35 research outputs found
Shape optimization of Stokesian peristaltic pumps using boundary integral methods
This article presents a new boundary integral approach for finding optimal
shapes of peristaltic pumps that transport a viscous fluid. Formulas for
computing the shape derivatives of the standard cost functionals and
constraints are derived. They involve evaluating physical variables (traction,
pressure, etc.) on the boundary only. By emplyoing these formulas in conjuction
with a boundary integral approach for solving forward and adjoint problems, we
completely avoid the issue of volume remeshing when updating the pump shape as
the optimization proceeds. This leads to significant cost savings and we
demonstrate the performance on several numerical examples
Simple and efficient representations for the fundamental solutions of Stokes flow in a half-space
We derive new formulas for the fundamental solutions of slow, viscous flow,
governed by the Stokes equations, in a half-space. They are simpler than the
classical representations obtained by Blake and collaborators, and can be
efficiently implemented using existing fast solvers libraries. We show, for
example, that the velocity field induced by a Stokeslet can be annihilated on
the boundary (to establish a zero slip condition) using a single reflected
Stokeslet combined with a single Papkovich-Neuber potential that involves only
a scalar harmonic function. The new representation has a physically intuitive
interpretation
Natural evolution strategies and variational Monte Carlo
A notion of quantum natural evolution strategies is introduced, which
provides a geometric synthesis of a number of known quantum/classical
algorithms for performing classical black-box optimization. Recent work of
Gomes et al. [2019] on heuristic combinatorial optimization using neural
quantum states is pedagogically reviewed in this context, emphasizing the
connection with natural evolution strategies. The algorithmic framework is
illustrated for approximate combinatorial optimization problems, and a
systematic strategy is found for improving the approximation ratios. In
particular it is found that natural evolution strategies can achieve
approximation ratios competitive with widely used heuristic algorithms for
Max-Cut, at the expense of increased computation time
Digital quantum simulation of Schr\"odinger dynamics using adaptive approximations of potential functions
Digital quantum simulation (DQS) of continuous-variable quantum systems in
the position basis requires efficient implementation of diagonal unitaries
approximating the time evolution operator generated by the potential energy
function. In this work, we provide efficient implementations suitable for
potential functions approximable by piecewise polynomials, with either uniform
or adaptively chosen subdomains. For a fixed precision of approximation, we
show how adaptive grids can significantly reduce the total gate count at the
cost of introducing a small number of ancillary qubits. We demonstrate the
circuit construction with both physically motivated and artificially designed
potential functions, and discuss their generalizations to higher dimensions
Toward Neural Network Simulation of Variational Quantum Algorithms
Variational quantum algorithms (VQAs) utilize a hybrid quantum-classical
architecture to recast problems of high-dimensional linear algebra as ones of
stochastic optimization. Despite the promise of leveraging near- to
intermediate-term quantum resources to accelerate this task, the computational
advantage of VQAs over wholly classical algorithms has not been firmly
established. For instance, while the variational quantum eigensolver (VQE) has
been developed to approximate low-lying eigenmodes of high-dimensional sparse
linear operators, analogous classical optimization algorithms exist in the
variational Monte Carlo (VMC) literature, utilizing neural networks in place of
quantum circuits to represent quantum states. In this paper we ask if classical
stochastic optimization algorithms can be constructed paralleling other VQAs,
focusing on the example of the variational quantum linear solver (VQLS). We
find that such a construction can be applied to the VQLS, yielding a paradigm
that could theoretically extend to other VQAs of similar form.Comment: To appear at the workshop on AI for Science: Progress and Promises at
NeurIPS 202
Spectrally Accurate Quadratures for Evaluation of Layer Potentials Close to the Boundary for the 2D Stokes and Laplace Equations
Dense particulate flow simulations using integral equation methods demand accurate evaluation of Stokes layer potentials on arbitrarily close interfaces. In this paper, we generalize techniques for close evaluation of Laplace double-layer potentials in [J. Helsing and R. Ojala, J. Comput. Phys., 227 (2008), pp. 2899--2921]. We create a “globally compensated” trapezoid rule quadrature for the Laplace single-layer potential on the interior and exterior of smooth curves. This exploits a complex representation, a product quadrature (in the style of Kress) for the sawtooth function, careful attention to branch cuts, and second-kind barycentric-type formulae for Cauchy integrals and their derivatives. Upon this we build accurate single- and double-layer Stokes potential evaluators by expressing them in terms of Laplace potentials. We test their convergence for vesicle-vesicle interactions, for an extensive set of Laplace and Stokes problems, and when applying the system matrix in a boundary value problem solver in the exterior of multiple close-to-touching ellipses. We achieve typically 12 digits of accuracy using small numbers of discretization nodes per curve. We provide documented codes for other researchers to use