Delft University of Technology

Our Random Walkers Induced temporal Graphs (RWIG) model generates temporal graph sequences based on M independent, random walkers that traverse an underlying graph as a function of time. Co-location of walkers at a given node and time defines an individual-level contact. RWIG is shown to be a realistic model for temporal human contact graphs.   

A key idea is that a random walk on a Markov graph executes the Markov process. Each of the M walkers traverses the same set of nodes (= states in the Markov graph), but with own transition probabilities (in discrete time) or rates (in continuous time). Hence, the Markov transition probability matrix P_j reflects the policy of motion of walker w_j. RWIG is analytically feasible: we derive closed form solutions for the probability distribution of contact graphs.

Usually, human mobility networks are inferred through measurements of timeseries of contacts between individuals. We also discuss this “inverse RWIG problem”, which aims to determine the parameters in RWIG (i.e. the set of probability transfer matrices P₁, P₂, ..., P_M and the initial probability state vectors s₁[0], ...,s_M[0] of walkers w₁,w₂, ...,w_M in discrete time), given a timeseries of contact graphs.

This talk is based on the article:
Almasan, A.-D., Shvydun, S., Scholtes, I. and P. Van Mieghem, 2025, "Generating Temporal Contact Graphs Using Random Walkers", IEEE Transactions on Network Science and Engineering, to appear.

In this talk I will consider the Burgers equation with a homogeneous Possion process as a forcing potential. In recent years, the randomly forced Burgers equation, with forcing that is ergodic in time, received a lot of attention, especially the almost sure existence of unique global solutions with given average velocity, that at each time only depend on the history up to that time. However, in all these results compactness in the space dimension of the forcing was essential. It was even conjectured that in the non-compact setting such unique global solutions would not exist. However, we have managed to use techniques developed for first and last passage percolation models to prove that in the case of Poisson forcing, these global solutions do exist almost surely, due to the existence of semi-infinite minimizers of the Lagrangian action. In this talk I will discuss this result and explain some of the techniques we have used.

This is joined work Yuri Bakhtin and Konstantin Khanin.

Pathwise Holder convergence with optimal rates is proved for the implicit Euler scheme associated with semilinear stochastic evolution equations with multiplicative noise. The results are applied to a class of second order parabolic SPDEs driven by space-time white noise. This is joint work with Sonja Cox.

The numerical study of lattice quantum chromodynamics (QCD) is an attempt to extract predictions about the world around us from the standard model of physics. Worldwide, there are several large collaborations on lattice QCD methods, with terascale computing power dedicated to these problems. Central to the computation in lattice QCD is the inversion of a series of fermion matrices, representing the interaction of quarks on a four-dimensional space-time lattice. In practical computation, this inversion may be approximated based on the solution of a set of linear systems.

In this talk, I will present a basic description of the linear algebra problems in lattice QCD and why we believe that multigrid methods are well-suited to effectively solving them. While multigrid methods are known to be efficient solution techniques for many operators, those arising in lattice QCD offer new challenges, not easily handled by classical multigrid and algebraic multigrid approaches. The role of adaptive multigrid techniques in addressing the fermion matrices will be highlighted, along with preliminary results for several model problems.

Joint work with Yogi Erlangga and Kees Vuik.

Shifted Laplace preconditioners have attracted considerable attention as a technique to speed up convergence of iterative solution methods for the Helmholtz equation. In this paper we present a comprehensive spectral analysis of the Helmholtz operator preconditioned with a shifted Laplacian. Our analysis is valid under general conditions. The propagating medium can be heterogeneous, and the analysis also holds for different types of damping, including a radiation condition for the boundary of the computational domain. By combining the results of the spectral analysis of the preconditioned Helmholtz operator with an upper bound on the GMRES-residual norm we are able to provide an optimal value for the shift, and to explain the mesh-depency of the convergence of GMRES preconditioned with a shifted Laplacian. We illustrate our results with a seismic test problem.

In this presentation we discuss the Heston model with stochastic interest rates driven by Hull-White or Cox-Ingersoll-Ross processes.

We present approximations in the Heston-Hull-White hybrid model, so that a characteristic function can be derived and derivative pricing can be efficiently done using the Fourier Cosine expansion technique.

This pricing method, called the COS method, is explained in some detail.

We furthermore discuss the effect of the approximations in the hybrid model on the instantaneous correlations, and check the influence of the correlation between stock and interest rate on the implied volatilities.

Shifted Laplace preconditioners have attracted considerable attention as

a technique to speed up convergence of iterative solution methods for the

Helmholtz equation. In the talk we present a comprehensive spectral

analysis of the discrete Helmholtz operator preconditioned with a shifted

Laplacian. Our analysis is valid under general conditions. The propagating

medium can be heterogeneous, and the analysis also holds for different types

of damping, including a radiation condition for the boundary of the computational

domain. By combining the results of the spectral analysis of the

preconditioned Helmholtz operator with an upper bound on the GMRES-residual

norm we are able to derive an optimal value for the shift, and to

explain the mesh-depency of the convergence of GMRES preconditioned

with a shifted Laplacian. We will illustrate our results with a seismic test

problem.

Joint work with: Yogi Erlangga (University of British Columbia) and Kees Vuik (TU Delft)