├── LICENSE.txt ├── README.md ├── display_figures.m ├── ell.m ├── gaussrnd.m ├── make_lvl.m ├── model_eit.m ├── model_gwf.m ├── model_id.m └── run_mcmc.m /LICENSE.txt: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. By contrast, 15 | the GNU General Public License is intended to guarantee your freedom to 16 | share and change all versions of a program--to make sure it remains free 17 | software for all its users. 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Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # bayes-hier 2 | MATLAB code to run dimension robust MCMC for hierarchical Bayesian inversion, as outlined 3 | in the paper _Hierarchical Bayesian Level Set Inversion_ by Dunlop, Iglesias and Stuart. 4 | Three example forward models are provided: direct point observations, a groundwater flow 5 | model and an electrical impedance tomography model. 6 | 7 | The following files are provided: 8 | 9 | `run_mcmc.m` 10 | Perform the MCMC, given the parameters (grid resolution, number of samples, prior 11 | smoothness, etc) defined at the start of the file. 12 | Optionally the output may be displayed as a figure. Saved in memory are the traces of 13 | the length scale parameter and a few Fourier modes. 14 | 15 | `gaussrnd.m` 16 | Generate a 2D sample from the Gaussian prior outlined in the paper, given a smoothness 17 | parameter alpha, inverse length scale parameter tau, and grid size N. 18 | The sample is provided in Fourier space, reshaped into an N^2*1 vector. 19 | 20 | `make_lvl.m` 21 | Take a square matrix representing a continuous function, and threshold at two levels, 22 | returning a matrix representing a piecewise constant function. 23 | The values that the thresholded function takes are defined in this file. 24 | 25 | `ell.m` 26 | Select the appropriate forward model, mapping the thresholded function to the output 27 | measurements. The three models are contained in model_id.m, model_gwf.m and 28 | model_eit.m. 29 | 30 | `model_id.m` 31 | Perform direct observations of the piecewise constant field. Observations are 32 | performed on a square grid of J points. J is defined in the file, and should be a 33 | perfect square. 34 | 35 | `model_gwf.m` 36 | Solve using the steady state Darcy flow model, as outlined in the paper. Observations 37 | are performed on a square grid of J points. J is defined in the file, and should be a 38 | perfect square. 39 | 40 | `model_eit.m` 41 | EIT forward model, using the complete electrode model. In the example provided, a 42 | a circular domain is used with 16 electrodes equally spaced around its boundary, and 43 | the maximum 15 linear independent current stimulation patterns are applied. 44 | 45 | NOTE: Requires EIDORS software (http://eidors3d.sourceforge.net/). The path to the 46 | software must be defined at the start of the file. 47 | 48 | `display_figures.m` 49 | Provide some visual output for the MCMC: the true field that generates the data, the 50 | current MCMC sample, and the trace of the length scale parameter. 51 | Note that, for the EIT example, output of the full sample on a square domain is shown, 52 | even though the simulation domain is circular. The code may be modified to use EIDORS 53 | built in image output tools (as was done in the paper). 54 | -------------------------------------------------------------------------------- /display_figures.m: -------------------------------------------------------------------------------- 1 | % Some example code for providing figure output. 2 | % 3 | % If the software for the forward map (e.g. EIDORS) has its own plotting 4 | % tools, it may be preferable to use those. 5 | 6 | % Define the output grids, and output the true image 7 | if s == output.thinning 8 | [X,Y] = meshgrid(1/(2*N):1/N:1-1/(2*N),1/(2*N):1/N:1-1/(2*N)); 9 | [XT,YT] = meshgrid(1/(2*NT):1/NT:1-1/(2*NT),1/(2*NT):1/NT:1-1/(2*NT)); 10 | 11 | subplot(221); 12 | surf(XT,YT,UT_phys,'EdgeColor','None');view(2);axis square; 13 | axis off; 14 | title('True field'); 15 | end 16 | 17 | % Output the current sample 18 | subplot(222); 19 | hh = surf(X,Y,make_lvl(idct2(reshape(U,N,N)),tau,prior.U.alpha),'EdgeColor','None');view(2);axis square; 20 | axis off; 21 | title('Current sample'); 22 | 23 | % Output the trace of tau 24 | subplot(224); 25 | plot(traceTau(1:s)); 26 | if s > 1 27 | xlim([1 s]); 28 | end 29 | axis square; 30 | title('Inverse length scale'); 31 | 32 | pause(0.01); -------------------------------------------------------------------------------- /ell.m: -------------------------------------------------------------------------------- 1 | % Define the forward map, mapping the geometric field into the observation 2 | % space. 3 | 4 | function l = ell(S,model) 5 | 6 | % Identity model 7 | if strcmp(model,'id') 8 | l = model_id(S); 9 | 10 | % Groundwater flow model 11 | elseif strcmp(model,'gwf') 12 | l = model_gwf(S); 13 | 14 | % EIT model 15 | % Note: requires EIDORS software 16 | elseif strcmp(model,'eit') 17 | l = model_eit(S); 18 | 19 | else 20 | error('Invalid model specified'); 21 | end 22 | end -------------------------------------------------------------------------------- /gaussrnd.m: -------------------------------------------------------------------------------- 1 | % Return a sample of a Gaussian random field on [0,1]^2 with: 2 | % mean 0 3 | % covariance operator C = (-Delta + tau^2)^(-alpha) 4 | % where Delta is the Laplacian with zero Neumann boundary conditions. 5 | % Returned is an N^2x1 vector of Fourier cosine coefficients, so that 6 | % U = idct2(reshape(L,N,N)) is the sample in physical space. 7 | 8 | function L = gaussrnd(alpha,tau,N) 9 | 10 | % Random variables in KL expansion 11 | xi = normrnd(0,1,N); 12 | 13 | % Define the (square root of) eigenvalues of the covariance operator 14 | [K1,K2] = meshgrid(0:N-1,0:N-1); 15 | coef = (pi^2*(K1.^2+K2.^2) + tau^2).^(-alpha/2); 16 | 17 | % Construct the KL coefficients 18 | L = N*coef.*xi; 19 | L(1,1) = 0; 20 | 21 | L = reshape(L,N^2,1); 22 | 23 | end -------------------------------------------------------------------------------- /make_lvl.m: -------------------------------------------------------------------------------- 1 | % The construction map taking a level set field U to a piecewise constant 2 | % geometric field V, with scaling of relative levels by the lengthscale 3 | % tau. 4 | % The example here is for three phases/two thresholding levels. 5 | 6 | function V = make_lvl(U,tau,alpha) 7 | 8 | d = 2; % Spatial dimension 9 | c = 0.1*tau^(d/2-alpha)*[1,-1]; % The thresholding levels, scaled by tau 10 | 11 | V = U; 12 | V(find(U>=c(1))) = 10; % Value of phase 1 13 | V(find(c(1)>U & U>=c(2))) = 5; % Value of phase 2 14 | V(find(c(2)>U)) = 1; % Value of phase 3 15 | end -------------------------------------------------------------------------------- /model_eit.m: -------------------------------------------------------------------------------- 1 | function l = model_eit(S) 2 | % Load EIDORS if not already loaded 3 | if exist('show_fem','file') == 0 4 | run('/path/to/eidors/startup.m'); 5 | end 6 | 7 | % Forward model parameters 8 | model_type = 'm2C'; % Choice of predefined 2D model (help mk_common_model) 9 | num_elec = 16; % Number of electrodes 10 | num_stim = num_elec - 1; % Number of lienarly independent stimulation patterns 11 | inj_pattern = [0,1]; % Injection pattern (help mk_stim_patterns) 12 | meas_pattern = '{mono}'; % Measurement pattern (help mk_stim_patterns) 13 | stim_current = 0.1; % Drive current levels (Amps) 14 | contact_imp = 0.01; % Contact impedences (Ohms) 15 | 16 | % Create the 2D model 17 | imdl= mk_common_model(model_type,num_elec); 18 | 19 | % Create homogeneous image 20 | img = mk_image(imdl); 21 | 22 | % Create stimulation patterns 23 | [stim,~] = mk_stim_patterns(num_elec,1,inj_pattern,meas_pattern,{'meas_current','balance_inj','balance_meas'},stim_current); 24 | 25 | % Set the contact impedences 26 | for j=1:num_elec 27 | img.fwd_model.electrode(j).z_contact = contact_imp; 28 | nodej = img.fwd_model.electrode(j).nodes; 29 | 30 | % Adjust electrode size (affected by mesh fidelity) 31 | img.fwd_model.electrode(j).nodes = [nodej,nodej+2,nodej+4]; 32 | end 33 | 34 | img.fwd_model.stimulation = stim; 35 | img.fwd_solve.get_all_meas = 1; 36 | 37 | N = size(S,1); 38 | [X,Y] = meshgrid(-1+1/N:2/N:1-1/N,-1+1/N:2/N:1-1/N); 39 | c_field = @(x,y,z) interp2(X,Y,S,x,y,'spline'); 40 | img.elem_data = elem_select(img.fwd_model,c_field); 41 | 42 | v = fwd_solve(img); 43 | l = v.meas(1:num_stim*num_elec); 44 | end -------------------------------------------------------------------------------- /model_gwf.m: -------------------------------------------------------------------------------- 1 | function data = model_gwf(U) 2 | 3 | % Model parameters 4 | Dx=6; % Domain = [0,Dx]*[0,Dy] 5 | Dy=6; 6 | J=64; % Number of measurements 7 | Nx=size(U,1); % Number of x-grid points (read from coefficient) 8 | Ny=Nx; % Number of y-grid points 9 | 10 | % Construct the recharge term 11 | Recharge = zeros(Nx,Ny); 12 | Recharge(ceil(4*Nx/6):Nx,:) = 137; 13 | Recharge(floor(5*Nx/6):Nx,:) = 274; 14 | Recharge = reshape(Recharge,1,Nx*Ny)/(Nx*Ny); 15 | 16 | 17 | [measure_x,measure_y]=meshgrid(Nx/(sqrt(J)+1):Nx/(sqrt(J)+1):Nx*(1-1/(sqrt(J)+1)),Ny/(sqrt(J)+1):Ny/(sqrt(J)+1):Ny*(1-1/(sqrt(J)+1))); 18 | measure_x=round(reshape(measure_x,1,J)); 19 | measure_y=round(reshape(measure_y,1,J)); 20 | Well_Location.pressure.x=measure_x; 21 | Well_Location.pressure.y=measure_y; 22 | 23 | Boundary.BN_x=zeros(Ny,2); 24 | Boundary.BN_y=zeros(Nx,2); 25 | for i=1:Ny 26 | Boundary.BN_x(i,2)=-Dy/Ny*500 ; %Dy=1; 27 | end 28 | for i=1:Nx 29 | Boundary.PD(i)=100; 30 | end 31 | 32 | 33 | 34 | 35 | %%%%%%%%%%%%%%%%%%Measurements%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 36 | Grid.Nx=Nx; Grid.hx=Dx/Nx; 37 | Grid.Ny=Ny; Grid.hy=Dy/Ny; 38 | Grid.V=Grid.hx*Grid.hy; 39 | Grid.N=Grid.Nx*Grid.Ny; 40 | Well_Data.Well_Location.pressure=Well_Location.pressure; 41 | Well_Data.Num_Press=length(Well_Data.Well_Location.pressure); 42 | Production.Q=Recharge; 43 | Production.BN_x=Boundary.BN_x; 44 | Production.BN_y=Boundary.BN_y; 45 | Production.PD=Boundary.PD; 46 | Model.Production=Production; 47 | Model.Grid=Grid; 48 | Model.Well_Data=Well_Data; 49 | 50 | 51 | Grid=Model.Grid; 52 | Nx=Grid.Nx; 53 | Ny=Grid.Ny; 54 | N=Nx*Ny; 55 | Model.Rock.K=reshape(U,Nx,Ny); 56 | Model.P=zeros(N,1); 57 | Model.Q=zeros(length(Model.Well_Data.Well_Location.pressure),1); 58 | 59 | 60 | Production=Model.Production; 61 | Rock=Model.Rock; 62 | 63 | % Compute TPFA 64 | Nx=Grid.Nx; Ny=Grid.Ny; N=Nx*Ny; 65 | hx=Grid.hx; hy=Grid.hy; 66 | L=(Rock.K); 67 | 68 | tx=hy/hx; TX=zeros(Nx+1,Ny); 69 | ty=hx/hy; TY=zeros(Nx,Ny+1); 70 | Average_x=0.5*(L(1:Nx-1,:)+L(2:Nx,:)); 71 | TX(2:Nx,:)=Average_x.*tx; 72 | Average_y=0.5*(L(:,1:Ny-1)+L(:,2:Ny)); 73 | TY(:,2:Ny)=Average_y.*ty; 74 | TY(1:Nx,1)=ty.*L(:,1)*2; 75 | 76 | x1=reshape(TX(1:Nx,:),N,1); x2=reshape(TX(2:Nx+1,:),N,1); 77 | y1=reshape(TY(:,1:Ny),N,1); y2=reshape(TY(:,2:Ny+1),N,1); 78 | 79 | for i=1:Nx 80 | f(i)=-y1(i)*Production.PD(i); 81 | end 82 | 83 | DiagVecs=[-y2,-x2,x1+x2+y1+y2,-x1,-y1]; 84 | DiagIndx=[-Nx,-1,0,1,Nx]; 85 | A=spdiags(DiagVecs,DiagIndx,N,N); 86 | 87 | 88 | for i=1:Grid.Nx 89 | Production.Q(i)=Production.Q(i)-f(i); 90 | end 91 | for j=1:Grid.Ny 92 | Production.Q(1+(j-1)*Grid.Nx)=Production.Q(1+(j-1)*Grid.Nx)-Production.BN_x(j,2); 93 | Production.Q(Grid.Nx+(j-1)*Grid.Nx)=Production.Q(Grid.Nx+(j-1)*Grid.Nx)-Production.BN_x(j,1); 94 | end 95 | for j=1:Grid.Nx 96 | Production.Q(j+(Grid.Ny-1)*Grid.Nx)=Production.Q(j+(Grid.Ny-1)*Grid.Nx)-Production.BN_y(j,2); 97 | end 98 | 99 | Model.P=A\Production.Q'; 100 | 101 | u=reshape(Model.P,Nx,Ny); 102 | data=[]; 103 | for i=1:1:length(Model.Well_Data.Well_Location.pressure.x) 104 | data=[data;u(Model.Well_Data.Well_Location.pressure.x(i),Model.Well_Data.Well_Location.pressure.y(i))]; 105 | end 106 | 107 | end 108 | -------------------------------------------------------------------------------- /model_id.m: -------------------------------------------------------------------------------- 1 | function l = model_id(S) 2 | 3 | % Number of observation points 4 | J = 100; 5 | 6 | % Find appropriate entries of S and return them 7 | N = size(S,1); 8 | step = round(N/(sqrt(J)+1)); 9 | 10 | obs_base = step:step:sqrt(J)*step; 11 | obs = obs_base + (step-1)*N; 12 | 13 | for p=2:sqrt(J); 14 | obs = [obs,obs_base + (step*p-1)*N]; 15 | end 16 | 17 | l = S(obs)'; 18 | 19 | end -------------------------------------------------------------------------------- /run_mcmc.m: -------------------------------------------------------------------------------- 1 | %% Define the parameters 2 | 3 | % Model choice 4 | model = 'id'; % Choice of model: 'id', 'gwf' or 'eit' 5 | 6 | % Sample resolution 7 | N = 80; % Sampling performed on NxN square grid 8 | NT = 320; % Data generated on NTxNT square grid 9 | 10 | % Prior parameters 11 | prior.U.alpha = 4; % (Conditional) field prior: N(0,(-Delta + tau^2)^(-alpha)) 12 | prior.tau.mean = 20; % Lengthscale prior: N(mean,std^2) 13 | prior.tau.std = 10; 14 | 15 | % Data parameters 16 | data.gamma = 0.3; % Noise standard deviation 17 | 18 | % MCMC parameters 19 | mcmc.beta = 0.05; % pCN jump for U 20 | mcmc.eps = 2; % RWM jump for tau 21 | mcmc.samples = 5000000; % Number of samples to generate 22 | 23 | % Output parameters 24 | output.figures = 1; % Display figures? 25 | output.thinning = 100; % How often to update output? 26 | 27 | %% Create the data (or alternatively import from a file) 28 | 29 | % Define the true geometric field 30 | tauT = 15; 31 | UT = gaussrnd(prior.U.alpha,tauT,NT); 32 | UT_phys = make_lvl(idct2(reshape(UT,NT,NT)),tauT,prior.U.alpha); 33 | 34 | % Create the data 35 | data.clean = ell(UT_phys,model); 36 | data.J = length(data.clean); 37 | eta = normrnd(0,1,data.J,1); 38 | data.noisy = data.clean + data.gamma*eta; 39 | 40 | 41 | %% Define the potential 42 | Phi = @(U,tau) norm((ell(make_lvl(idct2(reshape(U,N,N)),tau,prior.U.alpha),model) - data.noisy)./data.gamma)^2/2; 43 | 44 | 45 | %% Run the MCMC 46 | 47 | % Define the initial MCMC state 48 | tau = 60; 49 | U = gaussrnd(prior.U.alpha,tau,N); 50 | 51 | phiU = Phi(U,tau); 52 | 53 | % Track the acceptance rates for the proposals 54 | mcmc.accepted.field = 0; 55 | mcmc.accepted.tau = 0; 56 | 57 | % Store the traces of tau and the lowest frequency modes 58 | traceTau = zeros(mcmc.samples,1); 59 | traceU = zeros(mcmc.samples,8); 60 | 61 | % Perform the MCMC 62 | for s=1:mcmc.samples 63 | traceTau(s) = tau; 64 | traceU(s,:) = [U(2),U(3),U(N+1),U(N+2),U(N+3),U(2*N+1),U(2*N+2),U(2*N+3)]; 65 | 66 | % Provide output every output.thinning samples 67 | if mod(s,output.thinning) == 0 68 | fprintf('s = %i\n',s); 69 | fprintf('Acceptance rate for U:\t\t%f\n',mcmc.accepted.field/s); 70 | fprintf('Acceptance rate for tau:\t%f\n\n',mcmc.accepted.tau/s); 71 | 72 | if output.figures == 1 73 | display_figures; 74 | end 75 | end 76 | 77 | %% Update U | tau,y 78 | 79 | % Generate the pCN proposal 80 | UJump = gaussrnd(prior.U.alpha,tau,N); 81 | UProp = sqrt(1-mcmc.beta^2)*U + mcmc.beta*UJump; 82 | 83 | % Calculate the acceptance probability aProb 84 | phiUProp = Phi(UProp,tau); 85 | aProb = min(1,exp(phiU - phiUProp)); 86 | 87 | % Accept proposal with probability aProb 88 | if unifrnd(0,1) < aProb 89 | U = UProp; 90 | phiU = phiUProp; 91 | mcmc.accepted.field = mcmc.accepted.field + 1; 92 | end 93 | 94 | %% Update tau | U,y 95 | 96 | % Generate the RWM proposal 97 | tauJump = normrnd(0,1); 98 | tauProp = tau + mcmc.eps*tauJump; 99 | 100 | % Calculate the acceptance probability aProb 101 | phiUProp = Phi(U,tauProp); 102 | 103 | [K1,K2] = meshgrid(0:N-1,0:N-1); 104 | eigL = reshape((pi^2*(K1.^2+K2.^2) + tau^2).^prior.U.alpha,N^2,1); 105 | eigLProp = reshape((pi^2*(K1.^2+K2.^2) + tauProp^2).^prior.U.alpha,N^2,1); 106 | summand = (eigL - eigLProp).*U.^2./N^2 + log(eigLProp./eigL); 107 | 108 | summand(1) = 0; 109 | propSum = sum(summand); 110 | aProb = min(1,exp(phiU - phiUProp + ((tau-prior.tau.mean)^2 - (tauProp - prior.tau.mean)^2)/(2*prior.tau.std^2) + propSum/2)); 111 | 112 | % Accept proposal with probability aProb 113 | if unifrnd(0,1) < aProb 114 | tau = tauProp; 115 | phiU = phiUProp; 116 | mcmc.accepted.tau = mcmc.accepted.tau + 1; 117 | end 118 | 119 | end 120 | 121 | 122 | 123 | 124 | 125 | 126 | 127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 | 151 | 152 | 153 | 154 | --------------------------------------------------------------------------------