# Medium[ii,:] = smoothperturb_fn.vector().array() print '\t{:12s} {:10s} {:12s} {:12s} {:12s} {:10s} \t{:10s} {:12s} {:12s}'.format(\ 'iter', 'cost', 'misfit', 'reg', '|G|', 'medmisf', 'a_ls', 'tol_cg', 'n_cg') dtruenorm = a_target_fn.vector().inner(waveobj.Mass * a_target_fn.vector()) ######### Inverse problem #waveobj.update_PDE({'a':a_initial_fn}) waveobj.solvefwd_cost() myplot.set_varname('a0') myplot.plot_vtk(waveobj.PDE.a) tolgrad = 1e-10 tolcost = 1e-14 check = False for iter in xrange(50): # gradient waveobj.solveadj_constructgrad() gradnorm = waveobj.MGv.inner(waveobj.Grad.vector()) if iter == 0: gradnorm0 = gradnorm diff = waveobj.PDE.a.vector() - a_target_fn.vector() medmisfit = diff.inner(waveobj.Mass * diff) if check and iter % 5 == 1: checkgradfd_med(waveobj, Medium, 1e-6, [1e-4, 1e-5, 1e-6]) checkhessfd_med(waveobj, Medium, 1e-6, [1e-4, 1e-5, 1e-6]) print '{:12d} {:12.4e} {:12.2e} {:12.2e} {:11.4e} {:10.2e} ({:4.2f})'.\ format(iter, waveobj.cost, waveobj.cost_misfit, waveobj.cost_reg, \ gradnorm, medmisfit, medmisfit/dtruenorm), # plots #myplot.plot_timeseries(waveobj.solfwd, 'p'+str(iter), 0, skip, dl.Function(V)) myplot.set_varname('a' + str(iter)) myplot.plot_vtk(waveobj.PDE.a) myplot.set_varname('grad' + str(iter))
def run_test(fpeak, lambdamin, lambdamax, Nxy, tfilterpts, r, Dt, skip): h = 1. / Nxy checkdt(Dt, h, r, np.sqrt(lambdamax), True) mesh = dl.UnitSquareMesh(Nxy, Nxy) Vl = dl.FunctionSpace(mesh, 'Lagrange', 1) V = dl.FunctionSpace(mesh, 'Lagrange', r) fctV = dl.Function(V) # set up plots: filename, ext = splitext(sys.argv[0]) if isdir(filename + '/'): rmtree(filename + '/') myplot = PlotFenics(filename) # source: Ricker = RickerWavelet(fpeak, 1e-10) Pt = PointSources(V, [[0.5, 0.5]]) mydelta = Pt[0].array() def mysrc(tt): return Ricker(tt) * mydelta # target medium: lambda_target = dl.Expression('lmin + x[0]*(lmax-lmin)', \ lmin=lambdamin, lmax=lambdamax) lambda_target_fn = dl.interpolate(lambda_target, Vl) myplot.set_varname('lambda_target') myplot.plot_vtk(lambda_target_fn) # initial medium: lambda_init = dl.Constant(lambdamin) lambda_init_fn = dl.interpolate(lambda_init, Vl) myplot.set_varname('lambda_init') myplot.plot_vtk(lambda_init_fn) # observation operator: #obspts = [[0.2, 0.5], [0.5, 0.2], [0.5, 0.8], [0.8, 0.5]] obspts = [[0.2, ii/10.] for ii in range(2,9)] + \ [[0.8, ii/10.] for ii in range(2,9)] + \ [[ii/10., 0.2] for ii in range(3,8)] + \ [[ii/10., 0.8] for ii in range(3,8)] obsop = TimeObsPtwise({'V': V, 'Points': obspts}, tfilterpts) # define pde operator: wavepde = AcousticWave({'V': V, 'Vl': Vl, 'Vr': Vl}) wavepde.timestepper = 'backward' wavepde.lump = True wavepde.set_abc(mesh, LeftRight(), True) wavepde.update({'lambda':lambda_target_fn, 'rho':1.0, \ 't0':t0, 'tf':tf, 'Dt':Dt, 'u0init':dl.Function(V), 'utinit':dl.Function(V)}) wavepde.ftime = mysrc # define objective function: waveobj = ObjectiveAcoustic(wavepde) waveobj.obsop = obsop # data print 'generate data' waveobj.solvefwd() myplot.plot_timeseries(waveobj.solfwd, 'pd', 0, skip, fctV) dd = waveobj.Bp.copy() # gradient print 'generate observations' waveobj.dd = dd waveobj.update_m(lambda_init_fn) waveobj.solvefwd_cost() cost1 = waveobj.misfit print 'misfit = {}'.format(waveobj.misfit) myplot.plot_timeseries(waveobj.solfwd, 'p', 0, skip, fctV) # Plot data and observations fig = plt.figure() if len(obspts) > 9: fig.set_size_inches(20., 15.) for ii in range(len(obspts)): if len(obspts) == 4: ax = fig.add_subplot(2, 2, ii + 1) else: ax = fig.add_subplot(4, 6, ii + 1) ax.plot(waveobj.PDE.times, waveobj.dd[ii, :], 'k--') ax.plot(waveobj.PDE.times, waveobj.Bp[ii, :], 'b') ax.set_title('Plot' + str(ii)) fig.savefig(filename + '/observations.eps') print 'compute gradient' waveobj.solveadj_constructgrad() myplot.plot_timeseries(waveobj.soladj, 'v', 0, skip, fctV) MG = waveobj.MGv.array().copy() myplot.set_varname('grad') myplot.plot_vtk(waveobj.Grad) """
# check regularizations are the same regt = regulab.costab(at,bt) reg0 = regulab.costab(a0,b0) regta = regula.cost(at) reg0a = regula.cost(a0) if mpirank == 0: print 'Regularization at target={:.2e}, at initial state={:.2e} [ab]'.format(\ regt, reg0) print 'Regularization at target={:.2e}, at initial state={:.2e} [a]'.format(\ regta, reg0a) # check gradients are the same evaluationpoint = {'a':at, 'b':bt} waveobjab.update_PDE(evaluationpoint) waveobjab.solvefwd_cost() waveobjab.solveadj_constructgrad() MGa, MGb = waveobjab.MG.split(deepcopy=True) MGanorm = MGa.vector().norm('l2') MGbnorm = MGb.vector().norm('l2') if mpirank == 0: print '|MGa|={}, |MGb|={}'.format(MGanorm, MGbnorm) waveobjabnoregul.update_PDE(evaluationpoint) waveobjabnoregul.solvefwd_cost() waveobjabnoregul.solveadj_constructgrad() MGaa, MGba = waveobjabnoregul.MG.split(deepcopy=True) MGaa.vector().axpy(1.0, regula.grad(evaluationpoint['a'])) diffa = MGa.vector() - MGaa.vector() diffb = MGb.vector() - MGba.vector() MGaanorm = MGaa.vector().norm('l2') MGbanorm = MGba.vector().norm('l2')
def run_test(fpeak, lambdamin, lambdamax, Nxy, tfilterpts, r, Dt, skip): h = 1./Nxy checkdt(Dt, h, r, np.sqrt(lambdamax), True) mesh = dl.UnitSquareMesh(Nxy, Nxy) Vl = dl.FunctionSpace(mesh, 'Lagrange', 1) V = dl.FunctionSpace(mesh, 'Lagrange', r) fctV = dl.Function(V) # set up plots: filename, ext = splitext(sys.argv[0]) if isdir(filename + '/'): rmtree(filename + '/') myplot = PlotFenics(filename) # source: Ricker = RickerWavelet(fpeak, 1e-10) Pt = PointSources(V, [[0.5,0.5]]) mydelta = Pt[0].array() def mysrc(tt): return Ricker(tt)*mydelta # target medium: lambda_target = dl.Expression('lmin + x[0]*(lmax-lmin)', \ lmin=lambdamin, lmax=lambdamax) lambda_target_fn = dl.interpolate(lambda_target, Vl) myplot.set_varname('lambda_target') myplot.plot_vtk(lambda_target_fn) # initial medium: lambda_init = dl.Constant(lambdamin) lambda_init_fn = dl.interpolate(lambda_init, Vl) myplot.set_varname('lambda_init') myplot.plot_vtk(lambda_init_fn) # observation operator: #obspts = [[0.2, 0.5], [0.5, 0.2], [0.5, 0.8], [0.8, 0.5]] obspts = [[0.2, ii/10.] for ii in range(2,9)] + \ [[0.8, ii/10.] for ii in range(2,9)] + \ [[ii/10., 0.2] for ii in range(3,8)] + \ [[ii/10., 0.8] for ii in range(3,8)] obsop = TimeObsPtwise({'V':V, 'Points':obspts}, tfilterpts) # define pde operator: wavepde = AcousticWave({'V':V, 'Vl':Vl, 'Vr':Vl}) wavepde.timestepper = 'centered' wavepde.lump = True wavepde.set_abc(mesh, LeftRight(), True) wavepde.update({'lambda':lambda_target_fn, 'rho':1.0, \ 't0':t0, 'tf':tf, 'Dt':Dt, 'u0init':dl.Function(V), 'utinit':dl.Function(V)}) wavepde.ftime = mysrc # define objective function: waveobj = ObjectiveAcoustic(wavepde) waveobj.obsop = obsop # data print 'generate noisy data' waveobj.solvefwd() myplot.plot_timeseries(waveobj.solfwd, 'pd', 0, skip, fctV) dd = waveobj.Bp.copy() nbobspt, dimsol = dd.shape noiselevel = 0.1 # = 10% sigmas = np.sqrt((dd**2).sum(axis=1)/dimsol)*noiselevel rndnoise = np.random.randn(nbobspt*dimsol).reshape((nbobspt, dimsol)) waveobj.dd = dd + sigmas.reshape((len(sigmas),1))*rndnoise # gradient print 'generate observations' waveobj.update_m(lambda_init_fn) waveobj.solvefwd_cost() cost1 = waveobj.misfit print 'misfit = {}'.format(waveobj.misfit) myplot.plot_timeseries(waveobj.solfwd, 'p', 0, skip, fctV) # Plot data and observations fig = plt.figure() if len(obspts) > 9: fig.set_size_inches(20., 15.) for ii in range(len(obspts)): if len(obspts) == 4: ax = fig.add_subplot(2,2,ii+1) else: ax = fig.add_subplot(4,6,ii+1) ax.plot(waveobj.PDE.times, waveobj.dd[ii,:], 'k--') ax.plot(waveobj.PDE.times, waveobj.Bp[ii,:], 'b') ax.set_title('Plot'+str(ii)) fig.savefig(filename + '/observations.eps') print 'compute gradient' waveobj.solveadj_constructgrad() myplot.plot_timeseries(waveobj.soladj, 'v', 0, skip, fctV) MG = waveobj.MGv.array().copy() myplot.set_varname('grad') myplot.plot_vtk(waveobj.Grad) print 'check gradient with FD' Medium = np.zeros((5, Vl.dim())) for ii in range(5): smoothperturb = dl.Expression('sin(n*pi*x[0])*sin(n*pi*x[1])', n=ii+1) smoothperturb_fn = dl.interpolate(smoothperturb, Vl) Medium[ii,:] = smoothperturb_fn.vector().array() checkgradfd_med(waveobj, Medium, 1e-6, [1e-5, 1e-4]) print 'check Hessian with FD' checkhessfd_med(waveobj, Medium, 1e-6, [1e-1, 1e-2, 1e-3, 1e-4, 1e-5], False)