High-Dimensional Bayesian Inversion With Priors Far From .
High-Dimensional Bayesian Inversionwith Priors Far from GaussiansFelix LuckaUniversity College Londonf.lucka@ucl.ac.ukMartin BurgerUniversity of Münstermartin.burger@uni-muenster.deSIAM-UQ, April 6, 2016.
Bayesian Inference for Inverse ProblemsLinear ill-posed inverse problem with additive Gaussian noise:f Au εplike (f u) exp 21 kf A uk22 pprior (u) exp λ kD T uk22 ppost (u f ) exp 21 kf A uk22 λ kD T uk22 Probabilistic representation allows for a rigorous quantification of thesolution’s uncertainties.
Bayesian Inference for Inverse ProblemsLinear ill-posed inverse problem with additive Gaussian noise:f Au εplike (f u) exp 21 kf A uk22 pprior (u) exp λ kD T uk1 ppost (u f ) exp 21 kf A uk22 λ kD T uk1 Probabilistic representation allows for a rigorous quantification of thesolution’s uncertainties.
Sparsity / Compressible Representation(a) 100%(b) 10%(c) 1%Sparsity a-priori constraints are used in variational regularization,compressed sensing and variable selection: ûλ argmin 12 kf A uk22 λkD T uk1u(e.g. total variation, wavelet shrinkage, LASSO,.)Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians2
Sparsity / Compressible Representation(a) 100%(b) 10%(c) 1%Sparsity a-priori constraints are used in variational regularization,compressed sensing and variable selection: ûλ argmin 12 kf A uk22 λkD T uk1u(e.g. total variation, wavelet shrinkage, LASSO,.)How about sparsity as a-priori information in the Bayesian approach?Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians2
Bayesian Inference with 1 Priorsppost (u f ) exp 12 kf A uk22 λ kD T uk1 Aims: Bayesian inversion in high dimensions (n ):MAP vs. CM, characterization of posterior structure.Priors: Simple 1 , total variation (TV), Besov space priors.u†λ 10λ 40λ 160λ 640λ 25601Starting points:Lassas, Siltanen, 2004. Can one use total variation prior foredge-preserving Bayesian inversion?, Inverse Problems, 20.Lassas, Saksman, Siltanen, 2009. Discretization invariantBayesian inversion and Besov space priors, Inverse Problemsand Imaging, 3(1).Kolehmainen, Lassas, Niinimäki, Siltanen, 2012.Sparsity-promoting Bayesian inversion, Inverse Problems,28(2).001/32/31
Efficient MCMC Techniques for 1 PriorsTask: Monte Carlo integration by samples fromppost (u f ) exp 12 kf A uk22 λ kD T uk1 Problem: Standard Markov chain Monte Carlo (MCMC) sampler(Metropolis-Hastings) inefficient for large n or λ.Contributions:IDevelopment of explicit single component Gibbs sampler.ITedious implementation for different scenarios.IStill efficient in high dimensions (n 106 ).IDetailed evaluation and comparison to MH.L, 2012. Fast Markov chain Monte Carlo sampling for sparseBayesian inference in high-dimensional inverse problems usingL1-type priors, Inverse Problems, 28(12):125012.
Efficient MCMC Techniques for 1 Priors(a) Reference(b) MH-Iso, 1h(c) MH-Iso, 4h(d) MH-Iso, 16h(e) Reference(f) SC Gibbs, 1h(g) SC Gibbs, 4h(h) SC Gibbs, 16hDeconvolution, simple 1 prior, n 513 513 263 169.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians5
Recent Generalization: Slice-Within-Gibbs Samplingpprior (u) exp λkD T uk1 Limitations:ID must be diagonalizable (synthesis priors): qp -prior: exp λkD T ukqp ? TV in 2D/3D?INon-negativity or other hard-constraints?IContributions:IReplace explicit by generalized slice sampling.IImplementation & evaluation for most common priors.Neal, 2003. Slice Sampling, Annals of Statistics 31(3).L, 2016. Fast Gibbs sampling for high-dimensional Bayesianinversion, submitted, arXiv:1602.08595.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians6
Image Deblurring Example in 2D(a) Unknown function ũ(b) Data fDeconvolution, simple 1 prior, n 1023 1023 1 046 529.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians7
Image Deblurring Example in 2D(a) Unknown function ũ(b) CM estimate by our Gibbs samplerDeconvolution, simple 1 prior, n 1023 1023 1 046 529.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians8
Image Deblurring Example in 2D(a) Unknown function ũ(b) MAP estimate by ADMMDeconvolution, simple 1 prior, n 1023 1023 1 046 529.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians9
Discretization Dilemma of the TV prior (Lassas & Siltanen, 2004)”Can one use total variation prior for edge-preserving Bayesian inversion?”I For λn const. and n the TV prior diverges.I CM diverges.I MAP converges to edge-preserving limit.u †, n 63n 255n 1 023n 4 095n 16 383n 65 535u †, nnnnnn1 632551 0234 09516 38365 53500011/32/3(a) CM by our Gibbs Sampler1 01/32/3(b) MAP by ADMM1
Discretization Dilemma of the TV prior (Lassas & Siltanen, 2004)”Can one use total variation prior for edge-preserving Bayesian inversion?”I For λn const. and n the TV prior diverges.I CM diverges.I MAP converges to edge-preserving limit.u †, n 63n 255n 1 023n 4 095n 16 383n 65 535u †, û aCMû bCM111/32/3(a) Zoom into CM estimates1/32/3(b) MCMC convergence check
Discretization Dilemma of the TV prior (Lassas & Siltanen, 2004)”Can one use total variation prior for edge-preserving Bayesian inversion?” I For λn n 1 and n the TV prior converges to asmoothness prior.I CM converges to smooth limit.I MAP converges to constant.u †, 1nnnnnn 00u †, 1632551 0234 09516 38365 535nnnnnn 632551 0234 09516 38365 53501/32/3(a) CM by our Gibbs Sampler1 01/32/3(b) MAP by ADMM1
TV-p Priors as an Alternative? ppost (u) exp 12 kf A uk2Σ 1 λ kD T ukppεu †, p 1.4p 1.2p 1.0p 0.8100u †, p 1.4p 1.2p 1.0p 0.8101/32/3(a) CM (Gibbs-MCMC)1 01/32/31(b) MAP (Simulated Annealing)Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians13
Need for New Theoretical PredictionsFor images dimensions 1: No theory yet.but we can compute it.Test scenario:ICT using only 45 projection angles and 500 measurement pixel.real solutiondata fcolormapFelix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians14
Need for New Theoretical PredictionsFor images dimensions 1: No theory yet.but we can compute it.MAP, n 642 , λ 500CM, n 642 , λ 500Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians15
Need for New Theoretical PredictionsFor images dimensions 1: No theory yet.but we can compute it.MAP, n 1282 , λ 500CM, n 1282 , λ 500Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians15
Need for New Theoretical PredictionsFor images dimensions 1: No theory yet.but we can compute it.MAP, n 2562 , λ 500CM, n 2562 , λ 500cf. Louchet, 2008, Louchet & Moisan, 2013 for the denoising case, A I .Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians15
Examination of Besov Space Priors by MCMCAn 1 -type, wavelet-based prior:pprior (u) exp λkWV T uk1 motivated by:M. Lassas, E. Saksman, S. Siltanen, 2009.Discretization invariant Bayesian inversion andBesov space priors, Inverse Probl Imaging, 3(1).V. Kolehmainen, M. Lassas, K. Niinimäki, S.Siltanen, 2012. Sparsity-promoting Bayesianinversion, Inverse Probl, 28(2).K. Hämäläinen, A. Kallonen, V.Kolehmainen, M. Lassas, K. Niinimäki, S.Siltanen, 2013. Sparse Tomography, SIAM JSci Comput, 35(3).Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians16
Sparse-Angle-CT with Discretization Invariant Besov PriorsReconstructions for λ 2e4, n 64 64 4.096MAP estimate (by ADMM)CM estimate (by our Gibbs sampler)Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians17
Sparse-Angle-CT with Discretization Invariant Besov PriorsReconstructions for λ 2e4, n 128 128 16.384MAP estimate (by ADMM)CM estimate (by our Gibbs sampler)Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians18
Sparse-Angle-CT with Discretization Invariant Besov PriorsReconstructions for λ 2e4, n 256 256 65.536MAP estimate (by ADMM)CM estimate (by our Gibbs sampler)Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians19
Sparse-Angle-CT with Discretization Invariant Besov PriorsReconstructions for λ 2e4, n 512 512 262.144MAP estimate (by ADMM)CM estimate (by our Gibbs sampler)Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians20
Sparse-Angle-CT with Discretization Invariant Besov PriorsReconstructions for λ 2e4, n 1024 1024 1.048.576MAP estimate (by ADMM)CM estimate (by our Gibbs sampler)Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians21
Application to Experimental Data: Walnut-CTICooperation with Samuli Siltanen, Esa Niemi et al.IImplementation of MCMC methods for Fanbeam-CT.IBesov and TV prior; non-negativity constraints.IStochastic noise modeling.IBayesian perspective on limited angle CT.Use the data set for your own work:http://www.fips.fi/dataset.php (documentation: arXiv:1502.04064)
Walnut-CT with TV Prior: Full vs. Limited Angle(a) MAP, full(b) CM, full(c) CStd, full(d) MAP, limited(e) CM, limited(f) CStd, limited
New Light on an Old Debate: MAP vs. CM EstimatesZûMAP : argmax { ppost (u f )}OR ûCM : u ppost (u f ) duu RnObservations.IGaussian priors: MAP CM. Funny coincidence?IFor reasonable non-Gaussian priors, MAP are sparser,sharper, look and perform better.Ilf the CM looks good, it looks like the MAP.IUQ wrt the CM ( variance) might not be interesting.IGribonval, Marchart, Louchet and Moisan, 2011-2013:CM are MAP for different priors.CM.are in contradiction with classical Bayes cost formalismwhich discriminates MAP ( variational regularization) andadvocates CM.CMMAPMAPFelix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians24
The Classical Bayes Cost ArgumentIAn estimator is a random variable, as it relies on f and u.IHow does it perform on average? Which estimator is ”best”?IDefine a cost function Ψ (u, v ).IBayes cost is the expected cost:ZZBC (û) Ψ (u, û(f )) plike (f u) df pprior (u) duIBayes estimator ûBC for given Ψ minimizes Bayes cost. Turns out: Z ûBC (f ) argminΨ (u, û(f )) ppost (u f ) duûFelix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians25
The Classical Bayes Cost ArgumentICM is Bayes estimator for Ψ (u, û) ku ûk22 (MSE).IAlso the minimum variance estimator.IThe mean value is the intuitive ”average”, the ”center of mass”.IMAP is asymptotic Bayes estimator of(0, if ku ûk 6 Ψ (u, û) 1 otherwise,for 0 (uniform cost). Not a proper Bayes estimator.MAP and CM seem fundamentally different one should decide!I“A real Bayesian would not use the MAP estimate”IPeople feel ”ashamed” when they have to compute MAP estimates(even when their results are good).Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians26
A False Conclusion“A real Bayesian would not use the MAP estimate as it is not a properBayes estimator”.”MAP estimate can be seen as an asymptotic Bayes estimator of(0, if ku ûk Ψ (u, û) 1 otherwise,for 0. It is not a proper Bayes estimator.”Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians27
A False Conclusion“A real Bayesian would not use the MAP estimate as it is not a properBayes estimator”.”MAP estimate can be seen as an asymptotic Bayes estimator of(0, if ku ûk Ψ (u, û) 1 otherwise,for 0. ? ? It is not a proper Bayes estimator.”Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians27
A False Conclusion“A real Bayesian would not use the MAP estimate as it is not a properBayes estimator”.”MAP estimate can be seen as an asymptotic Bayes estimator of(0, if ku ûk Ψ (u, û) 1 otherwise,for 0. ? ? It is not a proper Bayes estimator.””MAP estimator is asymptotic Bayes estimator for some degenerate Ψ ”; “MAP can’t be Bayes estimator for some proper Ψ ” !!!!Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians27
A False Conclusion“A real Bayesian would not use the MAP estimate as it is not a properBayes estimator”.”MAP estimate can be seen as an asymptotic Bayes estimator of(0, if ku ûk Ψ (u, û) 1 otherwise,for 0. ? ? It is not a proper Bayes estimator.””MAP estimator is asymptotic Bayes estimator for some degenerate Ψ ”; “MAP can’t be Bayes estimator for some proper Ψ ” !!!!We need new cost functions!Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians27
Bregman distancesFor a proper, convex functional J : Rn R { }, the Bregmandistance DJp (u, v ) between u, v Rn for a subgradient p J (v ) isdefined asDJp (u, v ) J (u) J (v ) hp, u v i,DJ (u, v) J (u)J 0 (v)(uJ (v)p J (v )qDJ(u, v) J (u)v)J (v)q(uv)with q 2 @J (v)J (x)J (x)rDJ(w, v)DJ (u, v)00J (v) J 0 (v)(xu0v(g) J (x) x 2v)pDJ(u, v)wuvJ (v) p(xv)J (v) r(xv)p, r 2 @J (v) [ 1, 1](h) J (x) x Basically, DJ (u, v ) measures the difference between J and itslinearization in v at another point u.
Two New Bayes Cost Functions ppost (u f ) exp 12 kf A uk22 λ J (u)with J proper, convex (prior is log-concave).Definition:(a) ΨLS (u, û) : kA(û u)k22 βkL(û u)k22(b) ΨBrg (u, û) : kA(û u)k22 2λDJ (û, u)for a regular L, β 0.Properties:I Proper, convex cost functionsI For J (u) β/λkLuk2 (Gaussian case!) we have λDJ (û, u) 2βkL(û u)k22 , and ΨLS (u, û) ΨBrg (u, û)!Theorems:(I) The CM estimate is the Bayes estimator for ΨLS (u, û)(II) The MAP estimate is the Bayes estimator for ΨBrg (u, û)Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians29
Two New Bayes Cost Functions II ppost (u f ) exp 12 kf A uk22 λ J (u)Definition:(a) ΨLS (u, û) : kA(û u)k22 βkL(û u)k22(b) ΨBrg (u, û) : kA(û u)k22 2λDJ (û, u)for a regular L, β 0.Theorems:(I) The CM estimate is the Bayes estimator for ΨLS (u, û)(II) The MAP estimate is the Bayes estimator for ΨBrg (u, û)Non-Gaussian case:I dom(J ) usually defines a (subset of a) Banach space for n .I In such a space: No natural Hilbert space norm as limit of kLuk2 .I Hilbert space norm not meaningful measure, e.g. for functions in BV.I Only choice: L 0 ΨLS only measures in output space, bad forill-posed inverse problems!Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians30
Average Optimality of CM and MAP-Centered PosteriorAverage optimality condition for the CM estimate:Z A (AûCM f ) λp̂CM 0,p̂CM J 0 (u)ppost (u f )duA (AûMAP f ) λp̂MAP 0,p̂MAP J 0 (ûMAP )Difference: J 0 (E(u f ) [u]) 6 E(u f ) [J 0 (u)] (except for Gaussian prior).“The posterior is well centered around the CM but not around the MAPestimate.” Use optimality condition to rewrite posterior in terms of ûMAP : 1ppost (u f ) exp kA(u ûMAP )k22 λDJp̂MAP (u, ûMAP )2Posterior energy is sum of two convex functionals both minimized by ûMAP .Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians31
Uncertainty Quantification: Which Measure?Two new inequalities,E(u f ) kL(ûCM u)k22 6 E(u f ) kL(ûMAP u)k22E(u f ) DJ (ûMAP , u) 6 E(u f ) DJ (ûCM , u)indicate that the use of anisotropic priors calls for different uncertaintymeasures than variance or mean square risks.References:M. Burger, F.L., 2014. Maximum a posteriori estimates in linearinverse problems with log-concave priors are proper Bayes estimators,Inverse Problems, 30(11):114004.T. Helin, M. Burger, 2015. Maximum a posteriori probabilityestimates in infinite-dimensional Bayesian inverse problems, InverseProblems, 31(8):085009.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians32
Summary, Conclusions & OutlookBayesian Modeling:IModeling sparsity with 1 priors can fail: Sometimes, only the MAPis sparse, nothing else.IAlternatives include hierarchical Bayesian models and spike-and-slabpriors.Bayesian Computation:IElementary MCMC samplers may perform very differently.IContrary to common beliefs sample-based Bayesian inversion in highdimensions (n 106 ) is feasible if tailored samplers are developed.IReason for the efficiency of the Gibbs samplers is unclear.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians33
Summary, Conclusions & OutlookBayesian Estimation / Uncertainty QuantificationIMAP estimates are proper Bayes estimators, minimizing a costfunction potentially better suited to asymptotic Banach spacestructure.IBut: Everything beyond ”MAP or CM?” is far more interesting andcan really complement variational approaches.IHowever: Extracting information from posterior samples is anon-trivial (future research) topic.IThe anisotropic structure of the priors calls for different uncertaintymeasures than variance or mean square risks.IBregman distances are interesting tools for Bayesian inversion.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians34
L, 2016. Fast Gibbs sampling for high-dimensional Bayesianinversion. submitted, arXiv:1602.08595L., 2014. Bayesian Inversion in Biomedical ImagingPhD Thesis, University of Münster.M. Burger, L., 2014. Maximum a posteriori estimates in linearinverse problems with log-concave priors are proper Bayes estimatorsInverse Problems, 30(11):114004.L., 2012. Fast Markov chain Monte Carlo sampling for sparseBayesian inference in high-dimensional inverse problems usingL1-type priors.Inverse Problems, 28(12):125012.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians35
Thank you for your attention!L, 2016. Fast Gibbs sampling for high-dimensional Bayesianinversion. submitted, arXiv:1602.08595L., 2014. Bayesian Inversion in Biomedical ImagingPhD Thesis, University of Münster.M. Burger, L., 2014. Maximum a posteriori estimates in linearinverse problems with log-concave priors are proper Bayes estimatorsInverse Problems, 30(11):114004.L., 2012. Fast Markov chain Monte Carlo sampling for sparseBayesian inference in high-dimensional inverse problems usingL1-type priors.Inverse Problems, 28(12):125012.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians35
Efficient MCMC Techniques for 1 Priors1MH, λ 100MH, λ 200MH, λ 400Gibbs, λ 100Gibbs, λ 200Gibbs, λ 4000.8R*( t )0.60.40.2001t (sec)23Temporal autocorrelation R (t) for 1D TV-deblurring, n 63.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians36
Efficient MCMC Techniques for 1 Priors)1MH Iso,RnGibbs,MH Iso,RnGibbs,MH Iso,RnGibbsMH Iso,RnGibbs,0.8R*( t )0.6n 127, λ 280n 127, λ 280n 255, λ 400n 255, λ 400n 511, λ 560n 511, λ 560n 1023, λ 800n 1023, λ 8000.40.20010210t (sec)410610Temporal autocorrelation R (t) for 1D TV-deblurring.Felix Lucka, f.lucka@ucl.ac.uk - High-Dimensional Bayesian Inversion with Priors Far from Gaussians37
edge-preserving Bayesian inversion?, Inverse Problems, 20. Lassas, Saksman, Siltanen, 2009. Discretization invariant Bayesian inversion and Besov space priors, Inverse Problems and Imaging, 3(1). Kolehmainen, Lassas, Niinim aki, Siltanen, 2012 . Sparsity-promoting Bayesian inversion, Inverse Problems, 28(2). 0 1/3 2/3 1 0 1 uy 6 10 6 40 6 .
Computational Bayesian Statistics An Introduction M. Antónia Amaral Turkman Carlos Daniel Paulino Peter Müller. Contents Preface to the English Version viii Preface ix 1 Bayesian Inference 1 1.1 The Classical Paradigm 2 1.2 The Bayesian Paradigm 5 1.3 Bayesian Inference 8 1.3.1 Parametric Inference 8
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PRINCIPIOS PARA LA FORMULACIÓN DE POLÍTICAS DE INVERSIÓN. GUÍA PARA EL DISEÑO DE ACUERDOS INTERNACIONALES DE INVERSIÓN (AII) OPCIONES PARA EL DISEÑO DE AII. Extracto del Marco de Políticas de Inversión para el Desarrollo Sostenible. DE ACUERDOS . INVERSIÓN PARA EL DESARROLLO SOSTENIBLE .
Kalman Filter (SDKF), is used to transform the matrix inversion into an iterative scalar inversion. The proposed design acts as . ing matrix inversion is the modified Gram-Schmidt algorithm, which is based on QR decomposition. Irturk et al. took this approach in [10] and performed matrix inversion for up to an
Administrado por Scotia Fondos, S.A. DE C.V., Sociedad Operadora de Fondos de Inversión, Grupo Financiero Scotiabank Inverlat SCOTIA PREVISIONAL DE LIQUIDEZ RESTRINGIDA, S.A. DE C.V. Fondo de Inversión en Instrumentos de Deuda INSTRUMENTO DE DEUDA SCOTILP FONDOS DE INVERSIÓN
A Nonlinear Dynamic Inversion Predictor-Based Model Reference Adaptive Controller for a Generic Transport Model Stefan F. Campbell and John T. Kaneshige T. A non-linear dynamic inversion control law is then: . applied to the fast dynamic inversion illustrated in Fig. 2.
Unit 2 Phonics and reading 1.Choose the picture that matches the vowel team word CSK 2.Complete the vowel team words E68 3.Complete the word with the correct vowel team HTK 4.Choose the vowel team sentence that matches the picture DJD 5.Choose the r-control word that matches the picture VVD 6.Complete the word with the correct r-controlled vowel: ar, er, ir, or, ur PLR 7.Complete the word with .
