Related papers: Acoustic shape optimization using energy stable cu…
In this chapter, we investigate recently proposed nonlinear conjugate gradient (NCG) methods for shape optimization problems. We briefly introduce the methods as well as the corresponding theoretical background and investigate their…
Second order accurate Cartesian grid methods have been well developed for interface problems in the literature. However, it is challenging to develop third or higher order accurate methods for problems with curved interfaces and internal…
We present a fully discrete finite element method for the interior null controllability problem subject to the wave equation. For the numerical scheme, piece-wise affine continuous elements in space and finite differences in time are…
We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…
Finite-difference methods are widely used for zeroth-order optimization in settings where gradient information is unavailable or expensive to compute. These procedures mimic first-order strategies by approximating gradients through function…
We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is…
We present a first-order method for solving constrained optimization problems. The method is derived from our previous work, a modified search direction method inspired by singular value decomposition. In this work, we simplify its…
This paper is concerned with the optimal shape design of the newtonian viscous incompressible fluids driven by the stationary nonhomogeneous Navier-Stokes equations. We use three approaches to derive the structures of shape gradients for…
In this paper, we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains, which to the best of our knowledge is a new contribution to the literature.…
This paper studies finite element approximations of the stochastic Allen-Cahn equation with gradient-type multiplicative noises that are white in time and correlated in space. The sharp interface limit as the parameter $\epsilon \rightarrow…
A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full…
We introduce a convergent finite difference method for solving the optimal transportation problem on the sphere. The method applies to both the traditional squared geodesic cost (arising in mesh generation) and a logarithmic cost (arising…
We study space--time isogeometric discretizations of the linear acoustic wave equation that use splines of arbitrary degree p, both in space and time. We propose a space--time variational formulation that is obtained by adding a…
A characteristic feature of the state-of-the-art of real-space methods in electronic structure calculations is the diversity of the techniques used in the discretization of the relevant partial differential equations. In this context, the…
We propose a formal framework based on collective coordinates to reduce infinite-dimensional stochastic partial differential equations (SPDEs) with symmetry to a set of finite-dimensional stochastic differential equations which describe the…
Shape optimization based on surface gradients and the Hadarmard-form is considered for a compressible viscous fluid. Special attention is given to the difference between the 'function composition' approach involving local shape derivatives…
The stochastic composition optimization proposed recently by Wang et al. [2014] minimizes the objective with the compositional expectation form: $\min_x~(\mathbb{E}_iF_i \circ \mathbb{E}_j G_j)(x).$ It summarizes many important applications…
There are several applications of stochastic optimization where one can benefit from a robust estimate of the gradient. For example, domains such as distributed learning with corrupted nodes, the presence of large outliers in the training…
It seems that in the current age, computers, computation, and data have an increasingly important role to play in scientific research and discovery. This is reflected in part by the rise of machine learning and artificial intelligence,…
From ancient to modern times, acoustic structures have been used to control the propagation of acoustic waves. However, the design of the acoustic structures has remained widely a time-consuming and computational resource-consuming…