From 91069005ec9f5249efe64cb96fb84fbbd70aad25 Mon Sep 17 00:00:00 2001
From: Michael Orlitzky <michael@orlitzky.com>
Date: Fri, 7 Feb 2014 10:20:26 -0500
Subject: [PATCH] More progress on the 1d FEM, the "big F" vector is now
 computed correctly.

---
 .ghci         |   2 +
 src/FEM/R1.hs | 170 +++++++++++++++++++++++++++++++++++++++++++++-----
 2 files changed, 156 insertions(+), 16 deletions(-)

diff --git a/.ghci b/.ghci
index f42b7bb..ebf54d4 100644
--- a/.ghci
+++ b/.ghci
@@ -4,6 +4,7 @@
 -- Load everything.
 :{
 :load src/BigFloat.hs
+  src/FEM/R1.hs
   src/Integration/Gaussian.hs
   src/Integration/Simpson.hs
   src/Integration/Trapezoid.hs
@@ -21,6 +22,7 @@
 :}
 
 import BigFloat
+import FEM.R1
 import Integration.Gaussian
 import Integration.Simpson
 import Integration.Trapezoid
diff --git a/src/FEM/R1.hs b/src/FEM/R1.hs
index c6a76c8..02eab31 100644
--- a/src/FEM/R1.hs
+++ b/src/FEM/R1.hs
@@ -20,28 +20,34 @@ where
 import qualified Algebra.Algebraic as Algebraic ( C )
 import qualified Algebra.Field as Field ( C )
 import qualified Algebra.RealField as RealField ( C )
+import qualified Algebra.ToRational as ToRational ( C )
 import Data.Vector.Fixed ( Arity, S )
 import NumericPrelude
 import qualified Prelude as P
 
+import Integration.Gaussian ( gaussian )
 import Linear.Matrix (
   Col,
   Mat(..),
   (!!!),
   construct,
+  ifoldl2,
   nrows )
 import Polynomials.Orthogonal ( legendre )
 
 -- | Dirichlet boundary conditions. Since u(a)=u(b)=0 are fixed,
 --   there's no additional information conveyed by this type.
-data Dirichlet = Dirichlet
+data Dirichlet a = Dirichlet { domain_dirichlet :: Interval a }
 
 -- | Neumann boundary conditions. @alpha@ specifies A(a)u'(b) and
 --   @beta@ specifies A(b)u'(b).
-data Neumann a = Neumann { alpha :: a, beta :: a }
+data Neumann a =
+  Neumann { domain_neumann :: Interval a,
+            alpha :: a,
+            beta :: a }
 
 -- | Boundary conditions can be either Dirichlet or Neumann.
-type BoundaryConditions a = Either Dirichlet (Neumann a)
+type BoundaryConditions a = Either (Dirichlet a) (Neumann a)
 
 type Interval a = (a,a)
 
@@ -54,8 +60,9 @@ data PDE a =
     c      :: (a -> a),
     -- | f(x)
     f      :: (a -> a),
-    -- | The domain in R^1 as an interval
-    domain :: Interval a,
+
+    -- | The boundary conditions. The domain also specifies the
+    --   boundary in R^1.
     bdy    :: BoundaryConditions a }
 
 
@@ -63,8 +70,11 @@ data PDE a =
 -- | Non-PDE parameters for the finite element method. The additional
 --   type parameter @n@ should be a type-level representation of the
 --   largest element in @max_degrees@. It needs to be known statically
---   for the dimensions of the pointer matrix.
-data Params m n a =
+--   for the dimensions of the pointer matrix. The parameter @l@ is
+--   the number of global basis functions. It's equal to the number of
+--   /internal/ mesh nodes (i.e. m-1), plus the sum of (p_i - 1) for
+--   each p_i in max_degrees.
+data Params m n l a =
   Params {
     -- | A partition of the domain.
     mesh :: Col m (Interval a),
@@ -87,10 +97,11 @@ data Params m n a =
 --
 --   >>> import Data.Vector.Fixed ( N5, N6 )
 --   >>> import Linear.Matrix ( Col5, fromList )
+--   >>> import Naturals ( N19 )
 --
 --   >>> let p = fromList [[3],[3],[5],[4],[5]] :: Col5 Int
 --   >>> let mesh = undefined :: Col5 (Int,Int)
---   >>> let params = Params mesh p :: Params N5 N5 Int
+--   >>> let params = Params mesh p :: Params N5 N5 N19 Int
 --   >>> let row1 = [0,1,5,6,0,0] :: [Int]
 --   >>> let row2 = [1,2,7,8,0,0] :: [Int]
 --   >>> let row3 = [2,3,9,10,11,12] :: [Int]
@@ -100,7 +111,7 @@ data Params m n a =
 --   >>> pointer params == expected
 --   True
 --
-pointer :: (Arity m, Arity n) => Params m n a -> Mat m (S n) Int
+pointer :: (Arity m, Arity n, Arity l) => Params m n l a -> Mat m (S n) Int
 pointer params =
   construct lambda
   where
@@ -135,15 +146,31 @@ pointer params =
 --
 --   Examples:
 --
---   >>> let phi = affine (-6,9)
+--   >>> let phi = affine (-6,7)
 --   >>> phi (-6)
 --   -1.0
---   >>>  phi (9)
+--   >>>  phi 7
 --   1.0
 --
 affine :: Field.C a => (a,a) -> (a -> a)
 affine (x1,x2) x = (fromInteger 2)*(x - x1)/(x2 - x1) - (fromInteger 1)
 
+-- | The inverse of 'affine'. It should send [-1,1] into [x1,x2].
+--
+--   Examples:
+--
+--   >>> let phi = affine_inv (-6,7)
+--   >>> phi (-1)
+--   -6.0
+--   >>> phi 1
+--   7.0
+--
+affine_inv :: Field.C a => (a,a) -> (a -> a)
+affine_inv (x1,x2) x =
+  x*(x2 - x1)/two + (x1 + x2)/two
+  where
+    two = fromInteger 2
+
 
 -- | Orthonormal basis functions over [-1,1]. The test case below
 --   comes from Sage where the orthogonality of the polynomials'
@@ -153,17 +180,128 @@ affine (x1,x2) x = (fromInteger 2)*(x - x1)/(x2 - x1) - (fromInteger 1)
 --
 --   >>> import qualified Algebra.Absolute as Absolute ( abs )
 --
---   >>> let expected = 6.33910180790284
---   >>> let actual = big_N 3 1.5 :: Double
+--   >>> let expected = 2.99624907925257
+--   >>> let actual = big_N 4 1.5 :: Double
 --   >>> Absolute.abs (actual - expected) < 1e-12
 --   True
 --
 big_N :: forall a. (Algebraic.C a, RealField.C a) => Integer -> a -> a
 big_N k x
   | k < 0 = error "requested a negative basis function"
+  | k == 0 = (one - x) / (fromInteger 2)
+  | k == 1 = (one + x) / (fromInteger 2)
   | otherwise =
-      coeff * ( legendre (k+2) x - legendre k x )
+      coeff * ( legendre k x - legendre (k-2) x )
       where
+        two = fromInteger 2
         four = fromInteger 4
-        six = fromInteger 6
-        coeff = one / (sqrt (four*(fromInteger k) + six)) :: a
+        coeff = one / (sqrt (four*(fromInteger k) - two)) :: a
+
+
+-- | A matrix containing 'big_N' functions indexed by their
+--   element/number. Each row in the matrix represents a finite element
+--   (i.e. an interval in the mesh). Within row @i@, column @j@ contains
+--   the @j@th 'big_N' basis function.
+--
+--   Any given 'big_N' will probably wind up in this matrix multiple
+--   times; the basis functions don't change depending on the
+--   interval. Only the /number/ of basis functions does. Computing
+--   them this way allows us to easily construct a lookup \"table\" of
+--   the proper dimensions.
+--
+--   The second example below relies on the fact that @big_N 3@ and
+--   @big_N 6@ expand to Legendre polynomials (2,4) and (5,7)
+--   respectively and so should be orthogonal over [-1,1].
+--
+--   Examples:
+--
+--   >>> import Data.Vector.Fixed ( N5 )
+--   >>> import Integration.Gaussian ( gaussian )
+--   >>> import Linear.Matrix ( Col5, fromList )
+--   >>> import Naturals ( N19 )
+--
+--   >>> let p = fromList [[3],[3],[5],[4],[5]] :: Col5 Int
+--   >>> let mesh = undefined :: Col5 (Double,Double)
+--   >>> let params = Params mesh p :: Params N5 N5 N19 Double
+--   >>> let big_ns = big_N_matrix params
+--   >>> let n1 = big_ns !!! (1,0)
+--   >>> let n4 = big_ns !!! (4,0)
+--   >>> n1 1.5 == n4 1.5
+--   True
+--   >>> let n1 = big_ns !!! (1,3)
+--   >>> let n2 = big_ns !!! (2,4)
+--   >>> gaussian (\x -> (n1 x) * (n2 x)) < 1e-12
+--   True
+--
+big_N_matrix :: (Arity m, Arity n, Arity l, Algebraic.C a, RealField.C a)
+             => Params m n l a
+             -> Mat m (S n) (a -> a)
+big_N_matrix params =
+  construct lambda
+  where
+    maxdeg i = (max_degrees params) !!! (i,0)
+    lambda i j = if j > maxdeg i
+                 then (const $ fromInteger 0)
+                 else big_N (toInteger j)
+
+
+
+-- | Compute the global load vector F.
+--
+--   Examples:
+--
+--   >>> import Data.Vector.Fixed ( N3, N4 )
+--   >>> import Linear.Matrix ( Col4, frobenius_norm, fromList )
+--   >>> import Naturals ( N7 )
+--
+--   >>> let big_A = const (1::Double)
+--   >>> let c x = sin x
+--   >>> let f x = x*(sin x)
+--   >>> let bdy = Left (Dirichlet (0,1::Double))
+--   >>> let pde = PDE big_A c f bdy
+--
+--   >>> let i1 = (0.0,1/3)
+--   >>> let i2 = (1/3,2/3)
+--   >>> let i3 = (2/3,4/5)
+--   >>> let i4 = (4/5,1.0)
+--   >>> let mesh = fromList [[i1], [i2], [i3], [i4]] :: Col4 (Double,Double)
+--   >>> let pvec = fromList [[2],[3],[2],[1]] :: Col4 Int
+--   >>> let params = Params mesh pvec :: Params N4 N3 N7 Double
+--
+--   >>> let f1 = [0.0418]
+--   >>> let f2 = [0.0805]
+--   >>> let f3 = [0.1007]
+--   >>> let f4 = [-0.0045]
+--   >>> let f5 = [-0.0332]
+--   >>> let f6 = [-0.0054]
+--   >>> let f7 = [-0.0267]
+--   >>> let expected = fromList [f1,f2,f3,f4,f5,f6,f7] :: Col N7 Double
+--   >>> let actual = big_F pde params
+--   >>> frobenius_norm (actual - expected) < 1e-4
+--   True
+--
+big_F :: forall m n l a.
+         (Arity l, Arity m, Arity n,
+          Algebraic.C a, RealField.C a, ToRational.C a)
+      => PDE a
+      -> Params m n l a
+      -> Col l a
+big_F pde params =
+  ifoldl2 accum zero (big_N_matrix params)
+  where
+    accum :: Int -> Int -> Col l a -> (a -> a) -> Col l a
+    accum i j prev_F this_N =
+      prev_F + this_F
+      where
+        two = fromInteger 2
+        (x1,x2) = (mesh params) !!! (i,0)
+        q = affine_inv (x1,x2)
+        integrand x = ((f pde) (q x)) * (this_N x)
+
+        -- The pointer matrix numbers from 1 so subtract one here to
+        -- get the right index.
+        global_idx = ((pointer params) !!! (i,j)) - 1
+        this_F = construct lambda
+        lambda k _ = if k == global_idx
+                     then (gaussian integrand)*(x2 - x1) / two
+                     else fromInteger 0
-- 
2.43.2