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The Gottesman-Knill theorem established that stabilizer states and operations can be efficiently simulated classically. For qudits with dimension three and greater, stabilizer states and Clifford operations have been found to correspond to…
Common techniques for the spatial discretisation of PDEs on a macroscale grid include finite difference, finite elements and finite volume methods. Such methods typically impose assumed microscale structures on the subgrid fields, so…
When combining the numerical concept of variational discretization and semi-smooth Newton methods for the numerical solution of pde constrained optimization with control constraints, special emphasis has to be taken on the implementation,…
This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide…
We study the hard-core model of statistical mechanics on a unit cubic lattice $\mathbb{Z}^3$, which is intrinsically related to the sphere-packing problem for spheres with centers in $\mathbb{Z}^3$. The model is defined by the sphere…
We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. In a recent work, we established NP-hardness of…
We show that discrete quasiprobability distributions defined via the discrete Heisenberg-Weyl group can be obtained as discretizations of the continuous $SU(N)$ quasiprobability distributions. This is done by identifying the phase-point…
We investigate the approximation efficiency of score functions by deep neural networks in diffusion-based generative modeling. While existing approximation theories utilize the smoothness of score functions, they suffer from the curse of…
We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass,…
High-resolution simulations of particle-based kinetic plasma models typically require a high number of particles and thus often become computationally intractable. This is exacerbated in multi-query simulations, where the problem depends on…
In this article we apply a discrete action principle for the Vlasov--Maxwell equations in a structure-preserving particle-field discretization framework. In this framework the finite-dimensional electromagnetic potentials and fields are…
Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite…
Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase…
We consider a fully discrete loosely coupled scheme for incompressible fluid-structure interaction based on the time semi-discrete splitting method introduced in {\emph{[Burman, Durst \& Guzm\'an, arXiv:1911.06760]}}. The splittling method…
It is known that the solution of a conservative steady-state two-sided fractional diffusion problem can exhibit singularities near the boundaries. As consequence of this, and due to the conservative nature of the problem, we adopt a finite…
Operator splitting schemes are a class of powerful algorithms that solve complicated monotone inclusion and convex optimization problems that are built from many simpler pieces. They give rise to algorithms in which all simple pieces of the…
We prove the convergence of a spectral discretization of the Vlasov-Poisson system. The velocity term of the Vlasov equation is discretized using either Hermite functions on the infinite domain or Legendre polynomials on a bounded domain.…
Mixture models, such as Gaussian mixture models, are widely used in machine learning to represent complex data distributions. A key challenge, especially in high-dimensional settings, is to determine the mixture order and estimate the…
Dynamics simulations of constrained particles can greatly aid in understanding the temporal and spatial evolution of biological processes such as lateral transport along membranes and self-assembly of viruses. Most theoretical efforts in…
We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involves the conditional distribution of the marginals of the discrete Euler process. Analytically, the method…