X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FFEM%2FR1.hs;h=28317a6faecabaf0a079215a192f961a7d5e5be3;hb=45aa63a662556bc9ed0f6018f9f3f580586f38a9;hp=098dcf34530ed1ccb26c5f79532f1a9f4cfd9156;hpb=2b6f975610bcc83e0c4a2fd26cd20fbcec06248d;p=numerical-analysis.git

diff --git a/src/FEM/R1.hs b/src/FEM/R1.hs
index 098dcf3..28317a6 100644
--- a/src/FEM/R1.hs
+++ b/src/FEM/R1.hs
@@ -28,12 +28,13 @@
 module FEM.R1
 where
 
+import Algebra.Absolute ( abs )
 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 NumericPrelude hiding ( abs )
 import qualified Prelude as P
 
 import Integration.Gaussian ( gaussian )
@@ -43,9 +44,20 @@ import Linear.Matrix (
   Row,
   (!!!),
   construct,
+  dot,
+  element_sum2,
   fromList,
   ifoldl2,
-  nrows )
+  map2,
+  nrows,
+  rows2,
+  set_idx,
+  toList,
+  transpose,
+  zip2,
+  zipwith2 )
+import Linear.System ( solve_positive_definite )
+import Piecewise ( Piecewise(..), evaluate', from_intervals )
 import Polynomials.Orthogonal ( legendre )
 
 -- | Dirichlet boundary conditions. Since u(a)=u(b)=0 are fixed,
@@ -185,8 +197,10 @@ affine_inv (x1,x2) x =
     two = fromInteger 2
 
 
+-- * Load vector
+
 -- | Normalized integrals of orthogonal basis functions over
---   n[-1,1]. The test case below comes from Sage where the
+--   [-1,1]. The test case below comes from Sage where the
 --   orthogonality of the polynomials' derivatives can easily be
 --   tested.
 --
@@ -255,65 +269,22 @@ big_N_matrix =
     lambda _ j x = big_N (toInteger j) x
 
 
-
-
--- | Derivatives of the 'big_N's, that is, orthogonal basis functions
---   over [-1,1]. The test case below comes from Sage where the
---   orthogonality of the polynomials' derivatives can easily be
---   tested. The indices are shifted by one so that k=0 is the first
---   basis function.
---
---   Examples:
---
---   >>> import qualified Algebra.Absolute as Absolute ( abs )
---
---   >>> let expected = 11.5757525403319
---   >>> let actual = big_N' 3 1.5 :: Double
---   >>> Absolute.abs (actual - expected) < 1e-10
---   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 = negate ( one / (fromInteger 2))
-  | k == 1 = one / (fromInteger 2)
-  | otherwise = coeff * ( legendre k x )
-      where
-        two = fromInteger 2
-        coeff = sqrt ((two*(fromInteger k) + one) / two) :: a
-
-
--- | The matrix of (N_i' * N_j') functions used in the integrand of
---  the stiffness/mass matrices.
-big_N's_matrix :: (Arity m, Arity n, Algebraic.C a, RealField.C a)
+-- | The matrix of (N_i * N_j) functions used in the integrand of
+--   the mass matrices.
+big_Ns_matrix :: (Arity m, Arity n, Algebraic.C a, RealField.C a)
                => Mat m n (a -> a)
-big_N's_matrix =
+big_Ns_matrix =
   construct lambda
   where
-    lambda i j x = (big_N' (toInteger i) x) * (big_N' (toInteger j) x)
+    lambda i j x = (big_N (toInteger i) x) * (big_N (toInteger j) x)
 
 
 -- | 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
+--   >>> import Linear.Matrix ( Col7, frobenius_norm )
+--   >>> import FEM.R1.Example1 ( pde', params' )
 --
 --   >>> let f1 = [0.0418]
 --   >>> let f2 = [0.0805]
@@ -322,8 +293,8 @@ big_N's_matrix =
 --   >>> 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
+--   >>> let expected = fromList [f1,f2,f3,f4,f5,f6,f7] :: Col7 Double
+--   >>> let actual = big_F pde' params'
 --   >>> frobenius_norm (actual - expected) < 1e-4
 --   True
 --
@@ -347,12 +318,47 @@ big_F pde params =
 
         -- 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
+        k = ((pointer params) !!! (i,j)) - 1
+        integral = (gaussian integrand)*(x2 - x1) / two
+        this_F = set_idx zero (k,0) integral
+
 
+-- * Stiffness matrix
+
+-- | Derivatives of the 'big_N's, that is, orthogonal basis functions
+--   over [-1,1]. The test case below comes from Sage where the
+--   orthogonality of the polynomials' derivatives can easily be
+--   tested. The indices are shifted by one so that k=0 is the first
+--   basis function.
+--
+--   Examples:
+--
+--   >>> import qualified Algebra.Absolute as Absolute ( abs )
+--
+--   >>> let expected = 11.5757525403319
+--   >>> let actual = big_N' 3 1.5 :: Double
+--   >>> Absolute.abs (actual - expected) < 1e-10
+--   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 = negate ( one / (fromInteger 2))
+  | k == 1 = one / (fromInteger 2)
+  | otherwise = coeff * ( legendre k x )
+      where
+        two = fromInteger 2
+        coeff = sqrt ((two*(fromInteger k) + one) / two) :: a
+
+
+-- | The matrix of (N_i' * N_j') functions used in the integrand of
+--   the stiffness matrix.
+big_N's_matrix :: (Arity m, Arity n, Algebraic.C a, RealField.C a)
+               => Mat m n (a -> a)
+big_N's_matrix =
+  construct lambda
+  where
+    lambda i j x = (big_N' (toInteger i) x) * (big_N' (toInteger j) x)
 
 
 big_K_elem :: forall m n l a b.
@@ -366,7 +372,7 @@ big_K_elem :: forall m n l a b.
       -> b
       -> Mat l l a
 big_K_elem pde params _ k cur_K _ =
-  ifoldl2 accum cur_K (big_N's_matrix :: Mat m (S n) (a -> a))
+  ifoldl2 accum cur_K (big_N's_matrix :: Mat (S n) (S n) (a -> a))
   where
     accum :: Int -> Int -> Mat l l a -> (a -> a) -> Mat l l a
     accum i j prev_K these_N's =
@@ -376,14 +382,15 @@ big_K_elem pde params _ k cur_K _ =
         (x1,x2) = (mesh params) !!! (k,0)
         q = affine_inv (x1,x2)
         integrand x = ((big_A pde) (q x)) * (these_N's x)
-        -- The pointer matrix numbers from 1 so subtract one here to
-        -- get the right index.
-        global_row_idx = ((pointer params) !!! (k,i)) - 1
-        global_col_idx = ((pointer params) !!! (k,j)) - 1
-        this_K = construct lambda
-        lambda v w = if v == global_row_idx && w == global_col_idx
-                     then (two/(x2 - x1))* (gaussian integrand)
-                     else fromInteger 0
+        -- The pointer matrix numbers from 1 so subtract one below to
+        -- get the right index. The indices i,j have upper bounds
+        -- dependent on the element k. Since we statically create the
+        -- matrix of basis function derivatives, we have to check here
+        -- whether or not i,j exceed the max index.
+        row_idx = ((pointer params) !!! (k,i)) - 1
+        col_idx = ((pointer params) !!! (k,j)) - 1
+        integral = (two/(x2 - x1))* (gaussian integrand)
+        this_K = set_idx zero (row_idx, col_idx) integral
 
 
 
@@ -394,23 +401,8 @@ big_K_elem pde params _ k cur_K _ =
 --
 --   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
+--   >>> import Linear.Matrix ( Mat7, frobenius_norm )
+--   >>> import FEM.R1.Example1 ( pde', params' )
 --
 --   >>> let k1 = [6, -3, 0, 0, 0, 0, 0] :: [Double]
 --   >>> let k2 = [-3, 10.5, -7.5, 0, 0, 0, 0] :: [Double]
@@ -419,8 +411,8 @@ big_K_elem pde params _ k cur_K _ =
 --   >>> let k5 = [0, 0, 0, 0, 6, 0, 0] :: [Double]
 --   >>> let k6 = [0, 0, 0, 0, 0, 6, 0] :: [Double]
 --   >>> let k7 = [0, 0, 0, 0, 0, 0, 15] :: [Double]
---   >>> let expected = fromList [k1,k2,k3,k4,k5,k6,k7] :: Mat N7 N7 Double
---   >>> let actual = big_K pde params
+--   >>> let expected = fromList [k1,k2,k3,k4,k5,k6,k7] :: Mat7 Double
+--   >>> let actual = big_K pde' params'
 --   >>> frobenius_norm (actual - expected) < 1e-10
 --   True
 --
@@ -435,3 +427,196 @@ big_K pde params =
   where
     col_idxs = fromList [map fromInteger [0..]] :: Row m a
 
+
+-- * Mass matrix
+
+big_M_elem :: forall m n l a b.
+         (Arity l, Arity m, Arity n,
+          Algebraic.C a, RealField.C a, ToRational.C a)
+      => PDE a
+      -> Params m n l a
+      -> Int
+      -> Int
+      -> Mat l l a
+      -> b
+      -> Mat l l a
+big_M_elem pde params _ k cur_M _ =
+  ifoldl2 accum cur_M (big_Ns_matrix :: Mat (S n) (S n) (a -> a))
+  where
+    accum :: Int -> Int -> Mat l l a -> (a -> a) -> Mat l l a
+    accum i j prev_M these_Ns =
+      prev_M + this_M
+      where
+        two = fromInteger 2
+        (x1,x2) = (mesh params) !!! (k,0)
+        q = affine_inv (x1,x2)
+        integrand x = ((c pde) (q x)) * (these_Ns x)
+        -- The pointer matrix numbers from 1 so subtract one here to
+        -- get the right index.
+        row_idx = ((pointer params) !!! (k,i)) - 1
+        col_idx = ((pointer params) !!! (k,j)) - 1
+        integral = (x2 - x1)*(gaussian integrand) / two
+        this_M = set_idx zero (row_idx, col_idx) integral
+
+
+-- | Compute the \"big M\" mass matrix. There are three
+--   parameters needed for M, namely i,j,k so a fold over a matrix will
+--   not do. This little gimmick simulates a three-index fold by doing a
+--   two-index fold over a row of the proper dimensions.
+--
+--   Examples:
+--
+--   >>> import Linear.Matrix ( Mat7, frobenius_norm )
+--   >>> import FEM.R1.Example1 ( pde', params' )
+--
+--   >>> let m1 = [0.0723,0.0266,0,-0.0135,-0.0305,0.0058,0] :: [Double]
+--   >>> let m2 = [0.0266,0.0897,0.0149,0,-0.0345,-0.0109,-0.0179] :: [Double]
+--   >>> let m3 = [0,0.0149,0.0809,0,0,0,-0.0185] :: [Double]
+--   >>> let m4 = [-0.0135,0,0,0.0110,0,0,0] :: [Double]
+--   >>> let m5 = [-0.0305,-0.0345,0,0,0.0319,0.0018,0] :: [Double]
+--   >>> let m6 = [0.0058,-0.0109,0,0,0.0018,0.0076,0] :: [Double]
+--   >>> let m7 = [0,-0.0179,-0.0185,0,0,0,0.0178] :: [Double]
+--
+--   >>> let expected = fromList [m1,m2,m3,m4,m5,m6,m7] :: Mat7 Double
+--   >>> let actual = big_M pde' params'
+--   >>> frobenius_norm (actual - expected) < 1e-3
+--   True
+--
+big_M :: 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
+      -> Mat l l a
+big_M pde params =
+  ifoldl2 (big_M_elem pde params) zero col_idxs
+  where
+    col_idxs = fromList [map fromInteger [0..]] :: Row m a
+
+
+
+-- | Determine the coefficient vector @x@ from the system @(K + M)x = F@.
+--
+--   Examples:
+--
+--   >>> import Linear.Matrix ( Col7, frobenius_norm )
+--   >>> import FEM.R1.Example1 ( pde', params' )
+--
+--   >>> let c1 = [0.02366220347687] :: [Double]
+--   >>> let c2 = [0.03431630082636] :: [Double]
+--   >>> let c3 = [0.02841800893264] :: [Double]
+--   >>> let c4 = [-0.00069489654996] :: [Double]
+--   >>> let c5 = [-0.00518637005151] :: [Double]
+--   >>> let c6 = [-0.00085028505337] :: [Double]
+--   >>> let c7 = [-0.00170478210110] :: [Double]
+--   >>>
+--   >>> let expected = fromList [c1,c2,c3,c4,c5,c6,c7] :: Col7 Double
+--   >>> let actual = coefficients pde' params'
+--   >>> frobenius_norm (actual - expected) < 1e-8
+--   True
+--
+coefficients :: forall m n l a.
+                (Arity m, Arity n, Arity l,
+                 Algebraic.C a, Eq a, RealField.C a, ToRational.C a)
+             => PDE a
+             -> Params m n (S l) a
+             -> Col (S l) a
+coefficients pde params =
+  solve_positive_definite matrix b
+  where
+    matrix = (big_K pde params) + (big_M pde params)
+    b      = big_F pde params
+
+
+solution :: forall m n l a.
+            (Arity m, Arity n, Arity l,
+            Algebraic.C a, Eq a, RealField.C a, ToRational.C a)
+         => PDE a
+         -> Params m n (S l) a
+         -> Piecewise a
+solution pde params =
+  from_intervals $ map head $ toList $ solved_column
+  where
+    global_coeffs :: Col (S l) a
+    global_coeffs = coefficients pde params
+
+    ptr :: Mat m (S n) Int
+    ptr = pointer params
+
+    -- Each mesh element has an associated row in the pointer
+    -- matrix. Stick them together.
+    mesh_with_ptr_rows :: Col m (Interval a, Row (S n) Int)
+    mesh_with_ptr_rows = zip2 (mesh params) (rows2 ptr)
+
+    make_local_coeffs :: (Interval a, Row (S n) Int) -> Row (S n) a
+    make_local_coeffs (_, ptr_row) =
+      construct lambda
+        where
+          lambda _ j = if (ptr_row !!! (0,j)) == zero
+                       then zero
+                       else global_coeffs !!! ((ptr_row !!! (0,j)) - 1, 0)
+
+    -- Create a column vector for each mesh element containing the global
+    -- coefficients corresponding to that element.
+    local_coeffs :: Col m (Row (S n) a)
+    local_coeffs = map2 make_local_coeffs mesh_with_ptr_rows
+
+    global_basis_functions :: Col (S n) (a -> a)
+    global_basis_functions =
+      construct lambda
+        where lambda i _ = big_N (toInteger i)
+
+    mesh_with_coeffs :: Col m (Interval a, Row (S n) a)
+    mesh_with_coeffs = zip2 (mesh params) local_coeffs
+
+    solved_column :: Col m (Interval a, (a -> a))
+    solved_column = map2 solve_piece $ mesh_with_coeffs
+
+    solve_piece :: (Interval a, Row (S n) a) -> (Interval a, (a -> a))
+    solve_piece (interval, coeffs_row) = (interval, g)
+      where
+        coeffs_col = transpose coeffs_row
+
+        g x = element_sum2 $ zipwith2 combine coeffs_col global_basis_functions
+          where
+            xi = (affine interval) x
+            combine ci ni = ci*(ni xi)
+
+
+energy_fem :: (Arity m, Arity n, Arity l,
+                Algebraic.C a, Eq a, RealField.C a, ToRational.C a)
+           => PDE a
+           -> Params m n (S l) a
+           -> a
+energy_fem pde params =
+  (coefficients pde params) `dot` (big_F pde params)
+
+
+relative_error :: forall m n l a.
+                  (Arity m, Arity n, Arity l,
+                   Algebraic.C a, Eq a, RealField.C a, ToRational.C a)
+                => PDE a
+                -> Params m n (S l) a
+                -> a -- ^ The energy norm of the true solution @u@
+                -> a
+relative_error pde params energy_true =
+  cent * sqrt(energy_true - (energy_fem pde params)/energy_true)
+  where
+    cent = fromInteger 100
+
+
+
+relative_error_pointwise :: forall m n l a.
+                            (Arity m, Arity n, Arity l,
+                            Algebraic.C a, Eq a, RealField.C a, ToRational.C a)
+                         => PDE a
+                         -> Params m n (S l) a
+                         -> (a -> a) -- ^ The true solution @u@
+                         -> a -- ^ The point @x@ at which to compute the error.
+                         -> a
+relative_error_pointwise pde params u x =
+  cent * ( u_exact - u_fem ) / u_exact
+  where
+    u_exact = abs $ u x
+    u_fem = evaluate' (solution pde params) x
+    cent = fromInteger 100