Optimizing the Kelvin Force in a Moving Target Subdomain

In order to generate a desired Kelvin (magnetic) force in a target subdomain moving along a prescribed trajectory, we propose a minimization problem with a tracking type cost functional. We use the so-called dipole approximation to realize the magnetic field, where the location and the direction of the magnetic sources are assumed to be fixed. The magnetic field intensity acts as the control and exhibits limiting pointwise constraints. We address two specific problems: the first one corresponds to a fixed final time whereas the second one deals with an unknown force to minimize the final time. We prove existence of solutions and deduce local uniqueness provided that a second order sufficient condition is valid. We use the classical backward Euler scheme for time discretization. For both problems we prove the H^{1}-weak convergence of this semi-discrete numerical scheme. This result is motivated by Γ-convergence and does not require second order sufficient condition. If the latter holds then we prove H^{1}-strong local convergence. We report computational results to assess the performance of the numerical methods. As an application, we study the control of magnetic nanoparticles as those used in magnetic drug delivery, where the optimized Kelvin force is used to transport the drug to a desired location.

Forecasting without a model and with a partially known model

We will first review the diffusion maps algorithm, which can be interpreted as defining a generalized Fourier basis on a low-dimensional manifold described by a data set. We then present a recent application which uses this generalized Fourier basis to perform model-free probabilistic forecasting of time series. We also briefly discuss how to combine this model-free approach with an imperfect model in order to overcome model error.

Reconstructing the Transition Rate Function of a Broadwell Random Walk from Exit Times

In this presentation, we will show how one can utilize the layer stripping method to study a stochastic inverse problem arising from a one-dimensional Broadwell process. The Broadwell process is a random walk that transitions between two states, each associated with velocities of opposite sign, but having identical constant speed. Our goal is to reconstruct a spatially dependent transition rate function of the process from exit time distributions out of a finite interval. In practice, we are able to reconstruct flip rate functions with error within
O(10^{−2})from O(10^{5}) exit times when the particle speed is unity. For smaller particle speeds, the noise increases for a fixed number of exit times. This method is less time consuming compared with traditional projection methods which involve optimization.

Volume integral method for the Helmholtz equation with a variable coefficient

High frequency waves are used in magnetic confinement fusion to heat and probe the plasma. In the corresponding mathematical models for wave propagation, the inhomogeneity of the plasma density appears in variable coefficients. The talk will describe some mathematical and computational issues involved in developing a numerical method for such problems with variable coefficients, based on a volume integral formulation.

A Plane Wave Discontinuous Galerkin Method for Acoustic Scattering with Dirichlet-to-Neumann Boundary Conditions

Plane Wave Discontinuous Galerkin (PWDG) methods can be used to approximate the Helmholtz equation on a bounded domain. To approximate a scattering problem, the PWDG can be used on a bounded region of free space around the scatterer provided a suitable truncation condition is imposed on the artificial boundary. I shall present error estimates for using the Dirichlet to Neumann map to supply the truncating boundary conditions and show numerical results that demonstrate the use of this approach.

A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a multilevel method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the proposed method is illustrated by numerical experiments on benchmark problems.

Best approximation property of the finite element solutions for parabolic problems

Finite element error analysis of state constrained optimal control problem or optimal control problems with pointwise controls often require pointwise error estimates in form of the best approximations due to low regularity of the optimal state or adjoint variables. Such best approximation error estimates are well known for elliptic problems, but for the parabolic problems such results are only available for the semidiscrete and not for the fully discrete approximations. In my talk, after reviewing pointwise best approximation properties for elliptic problems and I show how such global and local best approximation results follow from our recently established results on discrete maximal parabolic regularity for a family of discontinuous Galerkin time discretization methods and the stability of the Ritz projection in L^{∞} norm.

Two-scale Method for the Monge-Ampere Equation: Convergence rates

We propose a two-scale finite element method for the numerical solution of the Monge-Ampère equation and prove that it has a rate of convergence of order h^{α/(α+2)} for solutions of class C^{2,α}. We also examine the rates for some weaker types of solutions. The method was inspired by the finite difference method of Froese and Oberman, but uses a different philosophy and tools. It relies on two scales: the first one is the mesh size h and the second one is a larger scale that controls appropriate directions and substitutes the need of a wide stencil. The proof hinges on a discrete Alexadroff estimate and a discrete continuous dependence on data.

Smoothing Point Clouds via Gradient Flows

In this talk we will discuss smoothing noisy point clouds sampled from manifolds in arbitrary dimensions using a gradient flow induced by the so called distance to measure function. Noisy point clouds arise in many applications, such as LiDAR and x-ray computed tomography. The objective of the gradient flow smoothing algorithm is to reduce the overall distance to measure of the point cloud. In doing so, we are reducing the noise in the sampled cloud. We will determine the effectiveness of the algorithm using both geometric and topological measures of accuracy.

PDEs on complex computational domains

In life sciences, such as neuroscience, PDE-based models are often defined on highly complex morphologies. Defining and handling the computational domain within hierarchical solvers, such as geometric multigrid methods, is one of the many challenges in efficiently solving systems of PDEs in life science applications. In this talk I will present new grid refinement methods for multigrid hierarchies, adaptive refinement through a posteriori error estimation, and multi-scale methods in the context of electrical and biochemical signaling in brain cells.

Numerical simulation of transient acoustic scattering by a piezoelectric obstacle.

A coupled BEM/FEM formulation for transient acoustic scattering by piezoelectric obstacles is proposed. The acoustic wave is represented by retarded layer potentials while the elastic displacement and electric potential are treated variationally, resulting in an integro-differential system. Well posedness of the Galerkin semi-discretization is shown in the Laplace domain and translated into time domain bounds. Trapezoidal-Rule and BDF2 Convolution Quadrature are combined with time stepping for time discretization obtaining a second order scheme.

A simple discontinuous Galerkin shock-capturing limiter with arbitrary-order accuracy

Shocks and discontinuities oftentimes appear in hyperbolic PDEs even when the initial conditions are smooth. In order to address this issue, shock-capturing methods are numerical methods that are specifically designed to handle discontinuities as well as smooth regimes in the solution. These type of methods are particularly important when designing high-order numerical methods, because these solvers produce Gibb’s phenomenon at locations of discontinuities. These artificial oscillations can create nonlinear instabilities that cause the solver to instantly fail. In this work, we present a novel shock capturing limiter for the high-order discontinuous Galerkin (DG) method. This limiter works by constructing local upper and lower bounds for the solution, and the solution is limited to stay within these bounds. The primary advantage of this limiter is that it is simple to implement (compared with classical WENO or artificial viscosity methods), it retains high-order accuracy in smooth regimes (in contrast with the original slope limiters when DG methods were first constructed), and perhaps more importantly, we find that it is effecting at resolving discontinuities without introducing unacceptable oscillations. In addition, we show how this limiter can be thought of as an extension of modern maximum principle preserving (MPP) limiters, wherein we are able to construct methods that also retain positivity of the solution with a negligible amount of additional work. We present one and two dimensional results on structured and unstructured grids that demonstrate the robustness of the solver and indicate the high-order accuracy of the solver.

Inverse subspace iteration for spectral stochastic finite element methods

We study random eigenvalue problems in the context of spectral stochastic finite elements. We assume that the matrix operator is given in the form of a polynomial chaos expansion, and we search for the coefficients of the polynomial chaos expansions of the eigenvectors and eigenvalues. We formulate a version of stochastic inverse subspace iteration, which is based on stochastic Galerkin finite element method, and we compare its accuracy with that of Monte Carlo and stochastic collocation methods.