Turing/LMS Workshop

Inverse Problems and Data Science

8-10 May 2017

Venue: Informatics Forum, 10 Crichton St, Edinburgh EH8 9AB, UK

Organisers: Natalia Bochkina (University of Edinburgh), Carola Schoenlieb (University of Cambridge), Marta Betcke (UCL), Sean Holman (University of Manchester)

The aim of the workshop is to bring together researchers on inverse problems working in different areas of mathematics, statistics and machine learning as well as from the applied disciplines where inverse problems arise, such as astronomy, biology, computer vision, geoscience and medicine. The topics of the workshop include nonlinear inverse problems, algorithms, inverse problems in machine learning, theoretical properties of statistical estimators in inverse problems, Bayesian inverse problems, applications in science and medicine.

Financial support: The Alan Turing Institute and London Mathematical Society

Sponsoring: if you would like to sponsor the event, please get in touch with one of the organisers. We are grateful to Schlumberger for sponsorship.

Times: the workshop will start at 10am on Monday 8th of May and concludes at 4:30pm on Wednesday 10th of May

Contact: events@turing.ac.uk

Registration: deadline is 1 May, registration fee is £60. Please register here.

There will also be a training course on Bayesian inverse problems on 11 May 2017 – registration is available here (deadline 4 May).

Programme

Monday

9:30 -10:00 registration and coffee

10:00 – 10:45 Sean Holman (University of Manchester, UK) “On the stability of the geodesic ray transform in the presence of caustics“

10:45 – 11:30 Gabriel Paternain (University of Cambridge, UK) ”Effective inversion of the attenuated X-ray transform associated with a connection”

11:30 – 11:45 discussion: interdisciplinary challenges

11:45 - 12:45 lunch

12:45 – 13:30 Andrew Curtis (University of Edinburgh, UK) “Nonlinear Travel-Time and Electrical Resistivity Tomography”

13:30 – 14:15 Nicholas Zabaras (University of Notre Dame, USA) “Inverse Problems with an Unknown Scale of Estimation”

14:15 – 15:00 Eduard Kontar (University of Glasgow, UK) TBC

15:00 - 15:15 discussion: interdisciplinary challenges

15:15-15:30 coffee break

15:30 – 16:15 Mike Christie (Heriot Watt University, UK) ”Bayesian Hierarchical Models for Measurement Error”

16:15-17:00 Botond Szabo (Leiden University, Netherlands) ”Confidence in Bayesian uncertainty quantification in inverse problems”

17:00-17:45 Anna Simoni (CREST, CNRS, Paris, France) ”Nonparametric Estimation in case of Endogenous Selection”

17:45 – 18:00 discussion: interdisciplinary challenges

18:00 - 20:00 poster session and reception

Tuesday

9:15 - 10:00 Michael Gutmann (University of Edinburgh, UK)” Bayesian Inference by Density Ratio Estimation”

10:00 - 10:30 Pol Moreno (University of Edinburgh) “Overcoming Occlusion with Inverse Graphics”

10:30 - 10:50 coffee break

10:50 – 11:35 Kyong Jin (EPFL, Switzerland) “Deep Convolutional Neural Network for Inverse Problems in Imaging”

11:35-12:05 Jonas Adler (KTH, Royal institute of Technology and Elekta) “Solving ill-posed inverse problems using learned iterative schemes”

12:05-12:20 discussion: interdisciplinary challenges

12:20-13:00 lunch

13:00 – 13:45 Christian Clason (Duisburg-Essen University, Germany)” A primal-dual extragradient method for nonlinear inverse problems for PDEs”

13:45 – 14:30 Carola Schoenlieb (University of Cambridge, UK) “Model-based learning in imaging”

14:30 - 14:45 discussion: interdisciplinary challenges

14:45 – 15:15 coffee break

15:15 - 16:00 Markus Reiss (Humboldt University, Berlin, Germany)” Optimal adaptation for early stopping in statistical inverse problems”

16:00 – 16:45 Axel Munk (Goettingen University, Germany) “Nanostatistics – Statistics for Nanoscopy”

16:45 - 17:15 Merle Behr (Goettingen University, Germany) “Multiscale Blind Source Separation”

17:15 – 18:00 discussion: interdisciplinary challenges

Tuesday: conference dinner

Wednesday

9:30 – 10:15 Aretha Teckentrup (University of Edinburgh, UK)” Gaussian process regression in Bayesian inverse problems”

10:15 – 11:00 Marcelo Pereyra (Heriot Watt University, UK) ”Bayesian inference by convex optimisation: theory, methods, and algorithms”

11:00-11:30 coffee break

11:30 – 12:15 Michal Branicki (University of Edinburgh, UK) “Information-based measures of skill and optimality in Bayesian filtering”

12:15 – 12:45 Nagoor Kani (Heriot Watt University, UK) “Model reduction using physics driven deep residual recurrent neural networks”

12:45 – 13:00 discussion: interdisciplinary challenges

13:00 – 14:00 lunch

14:00 – 14:30 Christopher Wallis (UCL) “Sparse image reconstruction on the sphere: Analysis and Synthesis”

14:30 – 15:00 Xiaohao Cai (UCL) “High-dimensional uncertainty estimation with sparse priors for radio interferometric imaging”

15:00 – 15:45 Marta Betcke (UCL) “Dynamic high-resolution photoacoustic tomography with optical flow constraint”

15:45 – 16:00 discussion: interdisciplinary challenges

16:00 closing remarks

Poster presentations

Shing Chan (Heriot-Watt University) A machine learning approach for efficient uncertainty quantification using multiscale methods.

Mohammad Golbabaee (University of Edinburgh) Inexact iterative projected gradient for fast compressed quantitative MRI

Hardial S Kalsi (King's College, London) Glottal Inverse Filtering using the Acoustical Klein-Gordon Equation

Dimitris Kamilis and Nick Polydorides (University of Edinburgh) A computational framework for uncertainty quantification for low-frequency, time-harmonic Maxwell equations with stochastic conductivity models

J.Nagoor Kani and Ahmed H. Elsheikh (Heriot Watt University) Model reduction using physics driven deep residual recurrent neural networks

Jon Cockayne, Chris Oates, Tim Sullivan, Mark Girolami Bayesian Probabilistic Numerical Methods

Alessandro Perelli (University of Edinburgh) Multi Denoising Approximate Message Passing for computational complexity reduction

Jenovah Rodrigues (University of Edinburgh) Bayesian Inverse Problems with Heterogeneous Noise

Ferdia Sherry (University of Cambridge) Learning a sampling pattern for MRI

Invited speaker abstracts.

Christian Clason (Mathematics, Duisburg-Essen University, Germany) ”A primal-dual extragradient method for nonlinear inverse problems for PDEs

This talk is concerned with the extension of the Chambolle--Pock primal-dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces. The proof of local convergence rests on verifying the Aubin property of the inverse of a monotone operator at the minimizer, which is difficult as it involves infinite-dimensional set-valued analysis. However, for nonsmooth functionals that are defined pointwise -- such as $L^1$ or $L^\infty$ norms -- it is possible to apply simpler tools from the finite-dimensional theory, which allows deriving explicit conditions for the convergence.  This is illustrated for the example of imaging problems with $L^1$- and $L^\infty$-fitting terms.

Mike Christie (Petroleum Institute, Heriot Watt University, UK) ”Bayesian Hierarchical Models for Measurement Error

The detailed geological description of oil reservoirs is always uncertain because of the large size and relatively small number of wells from which hard data can be obtained.  To handle this uncertainty, reservoir models are calibrated or ‘history matched’ to production data (oil rates, pressures etc). The quality of any reservoir forecasts depends not only on the quality of the match, but also how well understood the measurement errors are (or indeed the split between measurement and modelling errors).

This talk will look at hierarchical models for estimating measurement and modelling errors in reservoir model calibration, and compare maximum likelihood estimates of measurement errors with marginalisation over unknown errors.

Andrew Curtis (University of Edinburgh, UK) “Nonlinear Travel-Time and Electrical Resistivity Tomography”

We solve the fully nonlinear travel-time tomography problem using reversible-jump Markov chain Monte Carlo methods. The results motivate a conjecture that the uncertainty in general, non-linearised tomography problems may consist of loop-like topologies that can be interpreted similarly to linearised measures of spatial resolution. This is confirmed in a second example by applying the same inversion algorithm to estimate the electrical resistivity structure of a medium (the Earth) from dipole-dipole electrical resistivity measurements, a problem governed by quite different physics: uncertainty loops appear similarly. If time allows, I will then discuss a simple decomposition of tomography problems that can be shown to be unimodal in some important cases, leading to a different method of solution.

Michael Gutmann (Informatics, University of Edinburgh, UK)” Bayesian Inference by Density Ratio Estimation”

This talk is about Bayesian inference when the likelihood function cannot be computed but data can be generated from the model. The model's data generating process is allowed to be arbitrarily complex. Exact solutions are then not possible. But by re-formulating the original problem as a problem of estimating the ratio between two probability density functions, I show how e.g. logistic regression can be used to obtain approximate solutions. The proposed inference framework is illustrated on stochastic nonlinear dynamical models.

Reference: https://arxiv.org/abs/1611.10242

Kyong Jin (EPFL, Switzerland) “Deep Convolutional Neural Network for Inverse Problems in Imaging

This talk discusses a novel deep convolutional neural network (CNN)-based algorithm for solving ill-posed inverse problems. Regularized iterative algorithms have emerged as the standard approach to ill-posed inverse problems in the past few decades. These methods produce excellent results, but can be challenging to deploy in practice due to factors including the high computational cost of the forward and adjoint operators and the difficulty of hyper parameter selection. The starting point of our work is the observation that unrolled iterative methods have the form of a CNN (filtering followed by point-wise non-linearity) when the normal operator (H*H, the adjoint of H times H) of the forward model is a convolution. Based on this observation, we propose using direct inversion followed by a CNN to solve normal-convolutional inverse problems. The direct inversion encapsulates the physical model of the system, but leads to artifacts when the problem is ill-posed; the CNN combines multiresolution decomposition and residual learning in order to learn to remove these artifacts while preserving image structure. The performance of the proposed network will be demonstrated in sparse-view reconstruction on parallel beam X-ray computed tomography and accelerated MR imaging reconstruction on parallel MRI.

Axel Munk (Department of Mathematics and Computer Science, and Max-Planck Institute for Biophysical Chemistry, Goettingen University, Germany)

“Nanostatistics – Statistics for Nanoscopy”

Conventional light microscopes have been used for centuries for the study of small length scales down to approximately 250 nm. Images from such a microscope are typically blurred and noisy, and the measurement error in such images can often be well approximated by Gaussian or Poisson noise. In the past, this approximation has been the focus of a multitude of deconvolution techniques in imaging. However, conventional microscopes have an intrinsic physical limit of resolution. Although this limit remained unchallenged for a century, it was broken for the first time in the 1990s with the advent of modern superresolution fluorescence microscopy techniques. Since then, superresolution fluorescence microscopy has become an indispensable tool for studying the structure and dynamics of living organisms, recently acknowledged with the c Nobel prize in chemistry 2014. Current experimental advances go to the physical limits of imaging, where discrete quantum effects are predominant. Consequently, the data is inherently of a non-Gaussian statistical nature, and we argue that recent technological progress also challenges the long-standing Poisson assumption. Thus, analysis and exploitation of the discrete physical mechanisms of fluorescent molecules and light, as well as their distributions in time and space, have become necessary to achieve the highest resolution possible and to extract biologically relevant information.

In this talk we survey some modern fluorescence microscopy techniques from a statistical modeling and analysis perspective. In the first part we address spatially adaptive multiscale deconvolution estimation and testing methods for scanning type microscopy. We illustrate that such methods benefit from recent advances in large-scale computing, mainly from convex optimization. In the second part of the talk we address challenges of quantitative biology which require more detailed models that delve into sub-Poisson statistics. To this end we suggest a prototypical model for fluorophore dynamics and use it to quantify the number of proteins in a spot.

Marcelo Pereyra (Mathematics, Heriot Watt University, UK) ”Bayesian inference by convex optimisation: theory, methods, and algorithms.

Convex optimisation has become the main Bayesian computation methodology in many areas of data science such as mathematical imaging and machine learning, where high dimensionality is often addressed by using models that are log-concave and where maximum-a-posteriori (MAP) estimation can be performed efficiently by optimisation. The first part of this talk presents a new decision-theoretic derivation of MAP estimation and shows that, contrary to common belief, under log-concavity MAP estimators are proper Bayesian estimators. A main novelty is that the derivation is based on differential geometry. Following on from this, we establish universal theoretical guarantees for the estimation error involved and show estimation stability in high dimensions. Moreover, the second part of the talk describes a new general methodology for approximating Bayesian high-posterior-density regions in log-concave models.  The approximations are derived by using recent concentration of measure results related to information theory, and can be computed very efficiently, even in large-scale problems, by using convex optimisation techniques. The approximations also have favourable theoretical properties, namely they outer-bound the true high-posterior-density credibility regions, and they are stable with respect to model dimension. The proposed methodology is finally illustrated on two high-dimensional imaging inverse problems related to tomographic reconstruction and sparse deconvolution, where they are used to explore the uncertainty about the solutions, and where convex-optimisation-empowered proximal Markov chain Monte Carlo algorithms are used as benchmark to compute exact credible regions and measure the approximation error.

Related pre-prints:

https://arxiv.org/abs/1612.06149

https://arxiv.org/pdf/1602.08590.pdf

Markus Reiss (Humboldt University, Berlin, Germany)” Optimal adaptation for early stopping in statistical inverse problems

For linear inverse problems $Y=\mathsf{A}\mu+\xi$, it is classical to recover the unknown function $\mu$ by an iterative scheme $(\widehat \mu^{(m)}, m=0,1,\ldots)$ and to provide $\widehat\mu^{(\tau)}$ as a result, where $\tau$ is some stopping rule. Stopping should be decided adaptively, that is in a data-driven way independently of the true function $\mu$.  For deterministic noise $\xi$  the discrepancy principle is usually applied to determine $\tau$. In the context of  stochastic noise $\xi$, we study oracle adaptation (that is, compared to the best possible stopping iteration). For a stopping rule based on the residual process, oracle adaptation bounds within a certain domain are established.  For Sobolev balls, the domain of adaptivity matches a corresponding lower bound. The proofs use bias and variance transfer techniques from weak prediction error to strong $L^2$-error, as well as convexity arguments and concentration bounds for the stochastic part. The performance of our stopping rule for Landweber and spectral cutoff methods is illustrated numerically.(Joint work with Gilles Blanchard, Potsdam, and Marc Hoffmann, Paris)

Anna Simoni (CREST, CNRS, Paris, France) ”Nonparametric Estimation in case of Endogenous Selection”

This paper addresses the problem of estimation of a nonparametric regression function from selectively observed data when selection is endogenous. Our approach relies on independence between covariates and selection conditionally on potential outcomes. Endogeneity of regressors is also allowed for. In the exogenous and endogenous case, consistent two-step estimation procedures are proposed and their rates of convergence are derived which take into account the degree of ill-posedness. In the first stage we have to solve an ill-posed inverse problem to recover nonparametrically the inverse selection probability function. Moreover, when the covariates are endogenous an additional inverse problem has to be solved in the second step to recover the instrumental regression function. Pointwise asymptotic distribution of the estimators is established. In addition, bootstrap uniform confidence bands are derived. Finite sample properties are illustrated in a Monte Carlo simulation study and an empirical illustration. Joint work with Christoph Breunig (Humboldt University, Berlin) and Enno Mammen (Heidelberg University).

Botond Szabo (Leiden University, Netherlands) ”Confidence in Bayesian uncertainty quantification in inverse problems

In our work we investigate the frequentist coverage of Bayesian credible sets in the inverse Gaussian sequence model. We consider a scale of priors of varying regularity and choose the regularity by an empirical or a hierarchical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of “polished tail” parameters, in the sense of high probability of coverage of such parameters. On the negative side, we show by theory and example that adaptation of the prior necessarily leads to gross and haphazard uncertainty quantification for some true parameters that are still within the hyperrectangle regularity scale. The preceding results are based on semi-explicit computations on an optimised statistical model. In the end of the talk I will briefly discuss to possible extensions of our coverage results to more general, abstract settings.

The talk is based on the papers written together with Judith Rousseau, Aad van der Vaart and Harry van Zanten.

Aretha Teckentrup (Mathematics, University of Edinburgh, UK)” Gaussian process regression in Bayesian inverse problems”

A major challenge in the application of sampling methods in Bayesian inverse problems is the typically large computational cost associated with solving the forward problem. To overcome this issue, we consider using a Gaussian process surrogate model to approximate the forward map. This results in an approximation to the solution of the Bayesian inverse problem, and more precisely in an approximate posterior distribution.

In this talk, we analyse the error in the approximate posterior distribution, and show that the approximate posterior distribution tends to the true posterior as the accuracy of the Gaussian process surrogate model increases.

Nicholas Zabaras (Computational Science and Engineering, University of Notre Dame, USA) “Inverse Problems with an Unknown Scale of Estimation

The presentation will focus on the Bayesian estimation of spatially varying parameters of multiresolution/multiscale nature. In particular, the characteristic length scale(s) of the unknown property are not known a priori and need to be evaluated based on the fidelity of the given data across the domain. Our approach is based on representing the spatial field with a wavelet expansion. The intra-scale correlations between wavelet coefficients form a quadtree, and this structure is exploited to identify additional basis functions to refine the model. Bayesian inference is performed using a sequential Monte Carlo sampler with a MCMC transition kernel. The SMC sampler is used to move between posterior densities defined on different scales, thereby providing for adaptive refinement of the wavelet representation. The marginal likelihoods provide a termination criterion for the scale determination algorithm thus allowing model comparison and selection. The approach is demonstrated with permeability estimation for groundwater flow using pressure measurements.

https://www.zabaras.com/

Contributed talk abstracts.

Jonas Adler, KTH- Royal Institute of Technology and Elekta

Solving ill-posed inverse problems using learned iterative schemes

We present partially learned iterative schemes for the solution of ill posed inverse problems with not necessarily linear forward operators. The methods builds on ideas from classical regularization theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularizing functional.

We also present results for tomographic reconstruction of human head phantoms and discuss several possible future research areas.

Related pre-print: https://arxiv.org/abs/1704.04058

Merle Behr, Goettingen University, Germany

Multiscale Blind Source Separation

We discuss a new methodology for statistical recovery of single linear mixtures of piecewise constant signals (sources) with unknown mixing weights and change points in a multiscale fashion. Exact recovery within an \epsilon-neighborhood of the mixture is obtained when the sources take only values in a known finite alphabet. Based on this we provide estimators for the mixing weights and sources for Gaussian error. We obtain uniform confidence sets and optimal rates (up to log-factors) for all quantities.

This blind source separation problem is motivated by different applications in digital communication, but also in cancer genetics. In the latter one aims to assign copy-number variations from genetic sequencing data to different tumor-clones and their corresponding proportions in the tumor. We analyze such data using the proposed method in order to estimate their proportion in the tumor and the corresponding copy number variations.

This is joint work with Chris Holmes (University of Oxford, UK) and Axel Munk (University of Goettingen, Germany).

Xiaohao Cai, MSSL, University College London
High-dimensional uncertainty estimation with sparse priors for radio interferometric imaging

In many fields high-dimensional inverse imaging problems are encountered. For example, imaging the raw data acquired by radio interferometric telescopes involves solving an ill-posed inverse problem to recover an image of the sky from noisy and incomplete Fourier measurements. Future telescopes, such as the Square Kilometre Array (SKA), will usher in a new big-data era for radio interferometry, with data rates comparable to world-wide internet traffic today.  Sparse regularisation techniques are a powerful approach for solving these problems, typically yielding excellent reconstruction fidelity (e.g. Pratley et al. 2016). Moreover, by leveraging recent developments in convex optimisation, these techniques can be scaled to extremely large data-sets (e.g. Onose et al. 2016). However, such approaches typically recover point estimators only and uncertainty information is not quantified. Standard Markov Chain Monte Carlo (MCMC) techniques that scale to high-dimensional settings cannot support the sparse (non-differentiable) priors that
have been shown to be highly effective in practice. We present work adapting the proximal Metropolis adjusted Langevin algorithm (P-MALA), developed recently by Pereyra (2016a), for radio interferometric imaging with sparse priors (Cai, Pereyra & McEwen 2017a), leveraging proximity operators from convex optimisation in an MCMC framework to recover the full posterior distribution of the sky image. While such an approach provides critical uncertainty information, scaling to extremely large data-sets, such as those anticipated from the SKA, is challenging. To address this issue we develop a technique to compute approximate local Bayesian credible intervals by post-processing the point (maximum a-posteriori) estimator recovered by solving the associated sparse regularisation problem (Cai, Pereyra & McEwen 2017b), leveraging recent results by Pereyra (2016b). This approach inherits the computational scalability of sparse regularisation techniques, while also providing critical uncertainty information.  We demonstrate these techniques on simulated observations made by radio interferometric telescopes. Joint work with Marcelo Pereyra (from Heriot-Watt University) and Jason D. McEwen (from MSSL, University College London).

J.Nagoor Kani (and Ahmed H. Elsheikh), Heriot Watt University

Model reduction using physics driven deep residual recurrent neural networks

We introduce a deep residual recurrent neural network (DR-RNN) to emulate the dynamics of physical phenomena. The developed DR-RNN is inspired by the iterative steps of line search methods in finding the residual minimiser of numerically discretised differential equations. We formulate this iterative scheme as stacked recurrent neural network (RNN) embedded with the dynamical structure of the emulated differential equations. We provide empirical evidence showing that these residual driven deep RNN can effectively emulate the physical system with significantly lower number of parameters in comparison to standard RNN architectures. We also show the significant gains in accuracy by increasing the depth of RNN similar to other recent applications of deep learning. The applicability of the developed DR-RNN is demonstrated on uncertainty quantification tasks where a large number of forward simulation are required.

Pol Moreno, University of Edinburgh, UK

Overcoming Occlusion with Inverse Graphics

Scene understanding tasks such as the prediction of object pose, shape, appearance and illumination are hampered by the occlusions often found in images. We propose a vision-as-inverse-graphics approach to handle these occlusions by making use of a graphics renderer in combination with a robust generative model (GM). Since searching over scene factors to obtain the best match for an image is very inefficient, we make use of a recognition model (RM) trained on synthetic data to initialize the search. This paper addresses two issues: (i) We study how the inferences are affected by the degree of occlusion of the foreground object, and show that a robust GM which includes an outlier model to account for occlusions works significantly better than a non-robust model. (ii) We characterize the performance of the RM and the gains that can be made by refining the search using the GM, using a new dataset that includes background clutter and occlusions. We find that pose and shape are predicted very well by the RM, but appearance and especially illumination less so. However, accuracy on these latter two factors can be clearly improved with the generative model.

Christopher Wallis, University College London

Sparse image reconstruction on the sphere: Analysis and Synthesis

We develop techniques to solve a number of ill-posed inverse problems on the sphere by sparse regularisation, exploiting sparsity in directional wavelet space. Through numerical experiments we evaluate the effectiveness of the technique in solving inpainting, denoising and deconvolution problems. We consider solving the problems in both the analysis and synthesis settings, with a number of different sampling schemes, and show that the sampling scheme has a large impact on the quality of the reconstruction. This is due to more efficient sampling schemes constraining the solution space and improving sparsity in wavelet space. We adapt and apply the technique to the Planck 353GHz total intensity map, improving the ability to extract the structure of galactic dust emission.

Poster abstracts.

Shing Chan, Heriot-Watt University, UK

A machine learning approach for efficient uncertainty quantification using multiscale methods.

Several multiscale methods account for sub-grid scale features using coarse scale basis functions. For example, in the Multiscale Finite Volume method the coarse scale basis functions are obtained by solving a set of local problems over dual-grid cells. We introduce a data-driven approach for the estimation of these coarse scale basis functions. Specifically, we employ a neural network predictor fitted using a set of solution samples from which it learns to generate subsequent basis functions at a lower computational cost than solving the local problems. The computational advantage of this approach is realized for uncertainty quantification tasks where a large number of realizations has to be evaluated. We attribute the ability to learn these basis functions to the modularity of the local problems and the redundancy of the permeability patches between samples. The proposed method is evaluated on elliptic problems yielding very promising results.

Inexact iterative projected gradient for fast compressed quantitative MRI

We will present a compressed sensing perspective of a novel form of MR imaging called Magnetic Resonance Fingerprinting (MRF). This enables direct estimation of the T1, T2 and proton density parameter maps for a patient through undersampled k-space sampling and BLIP, a gradient projection algorithm that enforces the MR Bloch dynamics. One of the key bottlenecks in MRF is the projection onto the constraint set. We will present both theoretical and numerical results showing that significant computational savings are possible through the use of inexact projections and a fast approximate nearest neighbor search.

Hardial S Kalsi, King's College, London

Glottal Inverse Filtering using the Acoustical Klein-Gordon Equation

Inversion of the glottal-pulse waveform from a speech signal remains an active field of research although dating back over half a century. Despite multiple approaches to solve this important inverse problem, it cannot be said today that the field is in a satisfactory state. In the main, approaches use classical “inverse filtering” frequency-domain methods to estimate both the vocal-tract and glottal-pulse waveform. In this poster, we illustrate a new approach which takes advantage of two recent developments: firstly, the description of the speech wave by means of an analogue of the Klein-Gordon wave equation of relativistic quantum mechanics and, secondly, the solution of this equation to find its Green's function. This approach allows accurate parameterisation of the vocal tract which greatly simplifies the inversion.

Dimitris Kamilis and Nick Polydorides, University of Edinburgh

A computational framework for uncertainty quantification for low-frequency, time-harmonic Maxwell equations with stochastic conductivity models

We present a computational framework for uncertainty quantification (UQ) for the quasi-magnetostatic Maxwell equations using lognormal random field conductivity models. Our methodology combines elements of sparse quadrature (SQ) for the efficient calculation of the high-dimensional UQ integrals, as well as model reduction methods for expediting the model evaluations. Our analysis and numerical results show that subject to some mild assumptions on the smoothness of the random conductivity fields, sparse quadrature outperforms the convergence of the conventional Monte-Carlo method, while model reduction further reduces the computational cost. Numerical results to illustrate the method are presented from three-dimensional simulations that are representative of models appearing in the geophysical prospecting Controlled-Source Electromagnetic Method (CSEM).

J.Nagoor Kani and Ahmed H. Elsheikh, Heriot Watt University

Model reduction using physics driven deep residual recurrent neural networks

We introduce a deep residual recurrent neural network (DR-RNN) to emulate the dynamics of physical phenomena. The developed DR-RNN is inspired by the iterative steps of line search methods in finding the residual minimiser of numerically discretised differential equations. We formulate this iterative scheme as stacked recurrent neural network (RNN) embedded with the dynamical structure of the emulated differential equations. We provide empirical evidence showing that these residual driven deep RNN can effectively emulate the physical system with significantly lower number of parameters in comparison to standard RNN architectures. We also show the significant gains in accuracy by increasing the depth of RNN similar to other recent applications of deep learning. The applicability of the developed DR-RNN is demonstrated on uncertainty quantification tasks where a large number of forward simulation are required.

Jon Cockayne, Chris Oates, Tim Sullivan, Mark Girolami

"Bayesian Probabilistic Numerical Methods"

The emergent field of probabilistic numerics has thus far lacked rigorous statistical principals. This work establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain Bayesian inverse problems, albeit problems that are non-standard. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well-defined, encompassing both non-linear and non-Gaussian models. For general computation, a numerical approximation scheme is developed and its asymptotic convergence is established. The theoretical development is then extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, with some illustrative applications presented

Alessandro Perelli (University of Edinburgh) Multi Denoising Approximate Message Passing for computational complexity reduction

Denoising-AMP (D-AMP) [1] can be viewed as an iterative algorithm where at each iteration a non-linear denoising function is applied to the signal estimate. D-AMP algorithm has been analysed in terms of inferential accuracy without considering computational complexity. This is an important missing aspect since the denoising is often the computational bottleneck in the D-AMP reconstruction.

The approach that it is proposed in this work is different; we aim to design a mechanism for leveraging a hierarchy denoising models (MultiD-AMP) in order to minimize the overall complexity given the expected risk, i.e. the estimation error. The intuition comes from the observation that at earlier iteration, when the estimate is far according to some distance to the true signal, the algorithm does not need a complicated denoiser, since the structure of the signal is poor, but faster denoisers and this leads to the idea of defining a family/hierarchy of denoisers of increased complexity. The main challenge is to define a switching scheme which is based on the empirical finding that in MultiD-AMP we can predict exactly, in the large system limit, the evolution of the Mean Square Error. We can exploit the State Evolution, evaluated on a set of training images, to find a proper switching strategy. The proposed framework has been tested on i.i.d. random Gaussian measurements with Gaussian noise and for deconvolution problem. The results show the effectiveness of the proposed reconstruction algorithm.

[1] Metzler, C. A., Maleki, A., Baraniuk, R. G. From denoising to compressed sensing. IEEE Transactions on Information Theory, 62(9), 5117-5144, 2016

Jenovah Rodrigues, University of Edinburgh.

'Bayesian Inverse Problems with Heterogeneous Noise.'

We study linear, ill-posed inverse problems in separable Hilbert spaces with noisy observations. A Bayesian solution with Gaussian regularising priors will be studied; the aim being to select the prior distribution in such a way that the solution achieves the optimal rate of convergence, when the unknown function belongs to a Sobolev space. Consequently, we will focus on obtaining the rate of convergence, for the rate of contraction, of the whole posterior distribution to the forementioned unknown function. We consider a Gaussian noise error model with heterogeneous variance, which is investigated using the spectral decomposition of the operator defined in the inverse problem. This is joint work with Natalia Bochkina (University of Edinburgh).

Ferdia Sherry, University of Cambridge

Learning a sampling pattern for MRI

Taking measurements in MRI is a time-consuming procedure, so ideally one would take few samples and still recover a useable image. It is crucial that these samples are positioned in frequency space in a way that allows as much information to be extracted from the samples as possible. We consider the problem of determining a suitable sampling pattern for a class of images that are in some sense similar.  The problem of learning a sampling pattern can be formulated as a bilevel optimisation problem, in which the upper problem measures the reconstruction quality and penalises the lack of sparsity of the sampling pattern and in which the lower problem is the total variation MRI reconstruction problem. We study the use of stochastic optimisation methods (taking a random pair of ground truth and noisy measurements for each iteration) to solve the bilevel problem.