Inverse Problems And Uncertainty Quantification
Inverse Problems andUncertainty QuantificationPer Christian HansenProfessor, Villum InvestigatorSection for Scientific ComputingResearch questionHow to make UQ ageneral and easy-to-usetool for inverse problemsWhat it’s like to do researchHeian Shrine, Kyoto
Inverse Problem: Image DeblurringCamera blur.Rotational blur.2/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Inverse Problem: X-Ray CTImage reconstruction frommeasurements of X-rayattenuation in an object.Medical imagingMaterials science100 µm3/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
So What is an Inverse Problem?In a forward problem, we use a mathematical model to compute the outputfrom a “system” given the input.In an inverse problem we compute/estimate a quantity that is not directlyobservable, using indirect measurements and the forward model.4/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Solving CT Problems, the Algebraic Way5/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Large-Scale ProblemsHow to solve large-scale problems A x b efficiently? Use iterative methods that produce increasing better reconstructions.Computer simulationImage: 128 128.Data: 360 projection angles in 0 –360 , 181 detector pixels.k iteration number6/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Algebraic Iterative Reconstruction MethodsLots of software is available 7/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Dealing with an Unmatching TransposeReconstruction errors No convergence with unmatchedtranspose B AT. Convergence with matchedtranspose AT.k8/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Fixing the ConvergenceDong, H, Hochstenbach, Riis; SISC, 2019.9/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Towards the Villum ProjectUse X-ray scanning to compute cross-sectionalimages of oil pipes on the seabed.Detect defects, cracks, etc. in the pipe.Defect!How much canwe trust thesize and thelocation?Reinforcing bars10/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Computational Uncertainty Quantification for Inverse ProblemsInverse problem: compute hidden features from external data.Data:blurred imageModel ofblurringReconstructionw/ edge priorThe problems are hampered by: measurement errors in the data, errors/uncertainties in the mathematical model, uncertainties in our prior knowledge about the solution.Uncertainty Quantification (UQ) is the study of the impact of all formsof error and uncertainty in the data and models, through the posteriorobtained via Bayes’ rule.Sampling the posterior is computationally challenging and calls forhierarchical prior modeling, model reduction, and many other “tools.”11/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Example: Archeology as an Inverse ProblemWhat did buildings of former times look like?Data: whatever ruins are left.Prior: everything we know about theculture, building styles, aesthetics, etc.Model: a temple that is worn downby the elements over 2000 years.12/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Example: Reconstruction of Viking HallsVery limited data: traces of the sturdytimbers that the hall was built from showas dark patches in the light natural subsoil.There might be many possible solutions!13/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
The UQ Approach to Inverse ProblemsUncertainty Quantification (UQ) is based on Bayesian statistics.Instead of producing a single solution (i.e., x A-1 b) we obtain thedistribution (the posterior) of all possible solutions.Classical computationmethods produce asingle image; but canwe trust it?Black hole?Uncertainty Quantification(UQ) reveals the reliability ofall possible reconstructions.12%14/1917%P. C. Hansen – Inverse Problems and Uncertainty Quantification63%8%Klitgaarden, Nov. 2021
The LowdownMeasuredblurredimage bClassicalsolution xMAPestimateUQ shows the uncertainty(variance) in each pixel;white high uncertainty.15/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Vision: Computational UQ becomes an essential part ofsolving inverse problems in science and engineering.Ingredients Develop formulations of inverse problems that incorporate all uncertaintiesin the data, the models, the assumptions, the computations, etc. Develop mathematical & statistical methods andalgorithms suited for practical applications.Mathematics Create a modeling framework and a computationalplatform for non-experts.Comput. ”engine”16/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
The Computational AspectPhilosophy Hide mathematics, statistics and scientific computing from non-expert users. Give expert users full control of the UQ methods and computations. All users can focusing on their modeling of the inverse problem.Case: UQ for edge-preserving reconstruction Today: 500 lines of codeActual CUQYpy code17/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
dataCase: Goal-Oriented CUQIreconstructionReconstruct the desired quantity directly fromdata, and perform UQ on this quantity.imagesegmentationExample in X-ray imaging: Find inclusion boundaries withouta classical two-stage process & perform UQ on the boundaries.ground truthdataboundaryboundaries w/ UQexact boundarypredicted boundary6σ credibility band 18/192D 1D computational problem, no pixels, no error accumulation.Represent the inclusion boundaries as random-field functions.Assign a hyper-parameter that controls the boundary’s regularity.Perform UQ by assigning probabilities to the functions and their regularity.P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Thanks for your attentionAny questions or uncertainties?19/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Appendix: Fixing the Convergence20/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Appendix: Nonconvergence ConvergenceImage: 128 128. Data: 90 projection angles in 0 –180 , 80 detector pixels.Both A and B are from the GPU-version of the ASTRA toolbox.21/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Appendix: Gaussian Likelihood & Prior22/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Appendix: UQ with Non-Negative PriorIf the prior or likelihood is non-Gaussian, we must sample the posterior:we generate many random instances of the regularized solution with thespecified likelihood and prior.Bardsley, Hansen, MCMC Algorithms for Non-negativity Constrained Inverse Problems, 2019.Mean of samplesWe have an analyticalexpression for the prior,but no analytical expression for the posterior.Hist. of reg. parametersMAP estimateStandard deviationPositron Emission Tomography.Solutions sampled by a newPoisson Hierarchical Gibbs Sampler.23/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Appendix: UQ for Model DiscrepanciesDong, Riis, Hansen, Modeling of sound fields, joint with DTU Elektro, 2019.Actual field”Naive” point source model Point source & model discrep.Described by a Gaussian processMeasureddata PhysicalmodelCannot includeall possibleaspects24/19 ModeldiscrepancyAccounts forknown unknowns &unknown unknownsP. C. Hansen – Inverse Problems and Uncertainty Quantification DataerrorsKnownstatisticsKlitgaarden, Nov. 2021
HD-Tomo: High-Definition TomographyThe following examples are from the projectHD-Tomo, which was funded by an ERCAdvanced Research Grant, 2012–17.25/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
What is an Inverse Problem?In a forward problem, we use a mathematical model to compute the outputfrom a “system” given the input.In an inverse problem we compute/estimate a quantity that is not directlyobservable, using indirect measurements and the forward model.Data:blurred imageData:sinogram26/19Model ofblurringReconstructionw/ sharp edgesModel ofCT scanReconstructionw/ domainsP. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Analogy: the “Sudoku” Problem – 数独374637427/19P. C. Hansen – Inverse Problems and Uncertainty QuantificationKlitgaarden, Nov. 2021
Appendix: Project OverviewApplicationareasImage processingPlasma CTDecision toolsOther use-cases rrorsGoal oriented UQNonGaussianpriorsX-ray CTEIT & PDEsSampling methods:choice, convergence, Visiting professors, short-term visitors, collaborations with research teams aboradInternal software for the groupReadinggroups28/19Software for ext. usersStudygroupsP. C. Hansen – Inverse Problems and Uncertainty QuantificationCUQIToolsUser interfaceTextbookKlitgaarden, Nov. 2021?
Uncertainty Quantification (UQ) reveals the reliability of all possible reconstructions. Uncertainty Quantification (UQ) is based on Bayesian statistics. Instead of producing a single solution (i.e., x A-1. b) we obtain the . distribution (the posterior) of all possible solutions.
B.Inverse S-BOX The inverse S-Box can be defined as the inverse of the S-Box, the calculation of inverse S-Box will take place by the calculation of inverse affine transformation of an Input value that which is followed by multiplicative inverse The representation of Rijndael's inverse S-box is as follows Table 2: Inverse Sbox IV.
1.1 Measurement Uncertainty 2 1.2 Test Uncertainty Ratio (TUR) 3 1.3 Test Uncertainty 4 1.4 Objective of this research 5 CHAPTER 2: MEASUREMENT UNCERTAINTY 7 2.1 Uncertainty Contributors 9 2.2 Definitions 13 2.3 Task Specific Uncertainty 19 CHAPTER 3: TERMS AND DEFINITIONS 21 3.1 Definition of terms 22 CHAPTER 4: CURRENT US AND ISO STANDARDS 33
fractional uncertainty or, when appropriate, the percent uncertainty. Example 2. In the example above the fractional uncertainty is 12 0.036 3.6% 330 Vml Vml (0.13) Reducing random uncertainty by repeated observation By taking a large number of individual measurements, we can use statistics to reduce the random uncertainty of a quantity.
73.2 cm if you are using a ruler that measures mm? 0.00007 Step 1 : Find Absolute Uncertainty ½ * 1mm 0.5 mm absolute uncertainty Step 2 convert uncertainty to same units as measurement (cm): x 0.05 cm Step 3: Calculate Relative Uncertainty Absolute Uncertainty Measurement Relative Uncertainty 1
Inverse functions mc-TY-inverse-2009-1 An inverse function is a second function which undoes the work of the first one. In this unit we describe two methods for finding inverse functions, and we also explain that the domain of a function may need to be restricted before an inverse function can exist.
Inverse Problems 28 (2012) 055002 T Bui-Thanh and O Ghattas For the forward problem, n is given and we solve the forward equations (1a) and (1b)for the scattered fieldU.For the inverse problem, on the other hand, given observation dataUobs over some compact subset dobs R , we are asked to infer the distribution of the refractive index n.One way to solve the inverse problem is to cast it .
Uncertainty in volume: DVm 001. 3 or 001 668 100 0 1497006 0 1 3 3. %. % .% m m ª Uncertainty in density is the sum of the uncertainty percentage of mass and volume, but the volume is one-tenth that of the mass, so we just keep the resultant uncertainty at 1%. r 186 1.%kgm-3 (for a percentage of uncertainty) Where 1% of the density is .
ACCOUNTING 0452/22 Paper 2 October/November 2018 1 hour 45 minutes Candidates answer on the Question Paper. No Additional Materials are required. READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. You may use an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction .
