We are interested in quantifying uncertainties that appear in nonlinear hyperbolic partial differential equations arising in a variety of applications from fluid flow to traffic modeling. A common approach to treat the stochastic components of the solution is by using generalized polynomial chaos expansions. This method was successfully applied in particular for general elliptic and parabolic PDEs as well as linear hyperbolic stochastic equations. More recently, gPC methods have been successfully applied to particular hyperbolic PDEs using the explicit form of nonlinearity or the particularity of the studied system structure as, e.g., in the p-system. While such models arise in many applications, e.g., in atmospheric flows, fluid flows under uncertain gas compositions and shallow water flows, a general gPC theory with corresponding numerical methods are still at large. Typical analytical and numerical challenges that appear for the gPC expanded systems are loss of hyperbolicity and positivity of solutions (like gas density or water depth). Any of those effects might trigger severe instabilities within classical finite-volume or discontinuous Galerkin methods. We will discuss properties and conditions to guarantee stability and present numerical results on selected examples.

We are interested in quantifying uncertainties that appear in nonlinear hyperbolic partial differential equations arising in a variety of applications from fluid flow to traffic modeling. A common approach to treat the stochastic components of the solution is by using generalized polynomial chaos expansions. This method was successfully applied in particular for general elliptic and parabolic PDEs as well as linear hyperbolic stochastic equations. More recently, gPC methods have been successfully applied to particular hyperbolic PDEs using the explicit form of nonlinearity or the particularity of the studied system structure as, e.g., in the p-system. While such models arise in many applications, e.g., in atmospheric flows, fluid flows under uncertain gas compositions and shallow water flows, a general gPC theory with corresponding numerical methods are still at large. Typical analytical and numerical challenges that appear for the gPC expanded systems are loss of hyperbolicity and positivity of solutions (like gas density or water depth). Any of those effects might trigger severe instabilities within classical finite-volume or discontinuous Galerkin methods. We will discuss properties and conditions to guarantee stability and present numerical results on selected examples.

We work on a portfolio management problem for a representative agent and a group of people, forming a market under relative performance concerns in a continuous-time setting. Herein, we define two wealth dynamics: the agent’s and the market’s wealth. The wealth dynamics appear in jump-diffusion markets. In our setting, we measure the performances of the market and the individual agent with preferences linked to the market performance. Therefore, we have two classical Merton problems to determine what the market does and the agent’s optimal strategy relative to the market performance. Furthermore, our framework assumes that the agent’s utility performance does not affect the market, while the market affects the agent’s utility. We explore the optimal investment strategies for both the agent and the market.

This is a joint work with Mogens Steffensen in the University of Copenhagen.

High dimensional approximation problems commonly arise from parametric PDE problems in which the parametric input depends on very many independent univariate random variables.  Typically (as in the method of “generalized polynomial chaos”, or GPC) the dependence on these variables is modelled by multivariate polynomials, leading to exponentially increasing difficulty and cost (expressed as the “curse of dimensionality”) as the dimension increases.  For this reason sparsity of coefficients is a major focus in implementations of GPC.

In this lecture we develop a different approach to one version of GPC.  In  this method there is no need for sparsification, and no curse of dimensionality.    The method, proposed in a 2022 paper with Frances Kuo, Vesa Kaarnioja, Yoshihito Kazashi and Fabio Nobile, uses kernel interpolation with periodic kernels, with the kernels located at lattice points, as advocated long ago by Hickernell and colleagues.

The lattice points and the kernels depend on parameters called “weights”.  In the 2022 paper the recommended weights were “SPOD” weights, leading to a cost growing as the square of the number of lattice points.   A newer 2023 paper with Kuo and Kaarnioja introduced “serendipitous” weights, for which the cost grows only linearly with both dimension and number of lattice points, allowing practical computations in as many as 1,000 dimensions.

The rate of convergence proved in the above papers was of the order $n^{-\alpha/2}$, for interpolation using the reproducing kernel of a space with mixed smoothness of order $\alpha$.  A new result with Frances Kuo doubles the proven convergence rate to $n^{-\alpha}$.

We propose our Adaptive Multilevel Monte Carlo (AMLMC) [Beck, Joakim, et al. "Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates." Computer Methods in Applied Mechanics and Engineering (2022)] method to solve an elliptic partial differential equation (PDE) with lognormal random input data, where the PDE model is subject to geometry-induced singularity. 

The previous work [Moon, K-S., et al. "Convergence rates for an adaptive dual weighted residual finite element algorithm." BIT Numerical Mathematics 46.2 (2006)] developed convergence rates for a goal-oriented adaptive algorithm based on isoparametric d-linear quadrilateral finite element approximations and the dual weighted residual error representation in the deterministic setting. This algorithm refines the mesh based on the error contribution to the QoI. 

This work aims to combine MLMC and the adaptive finite element solver. Contrary to the standard Multilevel Monte Carlo methods, where each sample is computed using a discretization-based numerical method, whose resolution is linked to the level, our AMLMC algorithm uses a sequence of tolerances as the levels. Specifically, for a given realization of the input coefficient and a given accuracy level, the AMLMC constructs its approximate sample as the ones using the first mesh in the sequence of deterministic, non-uniform meshes generated by the above-mentioned adaptive algorithm that satisfies the sample-dependent bias constraint.

Stochastic optimization problems arise in many fields, like data sciences, reliability engineering, and finance. The stochastic gradient descent (SGD) method is a cheap approach to solving such problems, relying on noisy gradient estimates to converge to local optima. However, in the case of μ-convex, L-smooth problems, the convergence is deeply affected by the condition number of the problem, L/μ. Here, we propose a Bayesian approach to find a suitable matrix to pre-condition gradient estimates in stochastic optimization that reduces the effect of large conditioning numbers. We show that maximizing the posterior distribution to find a suitable pre-conditioning matrix is a constrained deterministic strongly convex problem that can be solved efficiently using the Newton-CG method with a path-following approach. Numerical results on stochastic problems with large condition numbers show that our Bayesian quasi-Newton pre-conditioner improves the convergence of SGD.

Solving high-dimensional differential equations is still a challenging field for researchers. In recent years, many works have been presented that provide approximation by training neural networks using loss functions based on the differential operator of the equation at hand, as well as its initial/terminal and boundary conditions. In this work, we use a machine learning approach for pricing European (basket) options written with respect to a set of correlated underlyings whose dynamics undertake random jumps. We approximate the solution of the corresponding partial integro-differential equation using a variant of the deep Ritz method that splits the differential operator into symmetric and asymmetric parts. The method is driven by a modified version of the neural network introduced in the deep Galerkin method. The structure of the proposed neural network ensures the asymptotic behavior of the solution for large values of the underlyings. Moreover, it leads the outputs of the network to be consistent with the prior known qualitative properties of the solution. We present results on the Merton jump-diffusion model.

We consider the computational complexity of approximating elliptic PDEs with random coefficients by sparse product polynomial expansions. Except for special cases (for instance, when the spatial discretisation limits the achievable overall convergence rate), previous approaches for a posteriori selection of polynomial terms and corresponding spatial discretizations do not guarantee optimal complexity in the sense of computational costs scaling linearly in the number of degrees of freedom. We show that one can achieve optimality of an adaptive Galerkin scheme for discretizations by spline wavelets in the spatial variable when a multiscale representation of the affinely parameterized random coefficients is used. This is joint work with Igor Voulis.

M. Bachmayr and I. Voulis, An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality, ESAIM:M2AN 56(6), pp. 1955-1992, 2022. Preprint: arXiv:2109:09136.

After this we will outline the use of neural networks for the calibration of a financial asset price model in the context of financial option pricing. To provide an efficient calibration framework, a data-driven approach is proposed to learn the solutions of financial models and to reduce the corresponding computation time significantly. 
Specifically, fitting model parameters is formulated as training hidden neurons within a machine-learning framework.
The rapid on-line computation of ANNs combined with a flexible optimization method (i.e. Differential Evolution) provides us fast calibration without getting stuck in local minima.

We are interested in Monte Carlo (MC) methods for estimating probabilities of rare events associated with solutions to the McKean-Vlasov stochastic differential equation (MV-SDE). MV-SDEs arise in the mean-field limit of stochastic interacting particle systems, which have many applications in pedestrian dynamics, collective animal behaviour and financial mathematics. Importance sampling (IS) is used to reduce high relative variance in MC estimators of rare event probabilities. Optimal change of measure is methodically derived from variance minimisation, yielding a high-dimensional partial differential control equation which is cumbersome to solve. This problem is circumvented by using a decoupling approach, resulting in a lower dimensional control PDE. The decoupling approach necessitates the use of a double Loop Monte Carlo (DLMC) estimator. We further combine IS with a novel multilevel DLMC estimator which not only reduces complexity from O(TOL-4) to O(TOL-3) but also drastically reduces associated constant, enabling computationally feasible estimation of rare event probabilities.

$$
d S_t= \sigma(t,S_t) S_t \frac{\sqrt v_t}{\sqrt {\mathbb{E}[v_t|S_t]}}dW_t,
$$
where $W$ is a Brownian motion and $v$ is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model.
Whilst such models are quite popular among practitioners, unfortunately, its well-posedness has not been fully understood yet and, in general, is possibly not guaranteed at all.
We develop a novel regularization approach based on the reproducing kernel Hilbert space (RKHS) technique and show that  the regularized  model is well-posed.  Furthermore, we prove propagation of chaos. We demonstrate numerically  that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models. Our results are also applicable to more general McKean--Vlasov equations.

(Joint work with Denis Belomestny, Oleg Butkovsky, and John Schoenmakers.)

We present a Bayesian inference approach to solving the joint inverse problem for the image and the view angles, while also providing uncertainty estimates. For the image, we impose a Laplace difference prior enabling the representation of sharp edges in the image; this prior has connections to total variation regularization. For the view angles, we use a von Mises prior which is a 2π-periodic continuous probability distribution.
Numerical results show that our algorithm can jointly identify the image and the view angles, while also providing uncertainty estimates of both. We demonstrate our method with simulations of a 2D X-ray computed tomography problems using fan beam configurations.

This is joint work with N. A. B. Riis and Y. Dong, Technical University of Denmark; F. Uribe, LUT University, Finland; and J. M. Bardsley, University of Montana.

The characterisations such as entropy or the $f$-divergences need the possibility to compute point-wise functions of the \pdf.  This normally rather trivial task becomes more difficult when the \pdf is approximated in a low-rank tensor format, as the point values are not directly accessible anymore. The data is considered as an element of a high order tensor space.   All we require is that the data can be considered as an element of an associative, commutative algebra with an inner product. Such an algebra is isomorphic to a commutative sub-algebra of the usual matrix algebra, allowing the use of matrix algorithms to accomplish the mentioned tasks. The algorithms to be described are iterative methods or truncated series expansions for particular functions of the tensor, which will then exhibit the desired result.

This is joint work with Fabian Wagner (TUM), Iason Papaioannou (TUM) and Jonas Latz (Cambridge).

Asymptotic-preserving particle schemes are able to avoid these time-step constraints by explicitly including information from models describing the infinite collision rate case. However, these schemes produce biased results when used with large simulation time steps. In recent years, we have shown that the multilevel Monte Carlo method can be used to reduce this bias by combining simulations with large and small time steps, computing accurate results with greatly reduced simulation cost. In this talk, I will present the current state of the art for this newly developed asymptotic-preserving multilevel Monte Carlo approach. This includes an overview of existing methods and numerical results. I will then conclude with a view on future prospects for these methods.