The subject of this article is the introduction of a new concept of well-posedness of Bayesian inverse problems. The conventional concept of (Lipschitz, Hellinger) well-posedness in [Stuart 2010, Acta Numerica 19, pp. 451-559] is difficult to verify in practice and may be inappropriate in some contexts. Our concept simply replaces the Lipschitz continuity of the posterior measure in the Hellinger distance by continuity in an appropriate distance between probability measures. Aside from the Hellinger distance, we investigate well-posedness with respect to weak convergence, the total variation distance, the Wasserstein distance, and also the Kullback-Leibler divergence. We demonstrate that the weakening to continuity is tolerable and that the generalisation to other distances is important. The main results of this article are proofs of wellposedness with respect to some of the aforementioned distances for large classes of Bayesian inverse problems. Here, little or no information about the underlying model is necessary; making these results particularly interesting for practitioners using black-box models. We illustrate our findings with numerical examples motivated from machine learning and image processing.

The identification of parameters in mathematical models using noisy observations is a common task in uncertainty quantification. We employ the framework of Bayesian inversion: we combine monitoring and observational data with prior information to estimate the posterior distribution of a parameter. Specifically, we are interested in the distribution of a diffusion coefficient of an elliptic PDE. In this setting, the sample space is high-dimensional, and each sample of the PDE solution is expensive. To address these issues we propose and analyse a novel Sequential Monte Carlo (SMC) sampler for the approximation of the posterior distribution. Classical, single-level SMC constructs a sequence of measures, starting with the prior distribution, and finishing with the posterior distribution. The intermediate measures arise from a tempering of the likelihood, or, equivalently, a rescaling of the noise. The resolution of the PDE discretisation is fixed. In contrast, our estimator employs a hierarchy of PDE discretisations to decrease the computational cost. We construct a sequence of intermediate measures by decreasing the temperature or by increasing the discretisation level at the same time. This idea builds on and generalises the multi-resolution sampler proposed in [P.S. Koutsourelakis, J. Comput. Phys., 228 (2009), pp. 6184-6211] where a bridging scheme is used to transfer samples from coarse to fine discretisation levels. Importantly, our choice between tempering and bridging is fully adaptive. We present numerical experiments in 2D space, comparing our estimator to single-level SMC and the multi-resolution sampler.In contrast to deterministic regularisation techniques, the Bayesian approach to inverse problems uses the probabilistic framework of Bayesian inference. Bayesian inference is built on Bayes' Formula in the formulation given by Laplace [34, II.1]. We remark that other formulations are possible, see e.g. the work by Matthies et al. [38]. We make use of the mathematical framework for Bayesian Inverse Problems (BIPs) given by Stuart [48]. Under weak assumptions -which we will give below -one can show that the BIP is well-posed. The solution of the BIP is the conditional probability measure of the unknown parameter given the observations.The Bayesian framework is very general and can handle different types of forward models. However, in this work we consider PDE-based forward models, and in particular an elliptic PDE. The exact solution of the associated BIP is often inaccessible for two reasons: (i) there is no closed form expression for the posterior measure, and (ii) the underlying PDE cannot be solved analytically. We focus on (i), and study efficient approximations to the full posterior measure. Alternatively, one could also only approximate the expectation of output quantities of interest with respect to the posterior measure, or estimate the model evidence, the normalization constant of the posterior measure.Typically, BIPs are approached with sampling based methods, such as Markov Chain Monte Carlo (M...

Stochastic gradient descent is an optimisation method that combines classical gradient descent with random subsampling within the target functional. In this work, we introduce the stochastic gradient process as a continuous-time representation of stochastic gradient descent. The stochastic gradient process is a dynamical system that is coupled with a continuous-time Markov process living on a finite state space. The dynamical system—a gradient flow—represents the gradient descent part, the process on the finite state space represents the random subsampling. Processes of this type are, for instance, used to model clonal populations in fluctuating environments. After introducing it, we study theoretical properties of the stochastic gradient process: We show that it converges weakly to the gradient flow with respect to the full target function, as the learning rate approaches zero. We give conditions under which the stochastic gradient process with constant learning rate is exponentially ergodic in the Wasserstein sense. Then we study the case, where the learning rate goes to zero sufficiently slowly and the single target functions are strongly convex. In this case, the process converges weakly to the point mass concentrated in the global minimum of the full target function; indicating consistency of the method. We conclude after a discussion of discretisation strategies for the stochastic gradient process and numerical experiments.

Gaussian random fields are popular models for spatially varying uncertainties, arising for instance in geotechnical engineering, hydrology or image processing. A Gaussian random field is fully characterised by its mean function and covariance operator. In more complex models these can also be partially unknown. In this case we need to handle a family of Gaussian random fields indexed with hyperparameters. Sampling for a fixed configuration of hyperparameters is already very expensive due to the nonlocal nature of many classical covariance operators. Sampling from multiple configurations increases the total computational cost severely. In this report we employ parameterised Karhunen-Loève expansions for sampling. To reduce the cost we construct a reduced basis surrogate built from snapshots of Karhunen-Loève eigenvectors. In particular, we consider Matérn-type covariance operators with unknown correlation length and standard deviation. We suggest a linearisation of the covariance function and describe the associated online-offline decomposition. In numerical experiments we investigate the approximation error of the reduced eigenpairs. As an application we consider forward uncertainty propagation and Bayesian inversion with an elliptic partial differential equation where the logarithm of the diffusion coefficient is a parameterised Gaussian random field. In the Bayesian inverse problem we employ Markov chain Monte Carlo on the reduced space to generate samples from the posterior measure. All numerical experiments are conducted in 2D physical space, with non-separable covariance operators, and finite element grids with ∼ 10 4 degrees of freedom.

We consider the inverse problem of parameter estimation in a diffuse interface model for tumour growth. The model consists of a fourth-order Cahn-Hilliard system and contains three phenomenological parameters: the tumour proliferation rate, the nutrient consumption rate, and the chemotactic sensitivity. We study the inverse problem within the Bayesian framework and construct the likelihood and noise for two typical observation settings. One setting involves an infinite-dimensional data space where we observe the full tumour. In the second setting we observe only the tumour volume, hence the data space is finite-dimensional. We show the well-posedness of the posterior measure for both settings, building upon and improving the analytical results in [C. Kahle and K.F. Lam, Appl. Math. Optim. (2018)]. A numerical example involving synthetic data is presented in which the posterior measure is numerically approximated by the sequential Monte Carlo approach with tempering. are able to ignore apoptosis (programmed cell death) signals, remain elusive to attacks from the immune system, and, most dangerously, have the ability to induce the growth of new blood vessels towards itself (angiogenesis). This leads to the spreading of cancer to other parts of the body, and the formation of secondary tumours (metastasis).The study of tumour growth can be roughly divided according to the physical and chemical phenomena occuring at three scales [47]: the tissue scale which is commonly observed in experiments involving movement of cells (such as metastasis and growth into the extracellular matrix) and nutrient diffusion; the cellular scale consisting of activities and interactions between individual cells such as mitosis and the activation of receptors; and sub-cellular scale where genetic mutations and DNA degradation occur. We focus on the tissue-scaled phenomena, as they are the first to be detected in a routine diagnosis, and can be described fairly well with help of continuum models consisting of differential equations.Since the seminal work in [11] and [27] where simple mathematical models for tumour growth are employed, there has been an explosion in the number of models proposed for modelling the multiscale nature of cancer, see for instance [19,21,47] and the references cited therein. The diversity of model variants reflects the difficulties when we try to identify key biological phenomena that are responsible for experimental observations.As metastasis is an important hallmark of cancer, we restrict our attention to continuum models that can capture such events. Continuum models often rely on a mathematical description to distinguish tumour tissue from healthy host tissues. To be able to capture metastasis the models have to allow for some form of topological change of the separation layers between the tumour and the host tissues. The classical description represents the separation layers as idealised hypersurfaces, known also as the sharp interface approach. In this case complicated boundary conditions have to be imposed t...