def __init__(self, beta_coeff, idx_set, jpdf): self.beta_coeff = beta_coeff self.idx_set = idx_set self.jpdf = jpdf self.N = jpdf.getDimension() # get the distribution type of each random variable dist_types = [] for i in range(self.N): dist_type = self.jpdf.getMarginal(i).getName() dist_types.append(dist_type) # create orthogonal univariate bases poly_collection = ot.PolynomialFamilyCollection(self.N) for i in range(self.N): pdf = jpdf.getDistributionCollection()[i] algo = ot.AdaptiveStieltjesAlgorithm(pdf) poly_collection[i] = ot.StandardDistributionPolynomialFactory(algo) # create multivariate basis multivariate_basis = ot.OrthogonalProductPolynomialFactory( poly_collection, ot.EnumerateFunction(self.N)) # get enumerate function (multi-index handling) enum_func = multivariate_basis.getEnumerateFunction() # get epansion self.expansion = multivariate_basis.getSubBasis( transform_multi_index_set(idx_set, enum_func)) # create openturns surrogate model sur_model = ot.FunctionCollection() for i in range(len(self.expansion)): multi = str(beta_coeff[i]) + '*x' help_function = ot.SymbolicFunction(['x'], [multi]) sur_model.add(ot.ComposedFunction(help_function, self.expansion[i])) self.surrogate_model = np.sum(sur_model)
myCollection[11] = distribution_J4 myCollection[12] = distribution_P1 myCollection[13] = distribution_P2 myCollection[14] = distribution_P3 myCollection[15] = distribution_P4 # Création d'une distribution ? en fonction de la collection myDistribution = ot.ComposedDistribution(myCollection) # ??? vectX = ot.RandomVector(myDistribution) ######################## ### Chaos Polynomial ### ######################## polyColl = ot.PolynomialFamilyCollection(dim) for i in range(dim): polyColl[i] = ot.HermiteFactory() enumerateFunction = ot.LinearEnumerateFunction(dim) multivariateBasis = ot.OrthogonalProductPolynomialFactory(polyColl, enumerateFunction) basisSequenceFactory = ot.LARS() fittingAlgorithm = ot.CorrectedLeaveOneOut() approximationAlgorithm = ot.LeastSquaresMetaModelSelectionFactory(basisSequenceFactory, fittingAlgorithm) # Génération du plan d'expériences N = 200 ot.RandomGenerator.SetSeed(77) Liste_test = ot.LHSExperiment(myDistribution, N) InputSample = Liste_test.generate()
def dali_pce(func, N, jpdf_cp, jpdf_ot, tol=1e-12, max_fcalls=1000, verbose=True, interp_dict={}): if not interp_dict: # if dictionary is empty --> cold-start idx_act = [] # M_activated x N idx_adm = [] # M_admissible x N fevals_act = [] # M_activated x 1 fevals_adm = [] # M_admissible x 1 coeff_act = [] # M_activated x 1 coeff_adm = [] # M_admissible x 1 # start with 0 multi-index knot0 = [] for n in range(N): # get knots per dimension based on maximum index kk, ww = seq_lj_1d(order=0, dist=jpdf_cp[n]) knot0.append(kk[0]) feval = func(knot0) # update activated sets idx_act.append([0] * N) coeff_act.append(feval) fevals_act.append(feval) # local error indicators local_error_indicators = np.abs(coeff_act) # get the OT distribution type of each random variable dist_types = [] for i in range(N): dist_type = jpdf_ot.getMarginal(i).getName() dist_types.append(dist_type) # create orthogonal univariate bases poly_collection = ot.PolynomialFamilyCollection(N) for i in range(N): if dist_types[i] == 'Uniform': poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily( ot.LegendreFactory()) elif dist_types[i] == 'Normal': poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily( ot.HermiteFactory()) elif dist_types[i] == 'Beta': poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily( ot.JacobiFactory()) elif dist_types[i] == 'Gamma': poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily( ot.LaguerreFactory()) else: pdf = jpdf_ot.getDistributionCollection()[i] algo = ot.AdaptiveStieltjesAlgorithm(pdf) poly_collection[i] = ot.StandardDistributionPolynomialFactory( algo) # create multivariate basis mv_basis = ot.OrthogonalProductPolynomialFactory( poly_collection, ot.EnumerateFunction(N)) # get enumerate function (multi-index handling) enum_func = mv_basis.getEnumerateFunction() else: idx_act = interp_dict['idx_act'] idx_adm = interp_dict['idx_adm'] coeff_act = interp_dict['coeff_act'] coeff_adm = interp_dict['coeff_adm'] fevals_act = interp_dict['fevals_act'] fevals_adm = interp_dict['fevals_adm'] mv_basis = interp_dict['mv_basis'] enum_func = interp_dict['enum_func'] # local error indicators local_error_indicators = np.abs(coeff_adm) # compute global error indicator global_error_indicator = local_error_indicators.sum() # max or sum # fcalls / M approx. terms up to now fcalls = len(idx_act) + len(idx_adm) # fcalls = M --> approx. terms # maximum index per dimension max_idx_per_dim = np.max(idx_act + idx_adm, axis=0) # univariate knots and polynomials per dimension knots_per_dim = {} for n in range(N): kk, ww = seq_lj_1d(order=max_idx_per_dim[n], dist=jpdf_cp[n]) knots_per_dim[n] = kk # start iterations while global_error_indicator > tol and fcalls < max_fcalls: if verbose: print(fcalls) print(global_error_indicator) # the index added last to the activated set is the one to be refined last_act_idx = idx_act[-1][:] # compute the knot corresponding to the lastly added index last_knot = [ knots_per_dim[n][i] for n, i in zip(range(N), last_act_idx) ] # get admissible neighbors of the lastly added index adm_neighbors = admissible_neighbors(last_act_idx, idx_act) for an in adm_neighbors: # update admissible index set idx_adm.append(an) # find which parameter/direction n (n=1,2,...,N) gets refined n_ref = np.argmin( [idx1 == idx2 for idx1, idx2 in zip(an, last_act_idx)]) # sequence of 1d Leja nodes/weights for the given refinement knots_n, weights_n = seq_lj_1d(an[n_ref], jpdf_cp[int(n_ref)]) # update max_idx_per_dim, knots_per_dim, if necessary if an[n_ref] > max_idx_per_dim[n_ref]: max_idx_per_dim[n_ref] = an[n_ref] knots_per_dim[n_ref] = knots_n # find new_knot and compute function on new_knot new_knot = last_knot[:] new_knot[n_ref] = knots_n[-1] feval = func(new_knot) fevals_adm.append(feval) fcalls += 1 # update function calls # create PCE basis idx_system = idx_act + idx_adm idx_system_single = transform_multi_index_set(idx_system, enum_func) system_basis = mv_basis.getSubBasis(idx_system_single) # get corresponding evaluations fevals_system = fevals_act + fevals_adm # multi-dimensional knots M = len(idx_system) # equations terms knots_md = [[knots_per_dim[n][idx_system[m][n]] for m in range(M)] for n in range(N)] knots_md = np.array(knots_md).T # design matrix D = get_design_matrix(system_basis, knots_md) # solve system of equaations Q, R = scl.qr(D, mode='economic') c = Q.T.dot(fevals_system) coeff_system = scl.solve_triangular(R, c) # find the multi-index with the largest contribution, add it to idx_act # and delete it from idx_adm coeff_act = coeff_system[:len(idx_act)].tolist() coeff_adm = coeff_system[-len(idx_adm):].tolist() help_idx = np.argmax(np.abs(coeff_adm)) idx_add = idx_adm.pop(help_idx) pce_coeff_add = coeff_adm.pop(help_idx) fevals_add = fevals_adm.pop(help_idx) idx_act.append(idx_add) coeff_act.append(pce_coeff_add) fevals_act.append(fevals_add) # re-compute coefficients of admissible multi-indices # local error indicators local_error_indicators = np.abs(coeff_adm) # compute global error indicator global_error_indicator = local_error_indicators.sum() # max or sum # store expansion data in dictionary interp_dict = {} interp_dict['idx_act'] = idx_act interp_dict['idx_adm'] = idx_adm interp_dict['coeff_act'] = coeff_act interp_dict['coeff_adm'] = coeff_adm interp_dict['fevals_act'] = fevals_act interp_dict['fevals_adm'] = fevals_adm interp_dict['enum_func'] = enum_func interp_dict['mv_basis'] = mv_basis return interp_dict
def alsace(func, N, jpdf, tol=1e-22, sample_type='R', limit_cond=5, max_fcalls=1000, seed=123, ed_file=None, ed_fevals_file=None, verbose=True, pce_dict={}): """ ALSACE - Approximations via Lower-Set and Least-Squares-based Adaptive Chaos Expansions func: function to be approximated. N: number of parameters. jpdf: joint probability density function. limit_cond: maximum allowed condition number of tr(inv(D.T*D)) sample_type: 'R'-random, 'L'-LHS seed: sampling seed tol, max_fcalls: exit criteria, self-explanatory. ed_file, ed_fevals_file: experimental design and corresponding evaluations 'act': activated, i.e. already part of the approximation. 'adm': admissible, i.e. candidates for the approximation's expansion. """ if not pce_dict: # if pce_dict is empty --> cold-start idx_act = [] idx_act.append([0] * N) # start with 0 multi-index idx_adm = [] # set seed ot.RandomGenerator.SetSeed(seed) ed_size = 2 * N # initial number of samples # initial experimental design and coresponding evaluations ed, ed_fevals = get_ed(func, jpdf, ed_size, sample_type=sample_type, knots=[], values=[], ed_file=ed_file, ed_fevals_file=ed_fevals_file) global_error_indicator = 1.0 # give arbitrary sufficiently large value # get the distribution type of each random variable dist_types = [] for i in range(N): dist_type = jpdf.getMarginal(i).getName() dist_types.append(dist_type) # create orthogonal univariate bases poly_collection = ot.PolynomialFamilyCollection(N) for i in range(N): pdf = jpdf.getDistributionCollection()[i] algo = ot.AdaptiveStieltjesAlgorithm(pdf) poly_collection[i] = ot.StandardDistributionPolynomialFactory(algo) # create multivariate basis mv_basis = ot.OrthogonalProductPolynomialFactory( poly_collection, ot.EnumerateFunction(N)) # get enumerate function (multi-index handling) enum_func = mv_basis.getEnumerateFunction() else: # get data from dictionary idx_act = pce_dict['idx_act'] idx_adm = pce_dict['idx_adm'] pce_coeff_act = pce_dict['pce_coeff_act'] pce_coeff_adm = pce_dict['pce_coeff_adm'] ed = pce_dict['ed'] ed_fevals = pce_dict['ed_fevals'] ed_size = len(ed_fevals) # compute local and global error indicators global_error_indicator = np.sum(np.array(pce_coeff_adm)**2) enum_func = pce_dict['enum_func'] mv_basis = pce_dict['mv_basis'] # while ed_size < max_fcalls and global_error_indicator > tol: # the index added last to the activated set is the one to be refined last_act_idx = idx_act[-1][:] # get admissible neighbors of the lastly added index adm_neighbors = admissible_neighbors(last_act_idx, idx_act) # update admissible indices idx_adm = idx_adm + adm_neighbors # get polynomial basis for the LS problem idx_ls = idx_act + idx_adm idx_ls_single = transform_multi_index_set(idx_ls, enum_func) ls_basis = mv_basis.getSubBasis(idx_ls_single) ls_basis_size = len(ls_basis) # construct the design matrix D and compute its QR decomposition D = get_design_matrix(ls_basis, ed) Q, R = sp.qr(D, mode='economic') # construct information matrix A= D^T*D A = np.matmul(D.T, D) / ed_size trAinv_test = np.sum(1. / np.linalg.eig(A)[0]) trAinv = np.trace(np.linalg.inv(A)) print('new trace ', trAinv_test) print('old trace ', trAinv) # If tr(A) becomes too large, enrich the ED until tr(A) becomes # acceptable or until ed_size reaches max_fcalls while (trAinv > limit_cond and ed_size < max_fcalls) or ed_size < ls_basis_size: # inform user if verbose: print('WARNING: tr(inv(A)) = ', trAinv) print('WARNING: cond(D) = ', np.linalg.cond(D)) print("") # select new size for the ED if ls_basis_size > ed_size: ed_size = ls_basis_size + N elif ed_size + N > max_fcalls: ed_size = max_fcalls else: ed_size = ed_size + N # expand ED ed, ed_fevals = get_ed(func, jpdf, ed_size, sample_type=sample_type, knots=ed, values=ed_fevals, ed_file=ed_file, ed_fevals_file=ed_fevals_file) # construct the design matrix D and compute its QR decomposition D = get_design_matrix(ls_basis, ed) Q, R = sp.qr(D, mode='economic') # construct information matrix A= D^T*D A = np.matmul(D.T, D) / ed_size trAinv = np.trace(np.linalg.inv(A)) # solve LS problem c = Q.T.dot(ed_fevals) pce_coeff_ls = sp.solve_triangular(R, c) # find the multi-index with the largest contribution, add it to idx_act # and delete it from idx_adm pce_coeff_act = pce_coeff_ls[:len(idx_act)].tolist() pce_coeff_adm = pce_coeff_ls[-len(idx_adm):].tolist() help_idx = np.argmax(np.abs(pce_coeff_adm)) idx_add = idx_adm.pop(help_idx) pce_coeff_add = pce_coeff_adm.pop(help_idx) idx_act.append(idx_add) pce_coeff_act.append(pce_coeff_add) # store expansion data in dictionary pce_dict = {} pce_dict['idx_act'] = idx_act pce_dict['idx_adm'] = idx_adm pce_dict['pce_coeff_act'] = pce_coeff_act pce_dict['pce_coeff_adm'] = pce_coeff_adm pce_dict['ed'] = ed pce_dict['ed_fevals'] = ed_fevals pce_dict['enum_func'] = enum_func pce_dict['mv_basis'] = mv_basis return pce_dict
distribution = ot.ComposedDistribution( [ot.Normal(), ot.Uniform(), ot.Gamma(2.75, 1.0), ot.Beta(2.5, 1.0, -1.0, 2.0)]) # %% inputDimension = distribution.getDimension() inputDimension # %% # STEP 1: Construction of the multivariate orthonormal basis # ---------------------------------------------------------- # %% # Create the univariate polynomial family collection which regroups the polynomial families for each direction. # %% polyColl = ot.PolynomialFamilyCollection(inputDimension) # %% # We could use the Krawtchouk and Charlier families (for discrete distributions). # %% polyColl[0] = ot.KrawtchoukFactory() polyColl[1] = ot.CharlierFactory() # %% # We could also use the automatic selection of the polynomial which corresponds to the distribution: this is done with the `StandardDistributionPolynomialFactory` class. # %% for i in range(inputDimension): marginal = distribution.getMarginal(i) polyColl[i] = ot.StandardDistributionPolynomialFactory(marginal)
from mero_pdf import jpdf from mero_pdf_ot import jpdf_ot sys.path.append("../") from pce_tools import transform_multi_index_set, get_design_matrix, PCE_Surrogate # construct joint pdf N = len(jpdf) # get the distribution type of each random variable dist_types = [] for i in range(N): dist_type = jpdf_ot.getMarginal(i).getName() dist_types.append(dist_type) # create orthogonal univariate bases poly_collection = ot.PolynomialFamilyCollection(N) for i in range(N): if dist_types[i] == 'Uniform': poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily( ot.LegendreFactory()) elif dist_types[i] == 'Normal': poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily( ot.HermiteFactory()) elif dist_types[i] == 'Beta': poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily( ot.JacobiFactory()) elif dist_types[i] == 'Gamma': poly_collection[i] = ot.OrthogonalUniVariatePolynomialFamily( ot.LaguerreFactory()) else: pdf = jpdf_ot.getDistributionCollection()[i]
def _compute_coefficients_legendre(self, sample_paths, legendre_quadrature_order=None): dimension = self._lower_bound.size truncation_order = self._truncation_order if legendre_quadrature_order is None: legendre_quadrature_order = self._legendre_quadrature_order elif type(legendre_quadrature_order) is not int \ or legendre_quadrature_order <= 0: raise ValueError('legendre_quadrature_order must be a positive ' + 'integer.') n_sample_paths = len(sample_paths) # Gauss-Legendre quadrature nodes and weights polyColl = ot.PolynomialFamilyCollection([ot.LegendreFactory()] * dimension) polynoms = ot.OrthogonalProductPolynomialFactory(polyColl) U, W = polynoms.getNodesAndWeights( ot.Indices([legendre_quadrature_order] * dimension)) W = np.ravel(W) U = np.array(U) scale = (self._upper_bound - self._lower_bound) / 2. shift = (self._upper_bound + self._lower_bound) / 2. X = scale * U + shift # Compute coefficients try: available_memory = int(.9 * get_available_memory()) except: if self.verbose: print('WRN: Available memory estimation failed! ' 'Assuming 1Gb is available (first guess).') available_memory = 1024**3 max_size = int(available_memory / 8 / truncation_order / n_sample_paths) batch_size = min(W.size, max_size) if self.verbose and batch_size < W.size: print('RAM: %d Mb available' % (available_memory / 1024**2)) print('RAM: %d allocable terms / %d total terms' % (max_size, W.size)) print('RAM: %d loops required' % np.ceil(float(W.size) / max_size)) while True: coefficients = np.zeros((n_sample_paths, truncation_order)) try: n_done = 0 while n_done < W.size: sample_paths_values = np.vstack([ np.ravel(sample_paths[i](X[n_done:(n_done + batch_size)])) for i in range(n_sample_paths) ]) mean_values = np.ravel( self._mean(X[n_done:(n_done + batch_size)]))[np.newaxis, :] centered_sample_paths_values = \ sample_paths_values - mean_values del sample_paths_values, mean_values eigenelements_values = np.vstack([ self._eigenfunctions[k]( X[n_done:(n_done + batch_size)]) / np.sqrt(self._eigenvalues[k]) for k in range(truncation_order) ]) coefficients += np.sum( W[np.newaxis, np.newaxis, n_done:(n_done + batch_size)] * centered_sample_paths_values[:, np.newaxis, :] * eigenelements_values[np.newaxis, :, :], axis=-1) del centered_sample_paths_values, eigenelements_values n_done += batch_size break except MemoryError: batch_size /= 2 coefficients *= np.prod(self._upper_bound - self._lower_bound) return coefficients
def _legendre_galerkin_scheme(self, legendre_galerkin_order=10, legendre_quadrature_order=None): # Input checks if legendre_galerkin_order <= 0: raise ValueError('legendre_galerkin_order must be a positive ' + 'integer!') if legendre_quadrature_order is not None: if legendre_quadrature_order <= 0: raise ValueError('legendre_quadrature_order must be a ' + 'positive integer!') # Settings dimension = self._lower_bound.size truncation_order = self._truncation_order galerkin_size = ot.EnumerateFunction( dimension).getStrataCumulatedCardinal(legendre_galerkin_order) if legendre_quadrature_order is None: legendre_quadrature_order = 2 * legendre_galerkin_order + 1 # Check if the current settings are compatible if truncation_order > galerkin_size: raise ValueError( 'The truncation order must be less than or ' + 'equal to the size of the functional basis in the chosen ' + 'Legendre Galerkin scheme. Current size of the galerkin basis ' + 'only allows to get %d terms in the KL expansion.' % galerkin_size) # Construction of the Galerkin basis: tensorized Legendre polynomials tensorized_legendre_polynomial_factory = \ ot.PolynomialFamilyCollection([ot.LegendreFactory()] * dimension) tensorized_legendre_polynomial_factory = \ ot.OrthogonalProductPolynomialFactory( tensorized_legendre_polynomial_factory) tensorized_legendre_polynomials = \ [tensorized_legendre_polynomial_factory.build(i) for i in range(galerkin_size)] # Compute matrix C coefficients using Gauss-Legendre quadrature polyColl = ot.PolynomialFamilyCollection([ot.LegendreFactory()] * dimension * 2) polynoms = ot.OrthogonalProductPolynomialFactory(polyColl) U, W = polynoms.getNodesAndWeights( ot.Indices([legendre_quadrature_order] * dimension * 2)) W = np.ravel(W) scale = (self._upper_bound - self._lower_bound) / 2. shift = (self._upper_bound + self._lower_bound) / 2. U = np.array(U) X = np.repeat(scale, 2) * U + np.repeat(shift, 2) if self.verbose: print('Computing matrix C...') try: available_memory = int(.9 * get_available_memory()) except: if self.verbose: print('WRN: Available memory estimation failed! ' 'Assuming 1Gb is available (first guess).') available_memory = 1024**3 max_size = int(available_memory / 8 / galerkin_size**2) batch_size = min(W.size, max_size) if self.verbose and batch_size < W.size: print('RAM: %d Mb available' % (available_memory / 1024**2)) print('RAM: %d allocable terms / %d total terms' % (max_size, W.size)) print('RAM: %d loops required' % np.ceil(float(W.size) / max_size)) while True: C = np.zeros((galerkin_size, galerkin_size)) try: n_done = 0 while n_done < W.size: covariance_at_X = self._covariance(X[n_done:(n_done + batch_size)]) H1 = np.vstack([ np.ravel(tensorized_legendre_polynomials[i]( U[n_done:(n_done + batch_size), :dimension])) for i in range(galerkin_size) ]) H2 = np.vstack([ np.ravel(tensorized_legendre_polynomials[i]( U[n_done:(n_done + batch_size), dimension:])) for i in range(galerkin_size) ]) C += np.sum(W[np.newaxis, np.newaxis, n_done:(n_done + batch_size)] * covariance_at_X[np.newaxis, np.newaxis, :] * H1[np.newaxis, :, :] * H2[:, np.newaxis, :], axis=-1) del covariance_at_X, H1, H2 n_done += batch_size break except MemoryError: batch_size /= 2 C *= np.prod(self._upper_bound - self._lower_bound)**2. # Matrix B is orthonormal up to some constant B = np.diag( np.repeat(np.prod(self._upper_bound - self._lower_bound), galerkin_size)) # Solve the generalized eigenvalue problem C D = L B D in L, D if self.verbose: print('Solving generalized eigenvalue problem...') eigenvalues, eigenvectors = linalg.eigh(C, b=B, lower=True) eigenvalues, eigenvectors = eigenvalues.real, eigenvectors.real # Sort the eigensolutions in the descending order of eigenvalues order = eigenvalues.argsort()[::-1] eigenvalues = eigenvalues[order] eigenvectors = eigenvectors[:, order] # Truncate the expansion eigenvalues = eigenvalues[:truncation_order] eigenvectors = eigenvectors[:, :truncation_order] # Eliminate unsignificant negative eigenvalues if eigenvalues.min() <= 0.: if eigenvalues.min() > .01 * eigenvalues.max(): raise Exception( 'The smallest significant eigenvalue seems ' + 'to be negative... Check the positive definiteness of the ' + 'covariance function.') else: truncation_order = np.nonzero(eigenvalues <= 0)[0][0] eigenvalues = eigenvalues[:truncation_order] eigenvectors = eigenvectors[:, :truncation_order] self._truncation_order = truncation_order print('WRN: truncation_order was too large.') print('It has been reset to: %d' % truncation_order) # Define eigenfunctions class LegendrePolynomialsBasedEigenFunction(): def __init__(self, vector): self._vector = vector def __call__(self, x): x = np.asanyarray(x) if x.ndim <= 1: x = np.atleast_2d(x).T u = (x - shift) / scale return np.sum([ np.ravel(tensorized_legendre_polynomials[i](u)) * self._vector[i] for i in range(truncation_order) ], axis=0) # Set attributes self._eigenvalues = eigenvalues self._eigenfunctions = [ LegendrePolynomialsBasedEigenFunction(vector) for vector in eigenvectors.T ] self._legendre_galerkin_order = legendre_galerkin_order self._legendre_quadrature_order = legendre_quadrature_order