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Numerically the reconstructability of unknown parameters in inverse problems heavily relies on the chosen data. Therefore, it is crucial to design an experiment that yields data that is sensitive to the parameters. We approach this problem from the perspective of a least squares optimization, and examine the positivity of the Gauss-Newton Hessian at the global minimum point of the objective function. We propose a general framework that provides an efficient down-sampling strategy that can select data that preserves the strict positivity of the Hessian. Matrix sketching techniques from randomized linear algebra is heavily leaned on to achieve this goal. The method requires drawing samples from a certain distribution, and gradient free sampling methods are integrated to execute the data selection. Numerical experiments demonstrate the effectiveness of this method in selecting sensor locations for Schr\"odinger potential reconstruction.

Chemotaxis describes the intricate interplay of cellular motion in response to a chemical signal. We here consider the case of slab geometry which models chemotactic motion between two infinite membranes. Like previous works, we are particularly interested in the asymptotic regime of high tumbling rates. We establish local existence and uniqueness of solutions to the kinetic equation and show their convergence towards solutions of a parabolic Keller-Segel model in the asymptotic limit. In addition, we prove convergence rates with respect to the asymptotic parameter under additional regularity assumptions on the problem data. Particular difficulties in our analysis are caused by vanishing velocities in the kinetic model as well as the occurrence of boundary terms.

Bacterial motion is steered by external stimuli (chemotaxis), and the motion described on the mesoscopic scale is uniquely determined by a parameter $K$ that models velocity change response from the bacteria. This parameter is called chemotaxis kernel. In a practical setting, it is inferred by experimental data. We deploy a PDE-constrained optimization framework to perform this reconstruction using velocity-averaged, localized data taken in the interior of the domain. The problem can be well-posed or ill-posed depending on the data preparation and the experimental setup. In particular, we propose one specific design that guarantees numerical reconstructability and local convergence. This design is adapted to the discretization of $K$ in space and decouples the reconstruction of local values of $K$ into smaller cell problems, opening up parallelization opportunities. Numerical evidences support the theoretical findings.

Chemotaxis is the physical phenomenon that bacteria adjust their motions according to chemical stimulus. A classical model for this phenomenon is a kinetic equation that describes the velocity jump process whose tumbling/transition kernel uniquely determines the effect of chemical stimulus on bacteria. The model has been shown to be an accurate model that matches with bacteria motion qualitatively. For a quantitative modeling, biophysicists and practitioners are also highly interested in determining the explicit value of the tumbling kernel. Due to the experimental limitations, measurements are typically macroscopic in nature. Do macroscopic quantities contain enough information to recover microscopic behavior? In this paper, we give a positive answer. We show that when given a special design of initial data, the population density, one specific macroscopic quantity as a function of time, contains sufficient information to recover the tumbling kernel and its associated damping coefficient. Moreover, we can read off the chemotaxis tumbling kernel using the values of population density directly from this specific experimental design. This theoretical result using kinetic theory sheds light on how practitioners may conduct experiments in laboratories.

In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller-Segel equation and a chemotaxis kinetic equation. These two equations describe the organism movement at the macro- and mesoscopic level respectively, and are asymptotically equivalent in the parabolic regime. How the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller-Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms' population level movement when reacting to certain stimulation. From this one infers the chemotaxis response, which constitutes an inverse problem. \\ In this paper we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated to two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought after. We prove the asymptotic equivalence of the two posterior distributions.