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diff --git a/DensityIntegrationUncertaintyQuantification.py b/DensityIntegrationUncertaintyQuantification.py
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+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Created on Mon Apr 3 15:51:35 2017
+
+@author: jiachengzhang
+"""
+
+import numpy as np
+import sys
+import scipy.sparse as scysparse
+import scipy.sparse.linalg as splinalg
+import scipy.linalg as linalg
+
+
+def Density_integration_Poisson_uncertainty(Xn,Yn,fluid_mask,grad_x,grad_y,dirichlet_label,dirichlet_value,
+ uncertainty_quantification=True, sigma_grad_x=None, sigma_grad_y=None, sigma_dirichlet=None):
+ # Evaluate the density field from the gradient fields by solving the Poisson equation.
+ # The uncertainty of the density field can be quantified.
+ """
+ Inputs:
+ Xn,Yn: 2d array of mesh grid.
+ fluid_mask: 2d array of binary mask of flow field. Boundary points are considered in the flow (mask should be True)
+ grad_x, grad_y: 2d array of gradient field.
+ dirichlet_label: 2d array of binary mask indicating the Dirichlet BC locations. Ohterwise Neumann. At least one point should be dirichlet.
+ dirichlet_value: 2d array of diriclet BC values.
+ uncertainty_quantification: True to perform the uncertainty quantification, False only perform integration.
+ sigma_grad_x, sigma_grad_y, sigma_dirichlet: the uncertainty given as std of the input fields. 2d array of fields.
+ Returns:
+ Pn: the integrated density field.
+ sigma_Pn: the uncertainty (std) of the integrated density field.
+ """
+
+ Ny, Nx = np.shape(Xn)
+ dx = Xn[1,1] - Xn[0,0]
+ dy = Yn[1,1] - Yn[0,0]
+ invdx = 1.0/dx
+ invdy = 1.0/dy
+ invdx2 = invdx**2
+ invdy2 = invdy**2
+
+ fluid_mask_ex = np.zeros((Ny+2,Nx+2)).astype('bool')
+ fluid_mask_ex[1:-1,1:-1] = fluid_mask
+
+ grad_x_ex = np.zeros((Ny+2,Nx+2))
+ grad_x_ex[1:-1,1:-1] = grad_x
+ grad_y_ex = np.zeros((Ny+2,Nx+2))
+ grad_y_ex[1:-1,1:-1] = grad_y
+ dirichlet_label_ex = np.zeros((Ny+2,Nx+2)).astype('bool')
+ dirichlet_label_ex[1:-1,1:-1] = dirichlet_label
+ dirichlet_value_ex = np.zeros((Ny+2,Nx+2))
+ dirichlet_value_ex[1:-1,1:-1] = dirichlet_value
+
+ sigma_grad_x_ex = np.zeros((Ny+2,Nx+2))
+ sigma_grad_y_ex = np.zeros((Ny+2,Nx+2))
+ sigma_dirichlet_ex = np.zeros((Ny+2,Nx+2))
+ if uncertainty_quantification==True:
+ sigma_grad_x_ex[1:-1,1:-1] = sigma_grad_x
+ sigma_grad_y_ex[1:-1,1:-1] = sigma_grad_y
+ sigma_dirichlet_ex[1:-1,1:-1] = sigma_dirichlet
+
+ # Generate the linear operator.
+ j,i = np.where(fluid_mask_ex==True)
+ Npts = len(j)
+ fluid_index = -np.ones(fluid_mask_ex.shape).astype('int')
+ fluid_index[j,i] = range(Npts)
+ iC = fluid_index[j,i]
+ iC_label = dirichlet_label_ex[j,i]
+ iE = fluid_index[j,i+1]
+ iW = fluid_index[j,i-1]
+ iN = fluid_index[j+1,i]
+ iS = fluid_index[j-1,i]
+
+ LaplacianOperator = scysparse.csr_matrix((Npts,Npts),dtype=np.float)
+ # RHS = np.zeros(Npts)
+ # Var_RHS = np.zeros(Npts) # variance
+
+ # Also generate the linear operator which maps from the grad_x, grad_y, dirichlet val to the rhs.
+ Map_grad_x = scysparse.csr_matrix((Npts,Npts),dtype=np.float)
+ Map_grad_y = scysparse.csr_matrix((Npts,Npts),dtype=np.float)
+ Map_dirichlet_val = scysparse.csr_matrix((Npts,Npts),dtype=np.float)
+
+ # First, construct the linear operator and RHS as if all they are all Nuemanan Bc
+
+ # if the east and west nodes are inside domain
+ loc = (iE!=-1)*(iW!=-1)
+ LaplacianOperator[iC[loc],iC[loc]] += -2.0*invdx2
+ LaplacianOperator[iC[loc],iE[loc]] += +1.0*invdx2
+ LaplacianOperator[iC[loc],iW[loc]] += +1.0*invdx2
+ # RHS[iC[loc]] += (grad_x_ex[j[loc],i[loc]+1] - grad_x_ex[j[loc],i[loc]-1])*(invdx*0.5)
+ # Var_RHS[iC[loc]] += (invdx*0.5)**2 * (sigma_grad_x_ex[j[loc],i[loc]+1]**2 + sigma_grad_x_ex[j[loc],i[loc]-1]**2)
+ Map_grad_x[iC[loc],iE[loc]] += invdx*0.5
+ Map_grad_x[iC[loc],iW[loc]] += -invdx*0.5
+
+ # if the east node is ouside domian
+ loc = (iE==-1)
+ LaplacianOperator[iC[loc],iC[loc]] += -2.0*invdx2
+ LaplacianOperator[iC[loc],iW[loc]] += +2.0*invdx2
+ # RHS[iC[loc]] += -(grad_x_ex[j[loc],i[loc]] + grad_x_ex[j[loc],i[loc]-1])*invdx
+ # Var_RHS[iC[loc]] += invdx**2 * (sigma_grad_x_ex[j[loc],i[loc]]**2 + sigma_grad_x_ex[j[loc],i[loc]-1]**2)
+ Map_grad_x[iC[loc],iC[loc]] += -invdx
+ Map_grad_x[iC[loc],iW[loc]] += -invdx
+
+ # if the west node is ouside domian
+ loc = (iW==-1)
+ LaplacianOperator[iC[loc],iC[loc]] += -2.0*invdx2
+ LaplacianOperator[iC[loc],iE[loc]] += +2.0*invdx2
+ # RHS[iC[loc]] += (grad_x_ex[j[loc],i[loc]] + grad_x_ex[j[loc],i[loc]+1])*invdx
+ # Var_RHS[iC[loc]] += invdx**2 * (sigma_grad_x_ex[j[loc],i[loc]]**2 + sigma_grad_x_ex[j[loc],i[loc]+1]**2)
+ Map_grad_x[iC[loc],iC[loc]] += invdx
+ Map_grad_x[iC[loc],iE[loc]] += invdx
+
+ # if the north and south nodes are inside domain
+ loc = (iN!=-1)*(iS!=-1)
+ LaplacianOperator[iC[loc],iC[loc]] += -2.0*invdy2
+ LaplacianOperator[iC[loc],iN[loc]] += +1.0*invdy2
+ LaplacianOperator[iC[loc],iS[loc]] += +1.0*invdy2
+ # RHS[iC[loc]] += (grad_y_ex[j[loc]+1,i[loc]] - grad_y_ex[j[loc]-1,i[loc]])*(invdy*0.5)
+ # Var_RHS[iC[loc]] += (invdy*0.5)**2 * (sigma_grad_y_ex[j[loc]+1,i[loc]]**2 + sigma_grad_y_ex[j[loc]-1,i[loc]]**2)
+ Map_grad_y[iC[loc],iN[loc]] += invdy*0.5
+ Map_grad_y[iC[loc],iS[loc]] += -invdy*0.5
+
+ # if the north node is ouside domian
+ loc = (iN==-1)
+ LaplacianOperator[iC[loc],iC[loc]] += -2.0*invdy2
+ LaplacianOperator[iC[loc],iS[loc]] += +2.0*invdy2
+ # RHS[iC[loc]] += -(grad_y_ex[j[loc],i[loc]] + grad_y_ex[j[loc]-1,i[loc]])*invdy
+ # Var_RHS[iC[loc]] += invdy**2 * (sigma_grad_y_ex[j[loc],i[loc]]**2 + sigma_grad_y_ex[j[loc]-1,i[loc]]**2)
+ Map_grad_y[iC[loc],iC[loc]] += -invdy
+ Map_grad_y[iC[loc],iS[loc]] += -invdy
+
+ # if the south node is ouside domian
+ loc = (iS==-1)
+ LaplacianOperator[iC[loc],iC[loc]] += -2.0*invdy2
+ LaplacianOperator[iC[loc],iN[loc]] += +2.0*invdy2
+ # RHS[iC[loc]] += (grad_y_ex[j[loc],i[loc]] + grad_y_ex[j[loc]+1,i[loc]])*invdy
+ # Var_RHS[iC[loc]] += invdy**2 * (sigma_grad_y_ex[j[loc],i[loc]]**2 + sigma_grad_y_ex[j[loc]+1,i[loc]]**2)
+ Map_grad_y[iC[loc],iC[loc]] += invdy
+ Map_grad_y[iC[loc],iN[loc]] += invdy
+
+ # Then change the boundary conidtion at locatiosn of Dirichlet.
+ loc = (iC_label==True)
+ LaplacianOperator[iC[loc],:] = 0.0
+ LaplacianOperator[iC[loc],iC[loc]] = 1.0*invdx2
+ # RHS[iC[loc]] = dirichlet_value_ex[j[loc],i[loc]] * invdx2
+ # Var_RHS[iC[loc]] = sigma_dirichlet_ex[j[loc],i[loc]]**2 * invdx2**2
+ Map_grad_x[iC[loc],:] = 0.0
+ Map_grad_y[iC[loc],:] = 0.0
+ Map_dirichlet_val[iC[loc],iC[loc]] = 1.0*invdx2
+
+ LaplacianOperator.eliminate_zeros()
+ Map_grad_x.eliminate_zeros()
+ Map_grad_y.eliminate_zeros()
+ Map_dirichlet_val.eliminate_zeros()
+
+ # Solve for the field.
+ grad_x_vect = grad_x_ex[j,i]
+ grad_y_vect = grad_y_ex[j,i]
+ dirichlet_val_vect = dirichlet_value_ex[j,i]
+ RHS = Map_grad_x.dot(grad_x_vect) + Map_grad_y.dot(grad_y_vect) + Map_dirichlet_val.dot(dirichlet_val_vect)
+ # Solve the linear system
+ Pn_ex = np.zeros((Ny+2,Nx+2))
+ p_vect = splinalg.spsolve(LaplacianOperator, RHS)
+ Pn_ex[j,i] = p_vect
+
+ # Uncertainty propagation
+ sigma_Pn_ex = np.zeros((Ny+2,Nx+2))
+ if uncertainty_quantification==True:
+ # Propagate to get the covariance matrix for RHS
+ Cov_grad_x = scysparse.diags(sigma_grad_x_ex[j,i]**2,shape=(Npts,Npts),format='csr')
+ Cov_grad_y = scysparse.diags(sigma_grad_y_ex[j,i]**2,shape=(Npts,Npts),format='csr')
+ Cov_dirichlet_val = scysparse.diags(sigma_dirichlet_ex[j,i]**2,shape=(Npts,Npts),format='csr')
+ Cov_RHS = Map_grad_x*Cov_grad_x*Map_grad_x.transpose() + Map_grad_y*Cov_grad_y*Map_grad_y.transpose() + \
+ Map_dirichlet_val*Cov_dirichlet_val*Map_dirichlet_val.transpose()
+
+ Laplacian_inv = linalg.inv(LaplacianOperator.A)
+ Cov_p = np.matmul(np.matmul(Laplacian_inv, Cov_RHS.A), Laplacian_inv.T)
+ Var_p_vect = np.diag(Cov_p)
+
+ sigma_Pn_ex[j,i] = Var_p_vect**0.5
+
+ return Pn_ex[1:-1,1:-1], sigma_Pn_ex[1:-1,1:-1]
+
+
+
+def Density_integration_WLS_uncertainty(Xn,Yn,fluid_mask,grad_x,grad_y,dirichlet_label,dirichlet_value,
+ uncertainty_quantification=True, sigma_grad_x=None, sigma_grad_y=None, sigma_dirichlet=None):
+ # Evaluate the density field from the gradient fields by solving the WLS system.
+ # The uncertainty of the density field is also quantified and returned.
+ """
+ Inputs:
+ Xn,Yn: 2d array of mesh grid.
+ fluid_mask: 2d array of binary mask of flow field. Boundary points are considered in the flow (mask should be True)
+ grad_x, grad_y: 2d array of gradient field.
+ dirichlet_label: 2d array of binary mask indicating the Dirichlet BC locations. Ohterwise Neumann. At least one point should be dirichlet.
+ dirichlet_value: 2d array of diriclet BC values.
+ uncertainty_quantification: True to perform the uncertainty quantification, False only perform integration.
+ sigma_grad_x, sigma_grad_y, sigma_dirichlet: the uncertainty given as std of the input fields. 2d array of fields.
+ Returns:
+ Pn: the integrated density field.
+ sigma_Pn: the uncertainty (std) of the integrated density field.
+ """
+
+ Ny, Nx = np.shape(Xn)
+ dx = Xn[1,1] - Xn[0,0]
+ dy = Yn[1,1] - Yn[0,0]
+ invdx = 1.0/dx
+ invdy = 1.0/dy
+ invdx2 = invdx**2
+ invdy2 = invdy**2
+
+ fluid_mask_ex = np.zeros((Ny+2,Nx+2)).astype('bool')
+ fluid_mask_ex[1:-1,1:-1] = fluid_mask
+
+ dirichlet_label_ex = np.zeros((Ny+2,Nx+2)).astype('bool')
+ dirichlet_label_ex[1:-1,1:-1] = dirichlet_label
+ dirichlet_value_ex = np.zeros((Ny+2,Nx+2))
+ dirichlet_value_ex[1:-1,1:-1] = dirichlet_value
+
+ grad_x_ex = np.zeros((Ny+2,Nx+2))
+ grad_x_ex[1:-1,1:-1] = grad_x
+ grad_y_ex = np.zeros((Ny+2,Nx+2))
+ grad_y_ex[1:-1,1:-1] = grad_y
+
+ sigma_grad_x_ex = np.zeros((Ny+2,Nx+2))
+ sigma_grad_y_ex = np.zeros((Ny+2,Nx+2))
+ sigma_dirichlet_ex = np.zeros((Ny+2,Nx+2))
+ sigma_grad_x_ex[1:-1,1:-1] = sigma_grad_x
+ sigma_grad_y_ex[1:-1,1:-1] = sigma_grad_y
+ sigma_dirichlet_ex[1:-1,1:-1] = sigma_dirichlet
+
+ # Generate the index for mapping the pressure and pressure gradients.
+ j,i = np.where(fluid_mask_ex==True)
+ Npts = len(j)
+ fluid_index = -np.ones(fluid_mask_ex.shape).astype('int')
+ fluid_index[j,i] = range(Npts)
+
+ # Generate the mask for the gradients
+ fluid_mask_Gx_ex = np.logical_and(fluid_mask_ex[:,1:],fluid_mask_ex[:,:-1])
+ fluid_mask_Gy_ex = np.logical_and(fluid_mask_ex[1:,:],fluid_mask_ex[:-1,:])
+
+ # Generate the linear operator and the mapping matrix for generating the rhs.
+ # For Gx
+ jx,ix = np.where(fluid_mask_Gx_ex==True)
+ Npts_x = len(jx)
+ iC = fluid_index[jx,ix]
+ iE = fluid_index[jx,ix+1]
+ Operator_Gx = scysparse.csr_matrix((Npts_x,Npts),dtype=np.float)
+ Map_Gx = scysparse.csr_matrix((Npts_x,Npts),dtype=np.float)
+ Operator_Gx[range(Npts_x),iC] += -invdx
+ Operator_Gx[range(Npts_x),iE] += invdx
+ Map_Gx[range(Npts_x),iC] += 0.5
+ Map_Gx[range(Npts_x),iE] += 0.5
+
+ # For Gy
+ jy,iy = np.where(fluid_mask_Gy_ex==True)
+ Npts_y = len(jy)
+ iC = fluid_index[jy,iy]
+ iN = fluid_index[jy+1,iy]
+ Operator_Gy = scysparse.csr_matrix((Npts_y,Npts),dtype=np.float)
+ Map_Gy = scysparse.csr_matrix((Npts_y,Npts),dtype=np.float)
+ Operator_Gy[range(Npts_y),iC] += -invdy
+ Operator_Gy[range(Npts_y),iN] += invdy
+ Map_Gy[range(Npts_y),iC] += 0.5
+ Map_Gy[range(Npts_y),iN] += 0.5
+
+ # For Dirichlet BC
+ j_d, i_d = np.where(dirichlet_label_ex==True)
+ Npts_d = len(j_d)
+ iC = fluid_index[j_d,i_d]
+ Operator_d = scysparse.csr_matrix((Npts_d,Npts),dtype=np.float)
+ Map_d = scysparse.eye(Npts_d,Npts_d,format='csr') * invdx
+ Operator_d[range(Npts_d),iC] += 1.0*invdx
+ # dirichlet value vector and cov
+ dirichlet_vect = dirichlet_value_ex[j_d,i_d]
+ dirichlet_sigma_vect = sigma_dirichlet_ex[j_d,i_d]
+ cov_dirichlet = scysparse.diags(dirichlet_sigma_vect**2, format='csr')
+ # Generate the vector and cov for pgrad.
+ pgrad_x_vect = grad_x_ex[j,i]
+ pgrad_y_vect = grad_y_ex[j,i]
+ pgrad_vect = np.concatenate((pgrad_x_vect,pgrad_y_vect))
+ cov_pgrad_x = sigma_grad_x_ex[j,i]**2
+ cov_pgrad_y = sigma_grad_y_ex[j,i]**2
+ cov_pgrad_vect = np.concatenate((cov_pgrad_x, cov_pgrad_y))
+ cov_pgrad = scysparse.diags(cov_pgrad_vect, format='csr')
+
+ # Construct the full operator.
+ Operator_GLS = scysparse.bmat([[Operator_Gx],[Operator_Gy],[Operator_d]])
+
+ # Construct the full mapping matrics and get the rhs.
+ Map_pgrad = scysparse.bmat([[Map_Gx, None],[None, Map_Gy],[scysparse.csr_matrix((Npts_d,Npts),dtype=np.float), None]])
+ Map_dirichlet = scysparse.bmat([[scysparse.csr_matrix((Npts_x+Npts_y,Npts_d),dtype=np.float)],[Map_d]])
+ rhs = Map_pgrad.dot(pgrad_vect) + Map_dirichlet.dot(dirichlet_vect)
+
+ # Evaluate the covriance matrix for the rhs
+ cov_rhs = Map_pgrad * cov_pgrad * Map_pgrad.transpose() + Map_dirichlet * cov_dirichlet * Map_dirichlet.transpose()
+
+ # Solve for the WLS solution
+ weights_vect = cov_rhs.diagonal()**(-1)
+ weights_matrix = scysparse.diags(weights_vect,format='csr')
+ # Operator_WLS = weights_matrix * Operator_GLS
+ # rhs_WLS = weights_matrix.dot(rhs)
+ sys_LHS = Operator_GLS.transpose() * weights_matrix * Operator_GLS
+ sys_rhs = (Operator_GLS.transpose() * weights_matrix).dot(rhs)
+
+ # Get the solution from lsqr
+ # p_vect_wls = splinalg.lsqr(Operator_WLS,rhs_WLS)[0]
+ # Pn_WLS = np.zeros(fluid_mask_ex.shape)
+ # Pn_WLS[j,i] = p_vect_wls
+
+ # Solve for the WLS solution
+ p_vect_wls = splinalg.spsolve(sys_LHS, sys_rhs)
+ Pn_WLS_ex = np.zeros(fluid_mask_ex.shape)
+ Pn_WLS_ex[j,i] = p_vect_wls
+
+ # Perform the uncertainty propagation
+ sigma_Pn_ex = np.zeros((Ny+2,Nx+2))
+ if uncertainty_quantification == True:
+ cov_sys_rhs = (Operator_GLS.transpose() * weights_matrix) * cov_rhs * (Operator_GLS.transpose() * weights_matrix).transpose()
+ sys_LHS_inv = linalg.inv(sys_LHS.A)
+ Cov_p = np.matmul(np.matmul(sys_LHS_inv, cov_sys_rhs.A), sys_LHS_inv.T)
+ Var_p_vect = np.diag(Cov_p)
+ sigma_Pn_ex[j,i] = Var_p_vect**0.5
+
+ return Pn_WLS_ex[1:-1,1:-1], sigma_Pn_ex[1:-1,1:-1]
+
+
+
+def Density_integration_WLS_uncertainty_weighted_average(Xn,Yn,fluid_mask,grad_x,grad_y,dirichlet_label,dirichlet_value,
+ uncertainty_quantification=True, sigma_grad_x=None, sigma_grad_y=None, sigma_dirichlet=None):
+ # Evaluate the density field from the gradient fields by solving the WLS system.
+ # The uncertainty of the density field is also quantified and returned.
+ # The gradient interpolation (from grid points to staggered location) is done by a weighted average approach
+ # which minimizes the sum of squared bias error and random error.
+ """
+ Inputs:
+ Xn,Yn: 2d array of mesh grid.
+ fluid_mask: 2d array of binary mask of flow field. Boundary points are considered in the flow (mask should be True)
+ grad_x, grad_y: 2d array of gradient field.
+ dirichlet_label: 2d array of binary mask indicating the Dirichlet BC locations. Ohterwise Neumann. At least one point should be dirichlet.
+ dirichlet_value: 2d array of diriclet BC values.
+ uncertainty_quantification: True to perform the uncertainty quantification, False only perform integration.
+ sigma_grad_x, sigma_grad_y, sigma_dirichlet: the uncertainty given as std of the input fields. 2d array of fields.
+ Returns:
+ Pn: the integrated density field.
+ sigma_Pn: the uncertainty (std) of the integrated density field.
+ """
+
+ Ny, Nx = np.shape(Xn)
+ dx = Xn[1,1] - Xn[0,0]
+ dy = Yn[1,1] - Yn[0,0]
+ invdx = 1.0/dx
+ invdy = 1.0/dy
+ invdx2 = invdx**2
+ invdy2 = invdy**2
+
+ fluid_mask_ex = np.zeros((Ny+2,Nx+2)).astype('bool')
+ fluid_mask_ex[1:-1,1:-1] = fluid_mask
+
+ dirichlet_label_ex = np.zeros((Ny+2,Nx+2)).astype('bool')
+ dirichlet_label_ex[1:-1,1:-1] = dirichlet_label
+ dirichlet_value_ex = np.zeros((Ny+2,Nx+2))
+ dirichlet_value_ex[1:-1,1:-1] = dirichlet_value
+
+ grad_x_ex = np.zeros((Ny+2,Nx+2))
+ grad_x_ex[1:-1,1:-1] = grad_x
+ grad_y_ex = np.zeros((Ny+2,Nx+2))
+ grad_y_ex[1:-1,1:-1] = grad_y
+
+ sigma_grad_x_ex = np.zeros((Ny+2,Nx+2))
+ sigma_grad_y_ex = np.zeros((Ny+2,Nx+2))
+ sigma_dirichlet_ex = np.zeros((Ny+2,Nx+2))
+ sigma_grad_x_ex[1:-1,1:-1] = sigma_grad_x
+ sigma_grad_y_ex[1:-1,1:-1] = sigma_grad_y
+ sigma_dirichlet_ex[1:-1,1:-1] = sigma_dirichlet
+
+ # Generate the index for mapping the pressure and pressure gradients.
+ j,i = np.where(fluid_mask_ex==True)
+ Npts = len(j)
+ fluid_index = -np.ones(fluid_mask_ex.shape).astype('int')
+ fluid_index[j,i] = range(Npts)
+
+ # Generate the mask for the gradients
+ fluid_mask_Gx_ex = np.logical_and(fluid_mask_ex[:,1:],fluid_mask_ex[:,:-1])
+ fluid_mask_Gy_ex = np.logical_and(fluid_mask_ex[1:,:],fluid_mask_ex[:-1,:])
+
+ # Generate the linear operator and the mapping matrix for generating the rhs.
+ # For Gx
+ jx,ix = np.where(fluid_mask_Gx_ex==True)
+ Npts_x = len(jx)
+ iC = fluid_index[jx,ix]
+ iE = fluid_index[jx,ix+1]
+ Operator_Gx = scysparse.csr_matrix((Npts_x,Npts),dtype=np.float)
+ Operator_Gx[range(Npts_x),iC] += -invdx
+ Operator_Gx[range(Npts_x),iE] += invdx
+ # The Mapping Gx that maps grided gradients to staggered location is by weighted average.
+ Map_Gx = scysparse.csr_matrix((Npts_x,Npts),dtype=np.float)
+ V_C = grad_x_ex[jx,ix]
+ V_E = grad_x_ex[jx,ix+1]
+ sigma_C = sigma_grad_x_ex[jx,ix]
+ sigma_E = sigma_grad_x_ex[jx,ix+1]
+ weight_C = ((V_C-V_E)**2 + 2*sigma_E**2) / (2*(V_C-V_E)**2 + 2*sigma_C**2 + 2*sigma_E**2)
+ weight_E = 1.0 - weight_C
+ Map_Gx[range(Npts_x),iC] += weight_C
+ Map_Gx[range(Npts_x),iE] += weight_E
+
+ # For Gy
+ jy,iy = np.where(fluid_mask_Gy_ex==True)
+ Npts_y = len(jy)
+ iC = fluid_index[jy,iy]
+ iN = fluid_index[jy+1,iy]
+ Operator_Gy = scysparse.csr_matrix((Npts_y,Npts),dtype=np.float)
+ Operator_Gy[range(Npts_y),iC] += -invdy
+ Operator_Gy[range(Npts_y),iN] += invdy
+ # The Mapping Gy that maps grided gradients to staggered location is by weighted average.
+ Map_Gy = scysparse.csr_matrix((Npts_y,Npts),dtype=np.float)
+ V_C = grad_y_ex[jy,iy]
+ V_N = grad_y_ex[jy+1,iy]
+ sigma_C = sigma_grad_y_ex[jy,iy]
+ sigma_N = sigma_grad_y_ex[jy+1,iy]
+ weight_C = ((V_C-V_N)**2 + 2*sigma_N**2) / (2*(V_C-V_N)**2 + 2*sigma_C**2 + 2*sigma_N**2)
+ weight_N = 1.0 - weight_C
+ Map_Gy[range(Npts_y),iC] += weight_C
+ Map_Gy[range(Npts_y),iN] += weight_N
+
+ # For Dirichlet BC
+ j_d, i_d = np.where(dirichlet_label_ex==True)
+ Npts_d = len(j_d)
+ iC = fluid_index[j_d,i_d]
+ Operator_d = scysparse.csr_matrix((Npts_d,Npts),dtype=np.float)
+ Map_d = scysparse.eye(Npts_d,Npts_d,format='csr') * invdx
+ Operator_d[range(Npts_d),iC] += 1.0*invdx
+ # dirichlet value vector and cov
+ dirichlet_vect = dirichlet_value_ex[j_d,i_d]
+ dirichlet_sigma_vect = sigma_dirichlet_ex[j_d,i_d]
+ cov_dirichlet = scysparse.diags(dirichlet_sigma_vect**2, format='csr')
+ # Generate the vector and cov for pgrad.
+ pgrad_x_vect = grad_x_ex[j,i]
+ pgrad_y_vect = grad_y_ex[j,i]
+ pgrad_vect = np.concatenate((pgrad_x_vect,pgrad_y_vect))
+ cov_pgrad_x = sigma_grad_x_ex[j,i]**2
+ cov_pgrad_y = sigma_grad_y_ex[j,i]**2
+ cov_pgrad_vect = np.concatenate((cov_pgrad_x, cov_pgrad_y))
+ cov_pgrad = scysparse.diags(cov_pgrad_vect, format='csr')
+
+ # Construct the full operator.
+ Operator_GLS = scysparse.bmat([[Operator_Gx],[Operator_Gy],[Operator_d]])
+
+ # Construct the full mapping matrics and get the rhs.
+ Map_pgrad = scysparse.bmat([[Map_Gx, None],[None, Map_Gy],[scysparse.csr_matrix((Npts_d,Npts),dtype=np.float), None]])
+ Map_dirichlet = scysparse.bmat([[scysparse.csr_matrix((Npts_x+Npts_y,Npts_d),dtype=np.float)],[Map_d]])
+ rhs = Map_pgrad.dot(pgrad_vect) + Map_dirichlet.dot(dirichlet_vect)
+
+ # Evaluate the covriance matrix for the rhs
+ cov_rhs = Map_pgrad * cov_pgrad * Map_pgrad.transpose() + Map_dirichlet * cov_dirichlet * Map_dirichlet.transpose()
+
+ # Solve for the WLS solution
+ weights_vect = cov_rhs.diagonal()**(-1)
+ weights_matrix = scysparse.diags(weights_vect,format='csr')
+ # Operator_WLS = weights_matrix * Operator_GLS
+ # rhs_WLS = weights_matrix.dot(rhs)
+ sys_LHS = Operator_GLS.transpose() * weights_matrix * Operator_GLS
+ sys_rhs = (Operator_GLS.transpose() * weights_matrix).dot(rhs)
+
+ # Get the solution from lsqr
+ # p_vect_wls = splinalg.lsqr(Operator_WLS,rhs_WLS)[0]
+ # Pn_WLS = np.zeros(fluid_mask_ex.shape)
+ # Pn_WLS[j,i] = p_vect_wls
+
+ # Solve for the WLS solution
+ p_vect_wls = splinalg.spsolve(sys_LHS, sys_rhs)
+ Pn_WLS_ex = np.zeros(fluid_mask_ex.shape)
+ Pn_WLS_ex[j,i] = p_vect_wls
+
+ # Perform the uncertainty propagation
+ sigma_Pn_ex = np.zeros((Ny+2,Nx+2))
+ if uncertainty_quantification == True:
+ cov_sys_rhs = (Operator_GLS.transpose() * weights_matrix) * cov_rhs * (Operator_GLS.transpose() * weights_matrix).transpose()
+ sys_LHS_inv = linalg.inv(sys_LHS.A)
+ Cov_p = np.matmul(np.matmul(sys_LHS_inv, cov_sys_rhs.A), sys_LHS_inv.T)
+ Var_p_vect = np.diag(Cov_p)
+ sigma_Pn_ex[j,i] = Var_p_vect**0.5
+
+ return Pn_WLS_ex[1:-1,1:-1], sigma_Pn_ex[1:-1,1:-1]
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/LICENSE.txt b/LICENSE.txt
new file mode 100755
index 0000000..e62ec04
--- /dev/null
+++ b/LICENSE.txt
@@ -0,0 +1,674 @@
+GNU GENERAL PUBLIC LICENSE
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+
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diff --git a/README.md b/README.md
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--- /dev/null
+++ b/README.md
@@ -0,0 +1,8 @@
+This code package implements the density integration methodology for BOS outlined in:
+
+Rajendran, L. K., Zhang, J., Bane, S., & Vlachos, P. (2020). Uncertainty-based weighted least squares density integration for background-oriented schlieren. Experiments in Fluids.
+
+Please cite the above paper if you use this code package for your work.
+
+sample_script.py is a sample python script that loads the sample dataset in 'sample-data.mat' and calls the density integration + uncertainty quantification function to perform the calculations.
+It saves the result to 'sample-result.mat' and a figure to 'sample-result.png'
diff --git a/__pycache__/DensityIntegrationUncertaintyQuantification.cpython-38.pyc b/__pycache__/DensityIntegrationUncertaintyQuantification.cpython-38.pyc
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diff --git a/loadmat_functions.py b/loadmat_functions.py
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--- /dev/null
+++ b/loadmat_functions.py
@@ -0,0 +1,35 @@
+# The following functions convert an object to a struct so that it can be saved to a mat file
+import scipy.io as sio
+
+def loadmat(filename):
+ '''
+ this function should be called instead of direct spio.loadmat
+ as it cures the problem of not properly recovering python dictionaries
+ from mat files. It calls the function check keys to cure all entries
+ which are still mat-objects
+ '''
+ data = sio.loadmat(filename, struct_as_record=False, squeeze_me=True)
+ return _check_keys(data)
+
+def _check_keys(dict):
+ '''
+ checks if entries in dictionary are mat-objects. If yes
+ todict is called to change them to nested dictionaries
+ '''
+ for key in dict:
+ if isinstance(dict[key], sio.matlab.mio5_params.mat_struct):
+ dict[key] = _todict(dict[key])
+ return dict
+
+def _todict(matobj):
+ '''
+ A recursive function which constructs from matobjects nested dictionaries
+ '''
+ dict = {}
+ for strg in matobj._fieldnames:
+ elem = matobj.__dict__[strg]
+ if isinstance(elem, sio.matlab.mio5_params.mat_struct):
+ dict[strg] = _todict(elem)
+ else:
+ dict[strg] = elem
+ return dict
\ No newline at end of file
diff --git a/sample-data.mat b/sample-data.mat
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diff --git a/sample-result.png b/sample-result.png
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diff --git a/sample_script.py b/sample_script.py
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--- /dev/null
+++ b/sample_script.py
@@ -0,0 +1,113 @@
+#!/usr/bin/env python2
+# -*- coding: utf-8 -*-
+
+import numpy as np
+import matplotlib
+# matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+import sys
+import os
+from scipy.io import savemat
+from scipy.io import loadmat
+import timeit
+
+from DensityIntegrationUncertaintyQuantification import Density_integration_Poisson_uncertainty
+from DensityIntegrationUncertaintyQuantification import Density_integration_WLS_uncertainty
+from DensityIntegrationUncertaintyQuantification import Density_integration_WLS_uncertainty_weighted_average
+
+# Load the data:
+# all loaded variables is in standard physical units: X, Y: [m], rho_x, rho_y: [kg/m^4]
+data = loadmat('sample-data.mat', squeeze_me=True)
+
+# set the density uncertainty at the boundary points [kg/m^3]
+sigma_rho_dirichlet = 0.01
+
+# calculate density and uncertainty (Poisson)
+rho_poisson, sigma_rho_poisson = Density_integration_Poisson_uncertainty(data['X'], data['Y'], data['mask'],
+ data['rho_x'], data['rho_y'],
+ data['dirichlet_label'], data['rho_dirichlet'],
+ uncertainty_quantification=True,
+ sigma_grad_x=data['sigma_rho_x'], sigma_grad_y=data['sigma_rho_y'],
+ sigma_dirichlet=sigma_rho_dirichlet)
+
+# calculate density and uncertainty (WLS)
+rho_wls, sigma_rho_wls = Density_integration_WLS_uncertainty(data['X'], data['Y'], data['mask'],
+ data['rho_x'], data['rho_y'],
+ data['dirichlet_label'], data['rho_dirichlet'],
+ uncertainty_quantification=True,
+ sigma_grad_x=data['sigma_rho_x'], sigma_grad_y=data['sigma_rho_y'],
+ sigma_dirichlet=sigma_rho_dirichlet)
+
+# save the results to file
+savemat(file_name='sample-result.mat', mdict={'X': data['X'], 'Y': data['Y'], 'rho_poisson': rho_poisson, 'sigma_rho_poisson': sigma_rho_poisson,
+ 'rho_wls': rho_wls, 'sigma_rho_wls': sigma_rho_wls}, long_field_names=True)
+
+# Plot the results
+fig1 = plt.figure(1, figsize=(12,8))
+plt.figure(1)
+ax1 = fig1.add_subplot(3,2,1)
+ax2 = fig1.add_subplot(3,2,2)
+ax3 = fig1.add_subplot(3,2,3)
+ax4 = fig1.add_subplot(3,2,4)
+ax5 = fig1.add_subplot(3,2,5)
+ax6 = fig1.add_subplot(3,2,6)
+
+# plot x gradient
+plt.axes(ax1)
+plt.pcolor(data['X'], data['Y'], data['rho_x'], vmin=-60, vmax=60)
+plt.colorbar()
+plt.title('rho_x')
+
+# plot y gradient
+plt.axes(ax2)
+plt.pcolor(data['X'], data['Y'], data['rho_y'], vmin=-60, vmax=60)
+plt.colorbar()
+plt.title('rho_y')
+
+# plot density (poisson)
+plt.axes(ax3)
+plt.pcolor(data['X'], data['Y'], rho_poisson, vmin=1.2, vmax=1.5)
+plt.colorbar()
+plt.title('rho, Poisson')
+
+# plot density uncertainty (poisson)
+plt.axes(ax4)
+plt.pcolor(data['X'], data['Y'], sigma_rho_poisson, vmin=0.0, vmax=0.01)
+plt.colorbar()
+plt.title('sigma rho, Poisson')
+
+# plot density (wls)
+plt.axes(ax5)
+plt.pcolor(data['X'], data['Y'], rho_wls, vmin=1.2, vmax=1.5)
+plt.colorbar()
+plt.title('rho, WLS')
+
+# plot density uncertainty (wls)
+plt.axes(ax6)
+plt.pcolor(data['X'], data['Y'], sigma_rho_wls, vmin=0.0, vmax=0.01)
+plt.colorbar()
+plt.title('sigma rho, WLS')
+
+plt.tight_layout()
+
+# save plot to file
+plt.savefig('sample-result.png')
+plt.close()
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+