def computeBilinearForm(self, grid): """ Compute bilinear form for the current grid @param grid: Grid @return DataMatrix """ # create bilinear form of the grid gs = grid.getStorage() A = DataMatrix(gs.size(), gs.size()) A.setAll(0.) createOperationLTwoDotExplicit(A, grid) gs = grid.getStorage() A = DataMatrix(gs.size(), gs.size()) createOperationLTwoDotExplicit(A, grid) # multiply the entries with the pdf at the center of the support p = DataVector(gs.dim()) q = DataVector(gs.dim()) for i in xrange(gs.size()): gpi = gs.get(i) gpi.getCoords(p) for j in xrange(gs.size()): gpj = gs.get(j) gpj.getCoords(q) y = float(A.get(i, j) * self._U.pdf(p)) A.set(i, j, y) A.set(j, i, y) self._map[self.getKey(gpi, gpj)] = A.get(i, j) return A
def computeBilinearForm(self, grid): """ Compute bilinear form for the current grid @param grid: Grid @return DataMatrix """ # create bilinear form of the grid gs = grid.getStorage() A = DataMatrix(gs.getSize(), gs.getSize()) A.setAll(0.) createOperationLTwoDotExplicit(A, grid) # multiply the entries with the pdf at the center of the support p = DataVector(gs.getDimension()) q = DataVector(gs.getDimension()) for i in range(gs.getSize()): gpi = gs.getPoint(i) gs.getCoordinates(gpi, p) for j in range(gs.getSize()): gpj = gs.getPoint(j) gs.getCoordinates(gpj, q) y = float(A.get(i, j) * self._U.pdf(p)) A.set(i, j, y) A.set(j, i, y) self._map[self.getKey([gpi, gpj])] = A.get(i, j) return A
def computeBilinearFormQuad(grid, U): gs = grid.getStorage() basis = getBasis(grid) A = DataMatrix(gs.size(), gs.size()) level = DataMatrix(gs.size(), gs.dim()) index = DataMatrix(gs.size(), gs.dim()) gs.getLevelIndexArraysForEval(level, index) s = np.ndarray(gs.dim(), dtype='float') # run over all rows for i in xrange(gs.size()): gpi = gs.get(i) # run over all columns for j in xrange(i, gs.size()): # print "%i/%i" % (i * gs.size() + j + 1, gs.size() ** 2) gpj = gs.get(j) for d in xrange(gs.dim()): # get level index lid, iid = level.get(i, d), index.get(i, d) ljd, ijd = level.get(j, d), index.get(j, d) # compute left and right boundary of the support of both # basis functions lb = max([(iid - 1) / lid, (ijd - 1) / ljd]) ub = min([(iid + 1) / lid, (ijd + 1) / ljd]) # same level, different index if lid == ljd and iid != ijd: s[d] = 0. # the support does not overlap elif lid != ljd and lb >= ub: s[d] = 0. else: lid, iid = gpi.getLevel(d), int(iid) ljd, ijd = gpj.getLevel(d), int(ijd) # ---------------------------------------------------- # use scipy for integration def f(x): return basis.eval(lid, iid, x) * \ basis.eval(ljd, ijd, x) * \ U[d].pdf(x) s[d], _ = quad(f, lb, ub, epsabs=1e-8) # ---------------------------------------------------- A.set(i, j, float(np.prod(s))) A.set(j, i, A.get(i, j)) return A
def computeBilinearFormQuad(grid, U): gs = grid.getStorage() basis = getBasis(grid) A = DataMatrix(gs.size(), gs.size()) level = DataMatrix(gs.size(), gs.dim()) index = DataMatrix(gs.size(), gs.dim()) gs.getLevelIndexArraysForEval(level, index) s = np.ndarray(gs.dim(), dtype='float') # run over all rows for i in xrange(gs.size()): gpi = gs.get(i) # run over all columns for j in xrange(i, gs.size()): # print "%i/%i" % (i * gs.size() + j + 1, gs.size() ** 2) gpj = gs.get(j) for d in xrange(gs.dim()): # get level index lid, iid = level.get(i, d), index.get(i, d) ljd, ijd = level.get(j, d), index.get(j, d) # compute left and right boundary of the support of both # basis functions lb = max([(iid - 1) / lid, (ijd - 1) / ljd]) ub = min([(iid + 1) / lid, (ijd + 1) / ljd]) # same level, different index if lid == ljd and iid != ijd: s[d] = 0. # the support does not overlap elif lid != ljd and lb >= ub: s[d] = 0. else: lid, iid = gpi.getLevel(d), int(iid) ljd, ijd = gpj.getLevel(d), int(ijd) # ---------------------------------------------------- # use scipy for integration def f(x): return basis.eval(lid, iid, x) * \ basis.eval(ljd, ijd, x) * \ U[d].pdf(x) s[d], _ = quad(f, lb, ub, epsabs=1e-8) # ---------------------------------------------------- A.set(i, j, float(np.prod(s))) A.set(j, i, A.get(i, j)) return A
def cdf(self, x): # convert the parameter to the right format if isList(x): x = DataVector(x) elif isNumerical(x): x = DataVector([x]) elif isMatrix(x): x = DataMatrix(x) if isinstance(x, DataMatrix): A = x B = DataMatrix(A.getNrows(), A.getNcols()) B.setAll(0.0) elif isinstance(x, DataVector): A = DataMatrix(1, len(x)) A.setRow(0, x) B = DataMatrix(1, len(x)) B.setAll(0) # do the transformation self.dist.cdf(A, B) # transform the outcome if isNumerical(x) or isinstance(x, DataVector): return B.get(0, 0) elif isinstance(x, DataMatrix): return B.array()
def computePiecewiseConstantBF(grid, U, admissibleSet): # create bilinear form of the grid gs = grid.getStorage() A = DataMatrix(gs.size(), gs.size()) createOperationLTwoDotExplicit(A, grid) # multiply the entries with the pdf at the center of the support p = DataVector(gs.dim()) q = DataVector(gs.dim()) B = DataMatrix(admissibleSet.getSize(), gs.size()) b = DataVector(admissibleSet.getSize()) # s = np.ndarray(gs.dim(), dtype='float') for k, gpi in enumerate(admissibleSet.values()): i = gs.seq(gpi) gpi.getCoords(p) for j in xrange(gs.size()): gs.get(j).getCoords(q) # for d in xrange(gs.dim()): # # get level index # xlow = max(p[0], q[0]) # xhigh = min(p[1], q[1]) # s[d] = U[d].cdf(xhigh) - U[d].cdf(xlow) y = float(A.get(i, j) * U.pdf(p)) B.set(k, j, y) if i == j: b[k] = y return B, b
def ppf(self, x): # convert the parameter to the right format if isList(x): x = DataVector(x) elif isNumerical(x): x = DataVector([x]) elif isMatrix(x): x = DataMatrix(x) if isinstance(x, DataMatrix): A = x B = DataMatrix(A.getNrows(), A.getNcols()) B.setAll(0.0) elif isinstance(x, DataVector): A = DataMatrix(1, len(x)) A.setRow(0, x) B = DataMatrix(1, len(x)) B.setAll(0) # do the transformation opInvRosen = createOperationInverseRosenblattTransformationKDE(self.dist) opInvRosen.doTransformation(A, B) # transform the outcome if isNumerical(x) or isinstance(x, DataVector): return B.get(0, 0) elif isinstance(x, DataMatrix): return B.array()
def computePiecewiseConstantBF(grid, U, admissibleSet): # create bilinear form of the grid gs = grid.getStorage() A = DataMatrix(gs.size(), gs.size()) createOperationLTwoDotExplicit(A, grid) # multiply the entries with the pdf at the center of the support p = DataVector(gs.getDimension()) q = DataVector(gs.getDimension()) B = DataMatrix(admissibleSet.getSize(), gs.size()) b = DataVector(admissibleSet.getSize()) # s = np.ndarray(gs.getDimension(), dtype='float') for k, gpi in enumerate(admissibleSet.values()): i = gs.getSequenceNumber(gpi) gs.getCoordinates(gpi, p) for j in range(gs.size()): gs.getCoordinates(gs.getPoint(j), q) # for d in xrange(gs.getDimension()): # # get level index # xlow = max(p[0], q[0]) # xhigh = min(p[1], q[1]) # s[d] = U[d].cdf(xhigh) - U[d].cdf(xlow) y = float(A.get(i, j) * U.pdf(p)) B.set(k, j, y) if i == j: b[k] = y return B, b
def ppf(self, x): # convert the parameter to the right format if isList(x): x = DataVector(x) elif isNumerical(x): x = DataVector([x]) if isinstance(x, DataMatrix): A = x B = DataMatrix(A.getNrows(), A.getNcols()) B.setAll(0.0) elif isinstance(x, DataVector): A = DataMatrix(1, len(x)) A.setRow(0, x) B = DataMatrix(1, len(x)) B.setAll(0) # do the transformation assert A.getNcols() == B.getNcols() == self.trainData.getNcols() op = createOperationInverseRosenblattTransformationKDE(self.trainData) op.doTransformation(A, B) # transform the outcome if isNumerical(x) or isinstance(x, DataVector): return B.get(0, 0) elif isinstance(x, DataMatrix): return B.array()
def ppf(self, x): # convert the parameter to the right format if isList(x): x = DataVector(x) elif isNumerical(x): x = DataVector([x]) elif isMatrix(x): x = DataMatrix(x) if isinstance(x, DataMatrix): A = x B = DataMatrix(A.getNrows(), A.getNcols()) B.setAll(0.0) elif isinstance(x, DataVector): A = DataMatrix(1, len(x)) A.setRow(0, x) B = DataMatrix(1, len(x)) B.setAll(0) # do the transformation self.dist.ppf(A, B) # transform the outcome if isNumerical(x) or isinstance(x, DataVector): return B.get(0, 0) elif isinstance(x, DataMatrix): return B.array()
def computeBFQuad(grid, U, admissibleSet, n=100): """ @param grid: Grid @param U: list of distributions @param admissibleSet: AdmissibleSet @param n: int, number of MC samples """ gs = grid.getStorage() basis = getBasis(grid) A = DataMatrix(admissibleSet.getSize(), gs.size()) b = DataVector(admissibleSet.getSize()) s = np.ndarray(gs.dim(), dtype='float') # run over all rows for i, gpi in enumerate(admissibleSet.values()): # run over all columns for j in xrange(gs.size()): # print "%i/%i" % (i * gs.size() + j + 1, gs.size() ** 2) gpj = gs.get(j) for d in xrange(gs.dim()): # get level index lid, iid = gpi.getLevel(d), gpi.getIndex(d) ljd, ijd = gpj.getLevel(d), gpj.getIndex(d) # compute left and right boundary of the support of both # basis functions xlow = max([(iid - 1) * 2 ** -lid, (ijd - 1) * 2 ** -ljd]) xhigh = min([(iid + 1) * 2 ** -lid, (ijd + 1) * 2 ** -ljd]) # same level, different index if lid == ljd and iid != ijd: s[d] = 0. # the support does not overlap elif lid != ljd and xlow >= xhigh: s[d] = 0. else: # ---------------------------------------------------- # use scipy for integration def f(x): return basis.eval(lid, iid, x) * \ basis.eval(ljd, ijd, x) * \ U[d].pdf(x) s[d], _ = quad(f, xlow, xhigh, epsabs=1e-8) # ---------------------------------------------------- A.set(i, j, float(np.prod(s))) if gs.seq(gpi) == j: b[i] = A.get(i, j) return A, b
def computeBFQuad(grid, U, admissibleSet, n=100): """ @param grid: Grid @param U: list of distributions @param admissibleSet: AdmissibleSet @param n: int, number of MC samples """ gs = grid.getStorage() basis = getBasis(grid) A = DataMatrix(admissibleSet.getSize(), gs.size()) b = DataVector(admissibleSet.getSize()) s = np.ndarray(gs.getDimension(), dtype='float') # run over all rows for i, gpi in enumerate(admissibleSet.values()): # run over all columns for j in range(gs.size()): # print "%i/%i" % (i * gs.size() + j + 1, gs.size() ** 2) gpj = gs.getPoint(j) for d in range(gs.getDimension()): # get level index lid, iid = gpi.getLevel(d), gpi.getIndex(d) ljd, ijd = gpj.getLevel(d), gpj.getIndex(d) # compute left and right boundary of the support of both # basis functions xlow = max([(iid - 1) * 2 ** -lid, (ijd - 1) * 2 ** -ljd]) xhigh = min([(iid + 1) * 2 ** -lid, (ijd + 1) * 2 ** -ljd]) # same level, different index if lid == ljd and iid != ijd: s[d] = 0. # the support does not overlap elif lid != ljd and xlow >= xhigh: s[d] = 0. else: # ---------------------------------------------------- # use scipy for integration def f(x): return basis.eval(lid, iid, x) * \ basis.eval(ljd, ijd, x) * \ U[d].pdf(x) s[d], _ = quad(f, xlow, xhigh, epsabs=1e-8) # ---------------------------------------------------- A.set(i, j, float(np.prod(s))) if gs.getSequenceNumber(gpi) == j: b[i] = A.get(i, j) return A, b
def computeBilinearForm(self, grid): """ Compute bilinear form for the current grid @param grid: Grid @return: DataMatrix """ gs = grid.getStorage() A = DataMatrix(gs.size(), gs.size()) A.setAll(0.) createOperationLTwoDotExplicit(A, grid) # store the result in the hash map for i in xrange(gs.size()): gpi = gs.get(i) for j in xrange(gs.size()): gpj = gs.get(j) key = self.getKey(gpi, gpj) self._map[key] = A.get(i, j) return A
def computePiecewiseConstantBilinearForm(grid, U): # create bilinear form of the grid gs = grid.getStorage() A = DataMatrix(gs.size(), gs.size()) createOperationLTwoDotExplicit(A, grid) # multiply the entries with the pdf at the center of the support p = DataVector(gs.getDimension()) q = DataVector(gs.getDimension()) for i in range(gs.size()): gs.getCoordinates(gs.getPoint(i), p) for j in range(gs.size()): gs.getCoordinates(gs.getPoint(j), q) # compute center of the support p.add(q) p.mult(0.5) # multiply the entries in A with the pdf at p y = float(A.get(i, j) * U.pdf(p)) A.set(i, j, y) A.set(j, i, y) return A
def computePiecewiseConstantBilinearForm(grid, U): # create bilinear form of the grid gs = grid.getStorage() A = DataMatrix(gs.size(), gs.size()) createOperationLTwoDotExplicit(A, grid) # multiply the entries with the pdf at the center of the support p = DataVector(gs.dim()) q = DataVector(gs.dim()) for i in xrange(gs.size()): gs.get(i).getCoords(p) for j in xrange(gs.size()): gs.get(j).getCoords(q) # compute center of the support p.add(q) p.mult(0.5) # multiply the entries in A with the pdf at p y = float(A.get(i, j) * U.pdf(p)) A.set(i, j, y) A.set(j, i, y) return A
def ppf(self, x): # convert the parameter to the right format if isList(x): x = DataVector(x) elif isNumerical(x): x = DataVector([x]) # do the transformation if self.grid.getStorage().dim() == 1: op = createOperationInverseRosenblattTransformation1D(self.grid) ans = np.ndarray(len(x)) for i, xi in enumerate(x.array()): ans[i] = op.doTransformation1D(self.alpha, xi) if len(ans) == 1: return ans[0] else: return ans else: if isinstance(x, DataMatrix): A = x B = DataMatrix(A.getNrows(), A.getNcols()) B.setAll(0.0) elif isinstance(x, DataVector): A = DataMatrix(1, len(x)) A.setRow(0, x) B = DataMatrix(1, len(x)) B.setAll(0) # do the transformation op = createOperationInverseRosenblattTransformation(self.grid) op.doTransformation(self.alpha, A, B) # extract the outcome if isNumerical(x) or isinstance(x, DataVector): return B.get(0, 0) elif isinstance(x, DataMatrix): return B.array()
def cdf(self, x, shuffle=True): # convert the parameter to the right format x = self._convertEvalPoint(x) # transform the samples to the unit hypercube if self.trans is not None: x_unit = self.trans.probabilisticToUnitMatrix(x) else: x_unit = x # do the transformation if self.dim == 1: op = createOperationRosenblattTransformation1D(self.grid) ans = np.ndarray(x.shape[0]) for i, xi in enumerate(x_unit[:, 0]): ans[i] = op.doTransformation1D(self.unnormalized_alpha_vec, xi) if len(ans) == 1: return ans[0] else: return ans else: A = DataMatrix(x_unit) B = DataMatrix(x_unit.shape[0], x_unit.shape[1]) B.setAll(0.0) # do the transformation op = createOperationRosenblattTransformation(self.grid) if shuffle: op.doTransformation(self.alpha_vec, A, B) else: op.doTransformation(self.alpha_vec, A, B, 0) # extract the outcome if x_unit.shape == (1, 1): return B.get(0, 0) else: return B.array()
def ppf(self, x): # convert the parameter to the right format if isList(x): x = DataVector(x) elif isNumerical(x): x = DataVector([x]) # do the transformation if self.grid.getStorage().dim() == 1: op = createOperationInverseRosenblattTransformation1D(self.grid) ans = np.ndarray(len(x)) for i, xi in enumerate(x.array()): ans[i] = op.doTransformation1D(self.alpha, xi) if len(ans) == 1: return ans[0] else: return ans else: if isinstance(x, DataMatrix): A = x B = DataMatrix(A.getNrows(), A.getNcols()) B.setAll(0.0) elif isinstance(x, DataVector): A = DataMatrix(1, len(x)) A.setRow(0, x) B = DataMatrix(1, len(x)) B.setAll(0) # do the transformation op = createOperationInverseRosenblattTransformation(self.grid) op.doTransformation(self.alpha, A, B) # extract the outcome if isNumerical(x) or isinstance(x, DataVector): return B.get(0, 0) elif isinstance(x, DataMatrix): return B.array()
def computeBilinearForm(grid, U): """ Compute bilinear form (A)_ij = \int phi_i phi_j dU(x) on measure U, which is in this case supposed to be a lebesgue measure. @param grid: Grid, sparse grid @param U: list of distributions, Lebeasgue measure @return: DataMatrix """ gs = grid.getStorage() basis = getBasis(grid) # interpolate phi_i phi_j on sparse grid with piecewise polynomial SG # the product of two piecewise linear functions is a piecewise # polynomial one of degree 2. ngrid = Grid.createPolyBoundaryGrid(1, 2) # ngrid = Grid.createLinearBoundaryGrid(1) ngrid.createGridGenerator().regular(gs.getMaxLevel() + 1) ngs = ngrid.getStorage() nodalValues = DataVector(ngs.size()) level = DataMatrix(gs.size(), gs.dim()) index = DataMatrix(gs.size(), gs.dim()) gs.getLevelIndexArraysForEval(level, index) A = DataMatrix(gs.size(), gs.size()) s = np.ndarray(gs.dim(), dtype='float') # run over all rows for i in xrange(gs.size()): gpi = gs.get(i) # run over all columns for j in xrange(i, gs.size()): # print "%i/%i" % (i * gs.size() + j + 1, gs.size() ** 2) gpj = gs.get(j) # run over all dimensions for d in xrange(gs.dim()): # get level index lid, iid = level.get(i, d), index.get(i, d) ljd, ijd = level.get(j, d), index.get(j, d) # compute left and right boundary of the support of both # basis functions lb = max([(iid - 1) / lid, (ijd - 1) / ljd]) ub = min([(iid + 1) / lid, (ijd + 1) / ljd]) # same level, different index if lid == ljd and iid != ijd: s[d] = 0. # the support does not overlap elif lid != ljd and lb >= ub: s[d] = 0. else: # ---------------------------------------------------- # do the 1d interpolation ... lid, iid = gpi.getLevel(d), int(iid) ljd, ijd = gpj.getLevel(d), int(ijd) for k in xrange(ngs.size()): x = ngs.get(k).getCoord(0) nodalValues[k] = max(0, basis.eval(lid, iid, x)) * \ max(0, basis.eval(ljd, ijd, x)) # ... by hierarchization v = hierarchize(ngrid, nodalValues) def f(x, y): return float(y * U[d].pdf(x[0])) g, w, _ = discretize(ngrid, v, f, refnums=0) # compute the integral of it s[d] = doQuadrature(g, w) # ---------------------------------------------------- # store result in matrix A.set(i, j, float(np.prod(s))) A.set(j, i, A.get(i, j)) return A
def computeBF(grid, U, admissibleSet): """ Compute bilinear form (A)_ij = \int phi_i phi_j dU(x) on measure U, which is in this case supposed to be a lebesgue measure. @param grid: Grid, sparse grid @param U: list of distributions, Lebeasgue measure @param admissibleSet: AdmissibleSet @return: DataMatrix """ gs = grid.getStorage() basis = getBasis(grid) # interpolate phi_i phi_j on sparse grid with piecewise polynomial SG # the product of two piecewise linear functions is a piecewise # polynomial one of degree 2. ngrid = Grid.createPolyBoundaryGrid(1, 2) ngrid.getGenerator().regular(2) ngs = ngrid.getStorage() nodalValues = DataVector(ngs.size()) A = DataMatrix(admissibleSet.getSize(), gs.size()) b = DataVector(admissibleSet.getSize()) s = np.ndarray(gs.getDimension(), dtype='float') # # pre compute basis evaluations # basis_eval = {} # for li in xrange(1, gs.getMaxLevel() + 1): # for i in xrange(1, 2 ** li + 1, 2): # # add value with it self # x = 2 ** -li * i # basis_eval[(li, i, li, i, x)] = basis.eval(li, i, x) * \ # basis.eval(li, i, x) # # # left side # x = 2 ** -(li + 1) * (2 * i - 1) # basis_eval[(li, i, li, i, x)] = basis.eval(li, i, x) * \ # basis.eval(li, i, x) # # right side # x = 2 ** -(li + 1) * (2 * i + 1) # basis_eval[(li, i, li, i, x)] = basis.eval(li, i, x) * \ # basis.eval(li, i, x) # # # add values for hierarchical lower nodes # for lj in xrange(li + 1, gs.getMaxLevel() + 1): # a = 2 ** (lj - li) # j = a * i - a + 1 # while j < a * i + a: # # center # x = 2 ** -lj * j # basis_eval[(li, i, lj, j, x)] = basis.eval(li, i, x) * \ # basis.eval(lj, j, x) # basis_eval[(lj, j, li, i, x)] = basis_eval[(li, i, lj, j, x)] # # left side # x = 2 ** -(lj + 1) * (2 * j - 1) # basis_eval[(li, i, lj, j, x)] = basis.eval(li, i, x) * \ # basis.eval(lj, j, x) # basis_eval[(lj, j, li, i, x)] = basis_eval[(li, i, lj, j, x)] # # right side # x = 2 ** -(lj + 1) * (2 * j + 1) # basis_eval[(li, i, lj, j, x)] = basis.eval(li, i, x) * \ # basis.eval(lj, j, x) # basis_eval[(lj, j, li, i, x)] = basis_eval[(li, i, lj, j, x)] # j += 2 # # print len(basis_eval) # run over all rows for i, gpi in enumerate(admissibleSet.values()): # run over all columns for j in range(gs.size()): # print "%i/%i" % (i * gs.size() + j + 1, gs.size() ** 2) gpj = gs.getPoint(j) for d in range(gs.getDimension()): # get level index lid, iid = gpi.getLevel(d), gpi.getIndex(d) ljd, ijd = gpj.getLevel(d), gpj.getIndex(d) # compute left and right boundary of the support of both # basis functions lb = max([(iid - 1) * 2 ** -lid, (ijd - 1) * 2 ** -ljd]) ub = min([(iid + 1) * 2 ** -lid, (ijd + 1) * 2 ** -ljd]) # same level, different index if lid == ljd and iid != ijd: s[d] = 0. # the support does not overlap elif lid != ljd and lb >= ub: s[d] = 0. else: # ---------------------------------------------------- # do the 1d interpolation ... # define transformation function T = LinearTransformation(lb, ub) for k in range(ngs.size()): x = ngs.getCoordinate(ngs.getPoint(k), 0) x = T.unitToProbabilistic(x) nodalValues[k] = basis.eval(lid, iid, x) * \ basis.eval(ljd, ijd, x) # ... by hierarchization v = hierarchize(ngrid, nodalValues) # discretize the following function def f(x, y): xp = T.unitToProbabilistic(x) return float(y * U[d].pdf(xp)) # sparse grid quadrature g, w, _ = discretize(ngrid, v, f, refnums=0, level=5, useDiscreteL2Error=False) s[d] = doQuadrature(g, w) * (ub - lb) # fig = plt.figure() # plotSG1d(ngrid, v) # x = np.linspace(xlow, ub, 100) # plt.plot(np.linspace(0, 1, 100), U[d].pdf(x)) # fig.show() # fig = plt.figure() # plotSG1d(g, w) # x = np.linspace(0, 1, 100) # plt.plot(x, # [evalSGFunction(ngrid, v, DataVector([xi])) * U[d].pdf(T.unitToProbabilistic(xi)) for xi in x]) # fig.show() # plt.show() # compute the integral of it # ---------------------------------------------------- A.set(i, j, float(np.prod(s))) if gs.getSequenceNumber(gpi) == j: b[i] = A.get(i, j) return A, b
def computeBF(grid, U, admissibleSet): """ Compute bilinear form (A)_ij = \int phi_i phi_j dU(x) on measure U, which is in this case supposed to be a lebesgue measure. @param grid: Grid, sparse grid @param U: list of distributions, Lebeasgue measure @param admissibleSet: AdmissibleSet @return: DataMatrix """ gs = grid.getStorage() basis = getBasis(grid) # interpolate phi_i phi_j on sparse grid with piecewise polynomial SG # the product of two piecewise linear functions is a piecewise # polynomial one of degree 2. ngrid = Grid.createPolyBoundaryGrid(1, 2) ngrid.createGridGenerator().regular(2) ngs = ngrid.getStorage() nodalValues = DataVector(ngs.size()) A = DataMatrix(admissibleSet.getSize(), gs.size()) b = DataVector(admissibleSet.getSize()) s = np.ndarray(gs.dim(), dtype='float') # # pre compute basis evaluations # basis_eval = {} # for li in xrange(1, gs.getMaxLevel() + 1): # for i in xrange(1, 2 ** li + 1, 2): # # add value with it self # x = 2 ** -li * i # basis_eval[(li, i, li, i, x)] = basis.eval(li, i, x) * \ # basis.eval(li, i, x) # # # left side # x = 2 ** -(li + 1) * (2 * i - 1) # basis_eval[(li, i, li, i, x)] = basis.eval(li, i, x) * \ # basis.eval(li, i, x) # # right side # x = 2 ** -(li + 1) * (2 * i + 1) # basis_eval[(li, i, li, i, x)] = basis.eval(li, i, x) * \ # basis.eval(li, i, x) # # # add values for hierarchical lower nodes # for lj in xrange(li + 1, gs.getMaxLevel() + 1): # a = 2 ** (lj - li) # j = a * i - a + 1 # while j < a * i + a: # # center # x = 2 ** -lj * j # basis_eval[(li, i, lj, j, x)] = basis.eval(li, i, x) * \ # basis.eval(lj, j, x) # basis_eval[(lj, j, li, i, x)] = basis_eval[(li, i, lj, j, x)] # # left side # x = 2 ** -(lj + 1) * (2 * j - 1) # basis_eval[(li, i, lj, j, x)] = basis.eval(li, i, x) * \ # basis.eval(lj, j, x) # basis_eval[(lj, j, li, i, x)] = basis_eval[(li, i, lj, j, x)] # # right side # x = 2 ** -(lj + 1) * (2 * j + 1) # basis_eval[(li, i, lj, j, x)] = basis.eval(li, i, x) * \ # basis.eval(lj, j, x) # basis_eval[(lj, j, li, i, x)] = basis_eval[(li, i, lj, j, x)] # j += 2 # # print len(basis_eval) # run over all rows for i, gpi in enumerate(admissibleSet.values()): # run over all columns for j in xrange(gs.size()): # print "%i/%i" % (i * gs.size() + j + 1, gs.size() ** 2) gpj = gs.get(j) for d in xrange(gs.dim()): # get level index lid, iid = gpi.getLevel(d), gpi.getIndex(d) ljd, ijd = gpj.getLevel(d), gpj.getIndex(d) # compute left and right boundary of the support of both # basis functions lb = max([(iid - 1) * 2 ** -lid, (ijd - 1) * 2 ** -ljd]) ub = min([(iid + 1) * 2 ** -lid, (ijd + 1) * 2 ** -ljd]) # same level, different index if lid == ljd and iid != ijd: s[d] = 0. # the support does not overlap elif lid != ljd and lb >= ub: s[d] = 0. else: # ---------------------------------------------------- # do the 1d interpolation ... # define transformation function T = LinearTransformation(lb, ub) for k in xrange(ngs.size()): x = ngs.get(k).getCoord(0) x = T.unitToProbabilistic(x) nodalValues[k] = basis.eval(lid, iid, x) * \ basis.eval(ljd, ijd, x) # ... by hierarchization v = hierarchize(ngrid, nodalValues) # discretize the following function def f(x, y): xp = T.unitToProbabilistic(x) return float(y * U[d].pdf(xp)) # sparse grid quadrature g, w, _ = discretize(ngrid, v, f, refnums=0, level=5, useDiscreteL2Error=False) s[d] = doQuadrature(g, w) * (ub - lb) # fig = plt.figure() # plotSG1d(ngrid, v) # x = np.linspace(xlow, ub, 100) # plt.plot(np.linspace(0, 1, 100), U[d].pdf(x)) # fig.show() # fig = plt.figure() # plotSG1d(g, w) # x = np.linspace(0, 1, 100) # plt.plot(x, # [evalSGFunction(ngrid, v, DataVector([xi])) * U[d].pdf(T.unitToProbabilistic(xi)) for xi in x]) # fig.show() # plt.show() # compute the integral of it # ---------------------------------------------------- A.set(i, j, float(np.prod(s))) if gs.seq(gpi) == j: b[i] = A.get(i, j) return A, b
def computeBilinearForm(grid, U): """ Compute bilinear form (A)_ij = \int phi_i phi_j dU(x) on measure U, which is in this case supposed to be a lebesgue measure. @param grid: Grid, sparse grid @param U: list of distributions, Lebeasgue measure @return: DataMatrix """ gs = grid.getStorage() basis = getBasis(grid) # interpolate phi_i phi_j on sparse grid with piecewise polynomial SG # the product of two piecewise linear functions is a piecewise # polynomial one of degree 2. ngrid = Grid.createPolyBoundaryGrid(1, 2) # ngrid = Grid.createLinearBoundaryGrid(1) ngrid.getGenerator().regular(gs.getMaxLevel() + 1) ngs = ngrid.getStorage() nodalValues = DataVector(ngs.size()) level = DataMatrix(gs.size(), gs.getDimension()) index = DataMatrix(gs.size(), gs.getDimension()) gs.getLevelIndexArraysForEval(level, index) A = DataMatrix(gs.size(), gs.size()) s = np.ndarray(gs.getDimension(), dtype='float') # run over all rows for i in range(gs.size()): gpi = gs.getPoint(i) # run over all columns for j in range(i, gs.size()): # print "%i/%i" % (i * gs.size() + j + 1, gs.size() ** 2) gpj = gs.getPoint(j) # run over all dimensions for d in range(gs.getDimension()): # get level index lid, iid = level.get(i, d), index.get(i, d) ljd, ijd = level.get(j, d), index.get(j, d) # compute left and right boundary of the support of both # basis functions lb = max([((iid - 1) / lid), ((ijd - 1) / ljd)]) ub = min([((iid + 1) / lid), ((ijd + 1) / ljd)]) # same level, different index if lid == ljd and iid != ijd: s[d] = 0. # the support does not overlap elif lid != ljd and lb >= ub: s[d] = 0. else: # ---------------------------------------------------- # do the 1d interpolation ... lid, iid = gpi.getLevel(d), int(iid) ljd, ijd = gpj.getLevel(d), int(ijd) for k in range(ngs.size()): x = ngs.getCoordinate(ngs.getPoint(k), 0) nodalValues[k] = max(0, basis.eval(lid, iid, x)) * \ max(0, basis.eval(ljd, ijd, x)) # ... by hierarchization v = hierarchize(ngrid, nodalValues) def f(x, y): return float(y * U[d].pdf(x[0])) g, w, _ = discretize(ngrid, v, f, refnums=0) # compute the integral of it s[d] = doQuadrature(g, w) # ---------------------------------------------------- # store result in matrix A.set(i, j, float(np.prod(s))) A.set(j, i, A.get(i, j)) return A