From 48e0c1d68bc6a7e3e067d6aab3642f187d41b7bd Mon Sep 17 00:00:00 2001 From: Oliver Schmidt Date: Fri, 13 Sep 2024 18:16:38 -0700 Subject: [PATCH] Adaptive sine-taper SPOD --- README.md | 7 + cavity_data/cavityPIV.mat | Bin 0 -> 3721470 bytes example_10_sineAdaptive.m | 108 ++++++++ spod_adapt.m | 515 ++++++++++++++++++++++++++++++++++++++ 4 files changed, 630 insertions(+) create mode 100644 cavity_data/cavityPIV.mat create mode 100644 example_10_sineAdaptive.m create mode 100644 spod_adapt.m diff --git a/README.md b/README.md index b1d09b7..72f6de3 100644 --- a/README.md +++ b/README.md @@ -4,6 +4,9 @@ SPOD() is a Matlab implementation of the frequency domain form of proper orthogo The large-eddy simulation data provided along with this example is a subset of the database of a Mach 0.9 turbulent jet described in [3] and was calculated using the unstructured flow solver Charles developed at Cascade Technologies. If you are using the database in your research or teaching, please include explicit mention of Brès et al. [3]. The test database consists of 5000 snapshots of the symmetric component (m=0) of a round turbulent jet. `spod.m` is a stand-alone Matlab function with no toolbox dependencies. All other Matlab files contained in this repository are related to the six examples that demonstrate the functionality of the code (see file descriptions below). A physical interpretation of the results obtained from the examples can be found in [[4](https://arxiv.org/abs/1711.06296)]. The reference for the frequency-time analsyis is [[5](https://arxiv.org/abs/2011.03644)]. + +`spod_adaptive.m` is the adaptive sine-taper SPOD algorithm for broadband-tonal flows by Yeung & Schmidt [[7](https://arxiv.org/abs/2312.02385)] + ## Download ### Using your browser @@ -19,6 +22,7 @@ git clone https://github.com/SpectralPOD/spod_matlab.git | File | Description | | ------------- |:-------------| | spod.m | Spectral proper orthogonal decomposition in Matlab | +| spod_adaptive.m | Adaptive sine-taper SPOD in Matlab | | example_1.m | Inspect data and plot SPOD spectrum | | example_2.m | Plot SPOD spectrum and inspect SPOD modes | | example_3.m | Specify spectral estimation parameters and use weighted inner product | @@ -28,6 +32,7 @@ git clone https://github.com/SpectralPOD/spod_matlab.git | example_7_FTanalysis.m | Frequency-time analysis | | example_8_invspod.m | Band-pass filtering using (inverse) SPOD | | example_9_multitaperWelch | SPOD using Multitaper-Welch estimators | +| example_10_sineAdaptive | Adaptive SPOD example| | tcoeffs.m | Time-continuous expansion coefficients via convolution | | invspod.m | Inversion of SPOD using block-wise expansion coefficients | | jet_data/getjet.m | Interfaces external data source with SPOD() (examples 4-5) | @@ -125,3 +130,5 @@ the SPOD. [5] Nekkanti, A. and Schmidt, O. T., *Frequency–time analysis, low-rank reconstruction and denoising of turbulent flows using SPOD* , J. of Fluid Mech. 926, A26, DOI 10.1017/jfm.2021.681, 2021 [6] Schmidt, O. T., *Spectral proper orthogonal decomposition using multitaper estimates*, Theor. Comput. Fluid Dyn., 1-14, DOI 10.1007/s00162-022-00626-x, https://rdcu.be/cUtP3, 2022 + +[7] Yeung, B. C. Y., Schmidt, O. T. *Adaptive spectral proper orthogonal decomposition of broadband-tonal flows*, Theor. Comput. 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z4oO*mU1v^!(C6C;oWoiW2z~bffqtLHn9H~zdSH7BR`9{UVjR98AQ2*v$o{W!7T!Va zVM{o_8axH!1{f}g^%VbkO{=GVbdaOU4x|*PI#X#ON0kfL;6-btG?1gpbDt8m=s!|J zjw*A#*7+toMFlykY@J2@&}yF&a#T4#WB?p_h7xjAdD13zvRIe`a#Z;rRlUf5HFC&N ze7{G%ZRW75Frl~o75^_{oV5H6zR}KeqRJkZmu0Y)t1Uag#GjwM{Snc3J!z0S7 z_IEwoD}R8GDy!^-pSEEB20p5+5EGrT6}ARGs_ZSpY{odg1U{FiZVWgH zcPA(wYrK7Vyvahdr%~h`qN|BFt1;B1 + disp('Computing multitaper SPOD') +else + disp('Computing SPOD') +end +disp('------------------------------------') +nSamples = nBlks*max(nWin); +if ~isfield(opts,'nsave'), opts.nsave = nBlks*min(nWin(1:opts.savefreqs(end))); end +L = zeros(nFreq,nSamples,precision); +A = zeros(nFreq,nSamples,nSamples,precision); +P = zeros(opts.savefreqs(end),nx,opts.nsave,precision); +for iFreq = opts.savefreqs + nTapers = nWin(iFreq); + nSamples = nBlks*nTapers; + + % parabolic weights penalize high-order sine windows + taperWeights = parabWeights(nTapers); + taperWelchWeights = repmat(taperWeights.',nBlks,1)/nBlks; + taperWelchWeights = taperWelchWeights(:).'; + + disp(['frequency ' num2str(iFreq) '/' num2str(nFreq) ' (f=' num2str(f(iFreq),'%.3g') ') with ' num2str(nTapers) ' windows']) + + % recover windowed DFT from unwindowed DFT using sine window central difference + Q_blk_hat_fi_l = Q_hat(:,:,wrapIdx((iFreq-1)*2+1-(1:nTapers),nDFT*2)); + Q_blk_hat_fi_u = Q_hat(:,:,wrapIdx((iFreq-1)*2+1+(1:nTapers),nDFT*2)); + Q_hat_f = sinetapNorm*(Q_blk_hat_fi_l-Q_blk_hat_fi_u)/(2i); + + Q_hat_f = reshape(Q_hat_f,[nx nSamples]); + Q_hat_f = Q_hat_f.*sqrt(taperWelchWeights); + + if ~opts.compress + M = Q_hat_f'*bsxfun(@times,Q_hat_f,weight); + else + M = Q_hat_f'*Q_hat_f; + end + [Theta,Lambda] = eig(M); + Lambda = diag(Lambda); + [Lambda,idx] = sort(Lambda,'descend'); + Theta = Theta(:,idx); + Psi = Q_hat_f*Theta(:,1:opts.nsave)*diag(1./sqrt(Lambda(1:opts.nsave))/sqrt(nSamples)); + P(iFreq,:,:) = Psi(:,1:opts.nsave); + A(iFreq,1:nSamples,1:nSamples) = diag(sqrt(Lambda))*Theta'.*sqrt(1./taperWelchWeights); + L(iFreq,1:nSamples) = abs(Lambda); + % correct for one-sided spectrum + if isrealx && iFreq~=1 && iFreq~=nFreq + L(iFreq,:) = 2*L(iFreq,:); + end +end +clear Q_hat + +% recover SPOD modes from compression basis +if opts.compress + nx = prod(dim(2:end)); + P = permute(P,[2 3 1]); + P = U*P(:,:)./sqrtW; + P = reshape(P,[nx opts.nsave opts.savefreqs(end)]); + P = permute(P,[3 1 2]); +end +P = reshape(P,[opts.savefreqs(end) dim(2:end) opts.nsave]); + +% return the same number of eigenvalues and coefficients for all frequencies +A = A(:,1:nBlks*min(nWin(1:opts.savefreqs(end))),:); +L = L(:,1:nBlks*min(nWin(1:opts.savefreqs(end)))); +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +function [weight,nOvlp,dt,nDFT,nBlks,nWin] = spod_parser(nt,nx,isrealx,varargin) +%SPOD_PARSER Parser for SPOD parameters + +% read input arguments from cell array +window = []; weight = []; nOvlp = []; dt = []; adaptive = false; +nvarargin = length(varargin); + +if nvarargin >= 1 + window = varargin{1}; + if nvarargin >= 2 + weight = varargin{2}; + if nvarargin >= 3 + nOvlp = varargin{3}; + if nvarargin >= 4 + dt = varargin{4}; + if nvarargin >= 5 + opts = varargin{5}; + if isfield(opts,'adaptive') + adaptive = opts.adaptive; + end + end + end + end + end +end + +% check window contains only integers +if ~isempty(window) + assert(sum(window~=round(window))==0,'Window must be one of: integer scalar | integer vector | [].') +end + +% check arguments and determine default spectral estimation parameters +% window size and type +window_name = 'sine'; +if isempty(window) + nDFT = nt; + nWin = 3*ones(nDFT,1); +elseif length(window)==1 + nDFT = window; + nWin = ones(nDFT,1); +elseif length(window)>=2 + nDFT = length(window); + nWin = window; +end + +if adaptive + nWin = []; +end + +weight = weight(:); + +% block overlap +if isempty(nOvlp) + nOvlp = 0; +elseif nOvlp > nDFT-1 + error('Overlap too large.') +end + +% time step between consecutive snapshots +if isempty(dt) + dt = 1; +end + +% inner product weight +if isempty(weight) + weight = ones(nx,1); + weight_name = 'uniform'; +elseif numel(weight) ~= nx + error('Weights must have the same spatial dimensions as data.'); +else + weight_name = 'user specified'; +end + +% number of blocks +nBlks = floor((nt-nOvlp)/(nDFT-nOvlp)); + +% test feasibility +if ~isempty(nWin) + if nDFT < 4 || nBlks*min(nWin(nWin>0)) < 3 + error('Spectral estimation parameters not meaningful.'); + end +end + +% display parameter summary +disp(' ') +if ~isempty(nWin) + if max(nWin)==1 + disp('SPOD parameters') + else + disp('Multitaper SPOD parameters') + end +else + disp('Adaptive SPOD parameters') +end +disp('------------------------------------') +if isrealx + disp('Spectrum type : one-sided (real-valued signal)') +else + disp('Spectrum type : two-sided (complex-valued signal)') +end +disp(['No. of snaphots per block : ' num2str(nDFT)]) +disp(['Block overlap : ' num2str(nOvlp)]) +disp(['No. of blocks : ' num2str(nBlks)]) +disp(['Windowing fct. (time) : ' window_name]) +disp(['Weighting fct. (space) : ' weight_name]) +if adaptive + disp(num2str(opts.tol,'No. of windows : adaptive tol = %g')) +elseif max(nWin)==min(nWin) + disp(['No. of windows : ' num2str(nWin(1))]) +else + disp(num2str([min(nWin) max(nWin)],'No. of windows : varies between %i and %i')) +end +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +function [mu_k] = parabWeights(K) +%PARABWEIGHTS Parabolic weights corresponding to K sine tapers + Nk = K*(4*K-1)*(K+1)/6; + k = (1:K).'; + mu_k = 1/Nk*(K^2-(k-1).^2); +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +function TF = anynan(A) +%ANYNAN checks if A contains NaNs +TF = sum(isnan(A))>0; +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +function similarity = calcModeSimilar(q1,q2) +%MODESIMILAR calculates the similarity between complex SPOD modes q1 and q2 + +q1 = double(q1(:)); +q2 = double(q2(:)); + +similarity = dot(q1,q2)/norm(q1)/norm(q2); +similarity = abs(similarity); +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +function nWin = adaptiveWindows(Q_hat,wrapIdx,sinetapNorm,opts) +%ADAPTIVEWINDOWS implements the adaptive algorithm and returns a frequency-dependent nWin + +% get data parameters +nx = size(Q_hat,1); +nBlks = size(Q_hat,2); +nDFT = size(Q_hat,3)/2; + +disp(' ') +disp('Adaptively converging nWin') +disp('------------------------------------') + +% store only the leading mode to track mode convergence +P = zeros(opts.savefreqs(end),nx,'like',Q_hat); + +% initialize window numbers +nWin = nan(size(opts.savefreqs)); +allConverged = false; +if nBlks>=3, nTapers_init = 1; else, nTapers_init = 3; end +nTapers = nTapers_init - 1; + +% increment window numbers until all frequencies are converged to within tolerance +while ~allConverged + nTapers = nTapers+1; + if nTapers>nTapers_init + P_prev = P; + end + unconvergedFreqs = opts.savefreqs(isnan(nWin)); + disp(['Converging ' num2str(length(unconvergedFreqs)) '/' num2str(length(opts.savefreqs)) ' frequencies using ' num2str(nTapers) ' windows']) + + % SPOD for unconverged frequencies + for iFreq = unconvergedFreqs + nSamples = nBlks*nTapers; + taperWeights = parabWeights(nTapers); + taperWelchWeights = repmat(taperWeights.',nBlks,1)/nBlks; + taperWelchWeights = taperWelchWeights(:).'; + + Q_blk_hat_fi_l = Q_hat(:,:,wrapIdx((iFreq-1)*2+1-(1:nTapers),nDFT*2)); + Q_blk_hat_fi_u = Q_hat(:,:,wrapIdx((iFreq-1)*2+1+(1:nTapers),nDFT*2)); + Q_hat_f = sinetapNorm*(Q_blk_hat_fi_l-Q_blk_hat_fi_u)/(2i); + Q_hat_f = reshape(Q_hat_f,[nx nSamples]); + Q_hat_f = Q_hat_f.*sqrt(taperWelchWeights); + + if nSamples<=nx + M = Q_hat_f'*Q_hat_f; + else + M = Q_hat_f*Q_hat_f'; + end + + if size(M,1)<60 + [Theta,Lambda] = eig(M); + Lambda = diag(Lambda); + [Lambda,idx] = max(Lambda); + Theta = Theta(:,idx); + else + [Theta,Lambda] = eigs(double(M),1); + end + if nSamples<=nx + Psi = Q_hat_f*Theta*diag(1./sqrt(Lambda)/sqrt(nSamples)); + else + Psi = Theta/sqrt(nSamples); + end + + P(iFreq,:) = Psi; + + % check if mode is converged + if nTapers>nTapers_init + similarity = calcModeSimilar(P_prev(iFreq,:),P(iFreq,:)); + if (1-similarity)<=opts.tol + nWin(iFreq) = nTapers; + else + if iFreq1 + if nTapers-nWin(iFreq-1)==1 + nWin(iFreq) = nTapers; + end + end + end + end + end + if ~anynan(nWin) + allConverged = true; + end +end +end \ No newline at end of file

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