PTAH

the Parallel Tensor Analysis Hybrid

The Problems

1. Curvature Invariants

   Riemannian Geometry

   Spacetime is defined by a metric; ie.,

   the Kerr-Newman metric in D = 4:

   g _{0, 0} = (-A ² - e ² + 2 M r - r ² + A ² Sin[th] ²) /

   (r ² + A ² Cos[th] ²)

   g _{0, 3} = -A (- e ² + 2 M r) Sin[th] ² / (r ² + A ² Cos[th] ²)

   g _{1, 1} = (r ² + A ² Cos[th] ²) / (A ² + e ² - 2 M r + r ²)

   g _{2, 2} = r ² + A ² Cos[th] ²

   g _{3, 3} = ((A ² + r ²) ² - A ²(A ² + e ² - 2 M r + r ²) Sin[th] ²)

   Sin[th] ² / (r ² + A ² Cos[th] ²)

   
   
   or its special cases:

   • e = 0 ~ the Kerr metric
   • A = 0 ~ the Reissner-Nordstrom metric
   • e = 0 and A = 0 ~ the Schwarzschild metric

   ---

   Quantities of interest are the Christoffel Connection:

   G ^a _{b c} = g ^{a d} ( d _b g _{c d} + d _c g _{b d} - d _d g _{b c} ) / 2

   
   the Riemann Curvature:

   R _{a b c} ^d = d _b G ^d _{a c} - d _a G ^d _{b c} + G ^f _{a c} G ^d _{f b} - G ^f _{b c} G ^d _{f a}

   
   and its covariant derivative:

   D _a R _{b c d e} = d _a R _{b c d e} - G ^f _{a b} R _{f c d e} -

   G ^f _{a c} R _{b f d e} - G ^f _{a d} R _{b c f e} - G ^f _{a e} R _{b c d f}

   
   the Ricci tensor:

   R _{a b} = R _{a c b} ^c

   
   and the scalar curvature:

   R = g ^{a b} R _{a b}

   
   which are always zero for vacuum solutions of Einstein's Equations:

   R _{a b} - g _{a b} R / 2 = T _{a b} = 0

   ---

   Consider the Schwarzschild solution in { r , q , f , t } coordinates:

   g _{0, 0} = 2 M / r - 1

   g _{1, 1} = 1 / ( 1 - 2 M / r )

   g _{2, 2} = r ²

   g _{3, 3} = r ² Sin ( q ) ²

   ( the Schwarzschild solution is known for all D;

   Kerr and Kerr-Newman are only known in D = 4 )

   The curvature tensor, for example, depends greatly on the coordinate system:

   R a b c d	{ r , q , f , t }	{ r , q , f - 2 t , t }
   ------	----------	--------------
   1 3 1 4	0	2 M Sin ( q ) 2 / ( r - 2 M )
   1 4 1 4	- 2 M / r 3	2 M ( - 2 M + r + 2 r 3 Sin ( q ) 2 ) /
   		(( 2 M - r ) r 3 )
   2 3 2 4	0	- 4 M r Sin ( q ) 2
   2 4 2 4	M ( r - 2 M ) /	M ( - 2 M + r + 8 r 3 Sin ( q ) 2 ) /
   	r 2	r 2

   In general, a tensor transforms under the following rule:

   T ' _b' ( x ' ) = T _a ( x ' ) d x ^a ( x ' ) / d x ' ^{b '}

   We would like to compute quantities whose form does not depend on the coordinate system:

   scalars constructed from the curvature tensor

   ---

   Curvature Invariants:

   R ^. R = R ^{a b c d} R _{a b c d}

   used to identify curvature singularities

   
   
   *R ^. R =e ^{a b c d} R _{a b e f} R _{c d} ^{e f}

   in D = 4, ~ angular momentum of the spacetime

   (I. Ciufolini & J. A. Wheeler, "Gravitation and Inertia", 1995)

   
   
   D R ^. D R = D _a R _{b c d e} D ^a R ^{b c d e}

   for some metrics, ~ norm of the timelike Killing vector

   (R. G. Gass, F. P. Esposito, L. C. R. Wijewardhana & L. Witten, gr-qc/9808055)

   ---

   D D R ^. D D R ^. D D R ^. D D R =

   D _a D _b R _{c d e f} D _g D _h R _{i j k l} D ^a D ^b R ^{c j e l} D ^g D ^h R ^{i d k f}

   which does not vanish identically for an expanding type-N spacetime

   (J. Bicak & V. Pravda, gr-qc/9804005)

   
   
   and many others: the number of curvature invariants of order 0 (no covariant derivatives) in D ( > 2 ) dimensions is

   ( D - 2 ) ( D - 1 ) D ( D + 3 ) / 12

   
   which is 14 for D = 4; for order k there are:

   D ( k + 1 ) ( D + k + 1 ) ! / ( 2 ( D - 2 ) ! ( k + 3 ) ! )

   (C. N. Haskins, Transactions of the American Mathematical Society 3, 71, 1902)

   
   

   which is 60 for k = 1 and 126 for k = 2

   ---

   Algebraic Simplification

   Using the coordinate transformation c = Cos(q), the Kerr-Newman metric can be written as

   g _{0, 0} = -(A ² c ² + e ² + r (-2 M + r)) / (A ² c ² + r ² )

   g _{0, 3} = -A (-1 + c ² )(e ² - 2 M r) / (A ² c ² + r ² )

   g _{1, 1} = (A ² c ² + r ² ) / (A ² + e ² - 2 M r + r ² )

   g _{2, 2} = (A ² c ² + r ² ) / (1 - c ² )

   g _{3, 3} = (1 - c ² )((A ² + r ² ) ² +

   A ² (-1 + c ² )(A ² + e ² + r (-2 M + r))) /

   (A ² c ² + r ² )
   

   
   
   
   R _{a b c d} has the following symmetries

   R _{a b c d} = - R _{b a c d}

   R _{a b c d} = - R _{a b d c}

   R _{a b c d} = R _{c d a b}

   ---

   Ways to increase Mathematica efficiency:

   • Avoid Trigonometric functions to reduce Simplify time
   • Use symmetries to reduce computation on contractions:

     R ^. R = 4 S _{a = 0}³ S_{b = 0}^{a - 1} S _{c = 0}³ S_{d = 0}^{c - 1} R ^{a b c d} R _{a b c d}

   • Use Factor and PowerExpand instead of Simplify
   • $HistoryLength = 0 to reduce memory requirments

     Problems which exceed Mathematica's capabilities almost always die thrashing

   
   
   
   

   Timings

   Metric	Dimension	R . R	*R . R	D R . D R
   		best time	best time	best time
   Schwarzschild	4	0:02	0:01	0:07
   Schwarzschild	5	0:07		0:31
   Schwarzschild	6	2:33		2:00 *
   Reissner-Nordstrom	4	0:03	0:01	0:10
   Kerr	4	0:18	0:02	1:06
   Kerr-Newman	4	0:38	0:04	3:13

   * used Factor[ ] instead of Simplify[PowerExpand[ ]]

   All times using Mathematica 4.0, sugra.m functions and the above coordinate system on a 450 MHz Pentium II with 128 MB RAM.

---

2. Supergravity Solitons

   D = 11 Supergravity

   The metric is "replaced" by the vielbein:

   g _{m, n} = h _{a b} e _m ^a e _n ^b

   which is used to move between spacetime and the flat tangent space

   
   Y _{m a} is the spin 3/2 gravitino

   spinors are 32-dimensional

   
   G denote the gamma matrices:

   { G _a , G _b } = 2 h _{a b}

   which also are used in totally antisymmetrized products:

   G _{a b} , G _{a b c} , etc.

   
   w _m ^{a b} is the spin connection, giving the covariant spinor derivative

   d _a ^b d _m + w _m ^{a b} G _{a b a} ^b / 4

   
   f _{n r s t} is the self-dual totally-antisymmetric gauge field strength

   ---

   Solitonic solutions

   are bosonic and preserve partial supersymmetry
   • Gravitino Variations:

     d Y _{m a} = ( d _a ^b d _m + w _m ^{a b} G _{a b a} ^b / 4 +

     f ^{n r s t} e _m ^a e _n ^b e _r ^c e _s ^d e _t ^e

     G _{a b c d e a} ^b / 144 -

     f _m ^{n r s} e _n ^a e _r ^b e _s ^c G _{a b c a} ^b / 18 ) e _b

   • Gauge Field Equations:

     D _m f ^{m n r s} = - e ^{n r s a b c d e f g h}

     f _{a b c d} f _{e f g h} / 576

   • Graviton Field Equations:

     R _{m n} - R g _{m n} / 2 = f _m ^{r s t} f _{n r s t} / 3 - g _{m n} f ² / 24

   (following K. S. Stelle, hep-th/9803116)

   Mosty approaches use highly specific ansatze, ie.:

   f ^{m n r s} = e ^{m n r s} f ( r ),

   where e ^{m n r s} is the volume form for a sub-manifold and r is a radial coordinate on the complementary submanifold

   But, one could start with a geometric solution and attempt to find a self-dual gauge field which satisfies the field equations

---

3. Dimensional Scaling

   
   

   • Naive scaling D R ^. D R has D ⁵ products;

     using symmetries has

     D ³ ( D ² - 1 ) / 12 ~ D ³ to D ⁴

   
   
   

   • e has D ! non-zero elements

     for D = 11, D ! = 39,916,800

   
   
   

   • Dual invariants exist in D mod 4 = 0:

     *R R

     *R R R R

     
     with a maximum of ( D ! ) ( D ( D - 1 ) / 2 ) ^{( D / 4 )} products

     
     and in D mod 2 = 0:

     *R *R R

     *R *R R R R

     
     with a maximum of ( D ! )^2 products

     
     

     D	number of products
     4	144
     6	518,400
     8	31,610,880
     10	13,168,189,440,000

---

Symbolic Computation without Mathematica

The Kerr-Newman metric can be parameterized as

g _{0, 0} = -A ² c ² / f - e ² / f + 2 M r / f - r ² / f

g _{0, 3} = -A e ² q / f + 2 A M q r / f

g _{1, 1} = -f / (2 h r)

g _{2, 2} = -f / q

g _{3, 3} = -A ⁴ q / f + 2 A ² h q ² r / f -

2 A ² q r ² / f - q r ⁴ / f

f = A ² c ² + r ²

h = (A ² + e ² + r (-2 M + r)) / (-2 r)

q = c ² - 1

Representation as polynomials with real coefficients, integer exponents:

• simplifies storage, multiplication, derivatives
• means most computations are integer
• sparse storage of terms on linked lists reduces memory requirements

  lists are doubly-linked to facilitate insertion and removal

Potential Problems:

1. Parameterization of fractional exponents:

   1 / ( 1 + k / r ³ ) ^{( 1 / n )} = 1 / q ->

   q ⁿ = 1 + k / r ³

   so that, for instance,

   n q ^{( n - 1 )} dq / dr = - 3 k / r ⁴

   and

   dq / dr = - 3 k / ( n q ^{( n - 1 )} r ⁴ )



2. Constants:

   A, M, e, I need to have zero derivatives



3. Recurring rationals

   to reduce floating point representational error



4. Ansatze:

   If u(r) is not known at computation time, neither are its derivatives - so we parametrize them:

   u'(r) = v(r) and

   u''(r) = w(r)

Classes

• term:

  double coefficient

  array of integer powers (coordinates, parameterization functions, ansatze, constants, rationals)

  
  subclass of element (linked list traversal, insertion, removal)




• sum:

  linked list of terms

  2 r ³ a + 3 r ³ a should be stored as 5 r ³ a

  -> exponents of terms must be unique, so lists must be searched

  
  List is split into an array of lists:

  exponents are hashed to select list

  list is then searched linearly

  
  subclass of hash (the array of linked lists)

  ---

  simplification:

  if p = r ² + a ², we would like

  ( r ⁴ + 2 a ² r ² + a ⁴ ) / p ³ -> 1 / p

  
  polynomial division is most effective when simplifying from high to low powers
  • each parameter function has associated "sort" variable or constant
  • terms are sorted from highest to lowest power of that variable
  • during simplification, each term is on one sort list for each function

  
  
  methods to

  zero, assign, search, add, multiply,

  take partial derivatives, simplify, output

---

• component:

  subclass of element

  
  indices are packed into 64 bit quantity for faster comparison (currently limited to D = 12, index values 0 to 62)

  contains pointer to a sum




• ntuple:

  subclass of hash, with components as elements

  
  specifies tensor rank

  flags for upper, antisymmetric, symmetric, spinor indices

  
  methods for sum-type operations plus

  raise and lower indices, contract indices,

  transpose indices, use symmetry to add components,

  take covariant derivative, take dual

Class Structure

Parallelization

SIMD:

• Single Instruction fetch + Multiple Data fetch -> multiple results
• characterized by topology (how data flows through processors)
• hardware implementation in array processors
• synchronization controlled by hardware
• useful for lattice, linear algebra, FFT, convolutions, image processing, pattern recognition

MIMD:

• Multiple Instruction fetch + Multiple Data fetch
• lends itself to general hardware approaches

  Symmetric Multiprocessors (SMP)

  Clusters (with network connections)

• work scheduling, synchronization and deadlock detection and prevention must be programmed for each application
• useful for iterative systems, systems of PDEs, zeroes of functions, cryptography

ideally n processors -> factor of n speedup

in practice, between log ₂ n and n / ln n due to internal waits for synchronization

PVM - Parallel Virtual Machine

• interrogating configuration
• spawning tasks
• sending and receiving messages (both data and control) between tasks
• notification when a task dies

make list of "atomic" operations
while ( there_is_more_to_be_done () )

do one atomic operation

replace it with "parallelize ()":

if ( starting computation )

break up computation into parcels
start scheduler process
get first parcel to work on

else

point to next calculation in parcel
if ( done with parcel )

if ( done with computation )

send results
synchronize with other tasks

else

get next parcel from scheduler

PTAH implementation

1. contractions (including dualizations) are done in parallel




2. scheduling by partitioning contractions

   lists of free indices and sum components are generated on all processors

   ie., for R _{a b} = R _{a c b} ^c

   free indices	summed indices
   0 0	0 1 0 1
   	0 2 0 2
   	0 3 0 3
   0 1	0 2 1 2
   	0 3 1 3
   0 2	0 1 2 1
   	0 3 2 3
   0 3	0 1 3 1
   	0 2 3 2
   etc.	etc.

   each processor is given a subset of the lists to process

   resultant terms or components are sent to all processors so that intermediate results are known to all

---

3. synchronization is by barrier

   all processors wait for all results in a given computation

   perform simplification themselves

4. fault tolerance

   processors may fail, so work must be reassignable

   PVM (Parallel Virtual Machine) does not guarantee message ordering

5. software metrics and sanity checks

   timings, number of entries into procedures

   depths of linked-lists

   magic numbers

   work scheduling

   PVM message tracing

Hybridization with Mathematica

• Mathematica functions to write C++ header files

  exportation of tensors and parameters

  C++ procedures to build sums and ntuples from the header files

• output is Mathematica FullForm

  easy importation into Mathematica for further analysis

Observations to date

• PTAH has been used to
  1. compute Kerr-Newman invariants both single-thread and parallel

  
  

  2. compute Greenberg's invariants (where C is the Weyl tensor, S is the traceless portion of the Ricci tensor and E is the Einstein curvature tensor):

     R ,
     C ^. C ,
     * C ^. C ,
     C ^. C ^. C ,
     * C ^. C ^. C ,
     S ^. S,
     S ^. S ^. S ,
     S ^. S ^. S ^. S ,
     C ^. E ^. E ,
     * C ^. E ^. E ,
     C ^. E ^. E ^. C ^. E ^. E ,
     * C ^. E ^. E ^. C ^. E ^. E ,
     C ^. E ^. E ^. C ^. E ^. E ^. C ^. E ^. E ,
     * C ^. E ^. E ^. C ^. E ^. E ^. C ^. E ^. E

     (P. J. Greenberg, Studies in Applied Mathematics, 51, 277-308, 1972)

  ---

  3. compute invariant decompositions based on Killing hypersurfaces:

     h _{i j} = g _{i j} - k _i k _j / | k |
     h ⁱ _j = g ^{i k} h _{k j}
     k ⁱ _j = I ⁱ _j - h ⁱ _j

     n = h ^e _i h ^f _j h ^g _k h ^h _l R _{e f g h} R ^{i j k l}

     (geodesic deviation within the normal hypersurface)

     h = 4 h ^f _j h ^h _l k ^e _i k ^g _k R _{e f g h} R ^{i j k l}

     (geodesic deviation which does not cross hypersurfaces)

     x = 4 h ^f _j h ^g _k h ^h _l k ^e _i R _{e f g h} R ^{i j k l} +

     4 h ^e _i k ^f _j k ^g _k k ^h _l R _{e f g h} R ^{i j k l}

     (geodesic deviation which crosses hypersurfaces)

     z = 2 h ^e _i h ^f _j k ^g _k k ^h _l R _{e f g h} R ^{i j k l}

     (does not contribute to geodesic deviation)

     k = k ^e _i k ^f _j k ^g _k k ^h _l R _{e f g h} R ^{i j k l}

     (geodesic deviation within the Killing hypersurface)

  ---

  4. work is in progress to search for D = 11 Supergravity solitons of the form

     Kerr x M ⁷

• for D = 4, single-thread is faster than parallel
• constant tensors and matrices should be computed once and non-symbolically:

  e , G _a , G _{a b} , etc.

• at least potentially, one can publish symbolic results which can never be checked by hand