , y ′ s , z ′ s ) S ′ (0 , 0 , 0) � R ( � R ( � M φ � M ) = φ � M ′ ) S (0 , 0 , 0) S ( − x ′ s , − y ′ s , − z ′ s ) He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame DA variables in S ′ frame. x ′ 2 + y ′ 2 + z ′ 2 = x ′ x ′ x ′ 2 + y ′ 2 + z ′ 2 = 1 1 d ′ d ′ 1 = r ′ , 2 = r ′ 2 , � x ′ 2 + y ′ 2 + z ′ 2 = y ′ y ′ x ′ 2 + y ′ 2 + z ′ 2 = z ′ z ′ d ′ d ′ 3 = r ′ 2 , 4 = r ′ 2 , Relation between the new and old DA variables ( M 1 ). d ′ 1 d 1 = � s ) d ′ 2 1 + ( x ′ 2 s + y ′ 2 s + z ′ 2 1 + 2 x ′ s d ′ 2 + 2 y ′ s d ′ 3 + 2 z s d ′ 4 � d ′ � d ′ � d ′ � � � 2 · d 2 3 · d 2 4 · d 2 + x ′ + y ′ + z ′ d 2 = 1 , d 3 = 1 , d 4 = 1 . s s s d ′ 2 d ′ 2 d ′ 2 1 1 1 Potential in S ′ frame is φ m 2 m = φ c 2 m ◦ M 1 . He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame Conversion of a Multipole Expansion (in the interaction list) into a Local Expansion � � R ( x, y, z ) R ( x ′ , y ′ , z ′ ) O ( x o , y o , z o ) O (0 , 0 , 0) � L R ( � R ( � M ) = φ � L ) φ � � M S (0 , 0 , 0) S ( − x o , − y o , − z o ) He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame DA variables in the observer frame O . d ′ 1 = x ′ , d ′ 2 = y ′ , d ′ 3 = z ′ . Relation between the DA variables in the source frame S and the observer frame O . ( M 2 ) 1 d 1 = 1 ) 2 + ( y o + d ′ 2 ) 2 + ( z o + d ′ � ( x o + d ′ 3 ) 2 x o + d ′ 1 d 2 = 3 ) 2 , 1 ) 2 + ( y o + d ′ 2 ) 2 + ( z o + d ′ ( x o + d ′ y o + d ′ 2 d 3 = 3 ) 2 , 1 ) 2 + ( y o + d ′ 2 ) 2 + ( z o + d ′ ( x o + d ′ z o + d ′ 3 d 4 = 3 ) 2 . 1 ) 2 + ( y o + d ′ 2 ) 2 + ( z o + d ′ ( x o + d ′ The potential in the observer frame O is φ m 2 l = φ c 2 m ◦ M 2 . He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame Local Expansion inherited from the parent box. � � � L L L � � L L He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame Translation of a Local Expansion � � R ( x ′ , y ′ , z ′ ) R ( x, y, z ) � L ′ L O ′ ( x ′ o , y ′ o , z ′ O ′ (0 , 0 , 0) o ) � R ( � R ( � O ( − x ′ o , − y ′ o , − z ′ o ) O (0 , 0 , 0) φ � L ) = φ � L ′ ) He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame DA variables in the new observer frame O ′ . d ′ 1 = x ′ , d ′ 2 = y ′ , d ′ 3 = z ′ . Relation between the DA variables in the old and new frame. ( M 3 ) d 1 = x ′ o + d ′ 1 , d 2 = y ′ o + d ′ 2 , d 3 = z ′ o + d ′ 3 . The potential in the observer frame O ′ is φ l 2 l = φ m 2 l ◦ M 3 . He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)

How we do it in the Differential Algebra (DA) frame Field is easy to calculate. In the local expansion, the potential is represented as a polynomial of the local coordinates. The calculate the field, one just need to take derivative of the the respect coordinate. This is also important in the high order map method. He Zhang, Martin Berz, Kyoko Makino Differential Algebra (DA) based Fast Multipole Method (FMM)