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We provide a clear and concise introduction to the subjects of inverse problems and data assimilation, and their inter-relations. The first part of our notes covers inverse problems; this refers to the study of how to estimate unknown model parameters from data. The second part of our notes covers data assimilation; this refers to a particular class of inverse problems in which the unknown parameter is the initial condition (and/or state) of a dynamical system, and the data comprises partial and noisy observations of the state. The third and final part of our notes describes the use of data assimilation methods to solve generic inverse problems by introducing an artificial algorithmic time. Our notes cover, among other topics, maximum a posteriori estimation, (stochastic) gradient descent, variational Bayes, Monte Carlo, importance sampling and Markov chain Monte Carlo for inverse problems; and 3DVAR, 4DVAR, extended and ensemble Kalman filters, and particle filters for data assimilation. Each of parts one and two starts with a chapter on the Bayesian formulation, in which the problem solution is given by a posterior distribution on the unknown parameter. Then the following chapter specializes the Bayesian formulation to a linear-Gaussian setting where explicit characterization of the posterior is possible and insightful. The next two chapters explore methods to extract information from the posterior in nonlinear and non-Gaussian settings using optimization and Gaussian approximations. The final two chapters describe sampling methods that can reproduce the full posterior in the large sample limit. Each chapter closes with a bibliography containing citations to alternative pedagogical literature and to relevant research literature. We also include a set of exercises at the end of parts one and two. Our notes are thus useful for both classroom teaching and self-guided study.

We analyse a class of time discretizations for solving the nonlinear Schr\"odinger equation with non-smooth potential and at low-regularity on an arbitrary Lipschitz domain $\Omega \subset \mathbb{R}^d$, $d \le 3$. We show that these schemes, together with their optimal local error structure, allow for convergence under lower regularity assumptions on both the solution and the potential than is required by classical methods, such as splitting or exponential integrator methods. Moreover, we show first and second order convergence in the case of periodic boundary conditions, in any fractional positive Sobolev space $H^{r}$, $r \ge 0$, beyond the more typical $L^2$ or $H^\sigma (\sigma>\frac{d}{2}$) -error analysis. Numerical experiments illustrate our results.

We study the numerical approximation by space-time finite element methods of a multi-physics system coupling hyperbolic elastodynamics with parabolic transport and modeling poro- and thermoelasticity. The equations are rewritten as a first-order system in time. Discretizations by continuous Galerkin methods in time and inf-sup stable pairs of finite element spaces for the spatial variables are investigated. Optimal order error estimates are proved by an analysis in weighted norms that depict the energy of the system's unknowns. A further important ingredient and challenge of the analysis is the control of the couplings terms. The techniques developed here can be generalized to other families of Galerkin space discretizations and advanced models. The error estimates are confirmed by numerical experiments, also for higher order piecewise polynomials in time and space. The latter lead to algebraic systems with complex block structure and put a facet of challenge on the design of iterative solvers. An efficient solution technique is referenced.

Mean field theory has provided theoretical insights into various algorithms by letting the problem size tend to infinity. We argue that the applications of mean-field theory go beyond theoretical insights as it can inspire the design of practical algorithms. Leveraging mean-field analyses in physics, we propose a novel algorithm for sparse measure recovery. For sparse measures over $\mathbb{R}$, we propose a polynomial-time recovery method from Fourier moments that improves upon convex relaxation methods in a specific parameter regime; then, we demonstrate the application of our results for the optimization of particular two-dimensional, single-layer neural networks in realizable settings.

Boolean functions are mathematical objects with numerous applications in domains like coding theory, cryptography, and telecommunications. Finding Boolean functions with specific properties is a complex combinatorial optimization problem where the search space grows super-exponentially with the number of input variables. One common property of interest is the nonlinearity of Boolean functions. Constructing highly nonlinear Boolean functions is difficult as it is not always known what nonlinearity values can be reached in practice. In this paper, we investigate the effects of the genetic operators for bit-string encoding in optimizing nonlinearity. While several mutation and crossover operators have commonly been used, the link between the genotype they operate on and the resulting phenotype changes is mostly obscure. By observing the range of possible changes an operator can provide, as well as relative probabilities of specific transitions in the objective space, one can use this information to design a more effective combination of genetic operators. The analysis reveals interesting insights into operator effectiveness and indicates how algorithm design may improve convergence compared to an operator-agnostic genetic algorithm.

In this paper we establish accuracy bounds of Prony's method (PM) for recovery of sparse measures from incomplete and noisy frequency measurements, or the so-called problem of super-resolution, when the minimal separation between the points in the support of the measure may be much smaller than the Rayleigh limit. In particular, we show that PM is optimal with respect to the previously established min-max bound for the problem, in the setting when the measurement bandwidth is constant, with the minimal separation going to zero. Our main technical contribution is an accurate analysis of the inter-relations between the different errors in each step of PM, resulting in previously unnoticed cancellations. We also prove that PM is numerically stable in finite-precision arithmetic. We believe our analysis will pave the way to providing accurate analysis of known algorithms for the super-resolution problem in full generality.

We propose and analyze a first-order finite difference scheme for the functionalized Cahn-Hilliard (FCH) equation with a logarithmic Flory-Huggins potential. The semi-implicit numerical scheme is designed based on a suitable convex-concave decomposition of the FCH free energy. We prove unique solvability of the numerical algorithm and verify its unconditional energy stability without any restriction on the time step size. Thanks to the singular nature of the logarithmic part in the Flory-Huggins potential near the pure states $\pm 1$, we establish the so-called positivity-preserving property for the phase function at a theoretic level. As a consequence, the numerical solutions will never reach the singular values $\pm 1$ in the point-wise sense and the fully discrete scheme is well defined at each time step. Next, we present a detailed optimal rate convergence analysis and derive error estimates in $l^{\infty}(0,T;L_h^2)\cap l^2(0,T;H^3_h)$ under a linear refinement requirement $\Delta t\leq C_1 h$. To achieve the goal, a higher order asymptotic expansion (up to the second order temporal and spatial accuracy) based on the Fourier projection is utilized to control the discrete maximum norm of solutions to the numerical scheme. We show that if the exact solution to the continuous problem is strictly separated from the pure states $\pm 1$, then the numerical solutions can be kept away from $\pm 1$ by a positive distance that is uniform with respect to the size of the time step and the grid. Finally, a few numerical experiments are presented. Convergence test is performed to demonstrate the accuracy and robustness of the proposed numerical scheme. Pearling bifurcation, meandering instability and spinodal decomposition are observed in the numerical simulations.

We present a new analytical and numerical framework for solution of Partial Differential Equations (PDEs) that is based on an exact transformation that moves the boundary constraints into the dynamics of the corresponding governing equation. The framework is based on a Partial Integral Equation (PIE) representation of PDEs, where a PDE equation is transformed into an equivalent PIE formulation that does not require boundary conditions on its solution state. The PDE-PIE framework allows for a development of a generalized PIE-Galerkin approximation methodology for a broad class of linear PDEs with non-constant coefficients governed by non-periodic boundary conditions, including, e.g., Dirichlet, Neumann and Robin boundaries. The significance of this result is that solution to almost any linear PDE can now be constructed in a form of an analytical approximation based on a series expansion using a suitable set of basis functions, such as, e.g., Chebyshev polynomials of the first kind, irrespective of the boundary conditions. In many cases involving homogeneous or simple time-dependent boundary inputs, an analytical integration in time is also possible. We present several PDE solution examples in one spatial variable implemented with the developed PIE-Galerkin methodology using both analytical and numerical integration in time. The developed framework can be naturally extended to multiple spatial dimensions and, potentially, to nonlinear problems.

In this work we consider stochastic gradient descent (SGD) for solving linear inverse problems in Banach spaces. SGD and its variants have been established as one of the most successful optimisation methods in machine learning, imaging and signal processing, etc. At each iteration SGD uses a single datum, or a small subset of data, resulting in highly scalable methods that are very attractive for large-scale inverse problems. Nonetheless, the theoretical analysis of SGD-based approaches for inverse problems has thus far been largely limited to Euclidean and Hilbert spaces. In this work we present a novel convergence analysis of SGD for linear inverse problems in general Banach spaces: we show the almost sure convergence of the iterates to the minimum norm solution and establish the regularising property for suitable a priori stopping criteria. Numerical results are also presented to illustrate features of the approach.

We study a class of McKean--Vlasov Stochastic Differential Equations (MV-SDEs) with drifts and diffusions having super-linear growth in measure and space -- the maps have general polynomial form but also satisfy a certain monotonicity condition. The combination of the drift's super-linear growth in measure (by way of a convolution) and the super-linear growth in space and measure of the diffusion coefficient require novel technical elements in order to obtain the main results. We establish wellposedness, propagation of chaos (PoC), and under further assumptions on the model parameters we show an exponential ergodicity property alongside the existence of an invariant distribution. No differentiability or non-degeneracy conditions are required. Further, we present a particle system based Euler-type split-step scheme (SSM) for the simulation of this type of MV-SDEs. The scheme attains, in stepsize, the strong error rate $1/2$ in the non-path-space root-mean-square error metric and we demonstrate the property of mean-square contraction. Our results are illustrated by numerical examples including: estimation of PoC rates across dimensions, preservation of periodic phase-space, and the observation that taming appears to be not a suitable method unless strong dissipativity is present.