Many important high-dimensional dynamical systems exhibit complex chaotic behaviour. Their complexity means that their dynamics are necessarily comprehended under strong reducing assumptions. It is therefore important to have a clear picture of these
reducing assumptions range of validity. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively hyperbolic, which implies many felicitous statistical properties. We demonstrate, contrary to the chaotic hypothesis, the existence of non-hyperbolic large-scale dynamics in a mean-field coupled system. To do this we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev Galerkin transfer operator discretisation. This enables us to obtain a high precision estimate of a homoclinic tangency, implying a failure of hyperbolicity. Robust non-hyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be assumed to hold in all systems, and a better understanding of the domain of its validity is required.
Given the Hamiltonian, the evaluation of unitary operators has been at the heart of many quantum algorithms. Motivated by existing deterministic and random methods, we present a hybrid approach, where Hamiltonians with large amplitude are evaluated a
t each time step, while the remaining terms are evaluated at random. The bound for the mean square error is obtained, together with a concentration bound. The mean square error consists of a variance term and a bias term, arising respectively from the random sampling of the Hamiltonian terms and the operator splitting error. Leveraging on the bias/variance trade-off, the error can be minimized by balancing the two. The concentration bound provides an estimate on the number of gates. The estimates are verified by using numerical experiments on classical computers.
Spatial symmetries and invariances play an important role in the description of materials. When modelling material properties, it is important to be able to respect such invariances. Here we discuss how to model and generate random ensembles of tenso
rs where one wants to be able to prescribe certain classes of spatial symmetries and invariances for the whole ensemble, while at the same time demanding that the mean or expected value of the ensemble be subject to a possibly higher spatial invariance class. Our special interest is in the class of physically symmetric and positive definite tensors, as they appear often in the description of materials. As the set of positive definite tensors is not a linear space, but rather an open convex cone in the linear vector space of physically symmetric tensors, it may be advantageous to widen the notion of mean to the so-called Frechet mean, which is based on distance measures between positive definite tensors other than the usual Euclidean one. For the sake of simplicity, as well as to expose the main idea as clearly as possible, we limit ourselves here to second order tensors. It is shown how the random ensemble can be modelled and generated, with fine control of the spatial symmetry or invariance of the whole ensemble, as well as its Frechet mean, independently in its scaling and directional aspects. As an example, a 2D and a 3D model of steady-state heat conduction in a human proximal femur, a bone with high material anisotropy, is explored. It is modelled with a random thermal conductivity tensor, and the numerical results show the distinct impact of incorporating into the constitutive model different material uncertainties$-$scaling, orientation, and prescribed material symmetry$-$on the desired quantities of interest, such as temperature distribution and heat flux.
We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational me
thod. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.
We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution $pi$ on R d , based on (over-damped) Langevin diffusions. This method both inspired by [PP18] and [GMS + 20] relies on a multilevel occupation measur
e, i.e. on an appropriate combination of R occupation measures of (constant-step) discretized schemes of the Langevin diffusion with respective steps $gamma$r = $gamma$02 --r , r = 0,. .. , R. For a given diffusion, we first state a result under general assumptions which guarantees an $epsilon$-approximation (in a L 2-sense) with a cost proportional to $epsilon$ --2 (i.e. proportional to a Monte-Carlo method without bias) or $epsilon$ --2 | log $epsilon$| 3 under less contractive assumptions. This general result is then applied to over-damped Langevin diffusions in a strongly convex setting, with a study of the dependence in the dimension d and in the spectrum of the Hessian matrix D 2 U of the potential U : R d $rightarrow$ R involved in the Gibbs distribution. This leads to strategies with cost in O(d$epsilon$ --2 log 3 (d$epsilon$ --2)) and in O(d$epsilon$ --2) under an additional condition on the third derivatives of U. In particular, in our last main result, we show that, up to universal constants, an appropriate choice of the diffusion coefficient and of the parameters of the procedure leads to a cost controlled by ($lambda$ U $lor$1) 2 $lambda$ 3 U d$epsilon$ --2 (where$lambda$U and $lambda$ U respectively denote the supremum and the infimum of the largest and lowest eigenvalue of D 2 U). In our numerical illustrations, we show that our theoretical bounds are confirmed in practice and finally propose an opening to some theoretical or numerical strategies in order to increase the robustness of the procedure when the largest and smallest eigenvalues of D 2 U are respectively too large or too small.
Under some regularity assumptions, we report an a priori error analysis of a dG scheme for the Poisson and Stokes flow problem in their dual mixed formulation. Both formulations satisfy a Babuv{s}ka-Brezzi type condition within the space H(div) x L2.
It is well known that the lowest order Crouzeix-Raviart element paired with piecewise constants satisfies such a condition on (broken) H1 x L2 spaces. In the present article, we use this pair. The continuity of the normal component is weakly imposed by penalizing jumps of the broken H(div) component. For the resulting methods, we prove well-posedness and convergence with constants independent of data and mesh size. We report error estimates in the methods natural norms and optimal local error estimates for the divergence error. In fact, our finite element solution shares for each triangle one DOF with the CR interpolant and the divergence is locally the best-approximation for any regularity. Numerical experiments support the findings and suggest that the other errors converge optimally even for the lowest regularity solutions and a crack-problem, as long as the crack is resolved by the mesh.
We study the problem of approximating the eigenspectrum of a symmetric matrix $A in mathbb{R}^{n times n}$ with bounded entries (i.e., $|A|_{infty} leq 1$). We present a simple sublinear time algorithm that approximates all eigenvalues of $A$ up to a
dditive error $pm epsilon n$ using those of a randomly sampled $tilde{O}(frac{1}{epsilon^4}) times tilde O(frac{1}{epsilon^4})$ principal submatrix. Our result can be viewed as a concentration bound on the full eigenspectrum of a random principal submatrix. It significantly extends existing work which shows concentration of just the spectral norm [Tro08]. It also extends work on sublinear time algorithms for testing the presence of large negative eigenvalues in the spectrum [BCJ20]. To complement our theoretical results, we provide numerical simulations, which demonstrate the effectiveness of our algorithm in approximating the eigenvalues of a wide range of matrices.
A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work
on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a natural generalization of the upwind flux can be evaluated directly on coarse levels using the precomputed coarse flux basis vectors. The resulting solver is applicable to problems discretized on general unstructured grids. The performance of the proposed nonlinear multigrid solver in comparison with the standard single level Newtons method is demonstrated through challenging numerical examples. It is observed that the proposed solver is robust for highly nonlinear problems and clearly outperforms Newtons method in the case of high Courant-Friedrichs-Lewy (CFL) numbers.
In this article, theory-based analytical methodologies of astrophysics employed in the modern era are suitably operated alongside a test research-grade telescope to image and determine the orbit of a near-earth asteroid from original observations, me
asurements, and calculations. Subsequently, its intrinsic orbital path has been calculated including the chance it would likely impact Earth in the time ahead. More so specifically, this case-study incorporates the most effective, feasible, and novel Gausss Method in order to maneuver the orbital plane components of a planetesimal, further elaborating and extending our probes on a selected near-earth asteroid (namely the 12538-1998 OH) through the observational data acquired over a six week period. Utilizing the CCD (Charge Coupled Device) snapshots captured, we simulate and calculate the orbit of our asteroid as outlined in quite detailed explanations. The uncertainties and deviations from the expected values are derived to reach a judgement whether our empirical findings are truly reliable and representative measurements by partaking a statistical analysis based systematic approach. Concluding the study by narrating what could have caused such discrepancy of findings in the first place, if any, measures are put forward that could be undertaken to improve the test-case for future investigations. Following the calculation of orbital elements and their uncertainties using Monte Carlo analysis, simulations were executed with various sample celestial bodies to derive a plausible prediction regarding the fate of Asteroid 1998 OH. Finally, the astrometric and photometric data, after their precise verification, were officially submitted to the Minor Planet Center: an organization hosted by the Center for Astrophysics, Harvard and Smithsonian and funded by NASA, for keeping track of the asteroids potential trajectories.
We present a novel computational modeling framework to numerically investigate fluid-structure interaction in viscous fluids using the phase field embedding method. Each rigid body or elastic structure immersed in the incompressible viscous fluid mat
rix, grossly referred to as the particle in this paper, is identified by a volume preserving phase field. The motion of the particle is driven by the fluid velocity in the matrix for passive particles or combined with its self-propelling velocity for active particles. The excluded volume effect between a pair of particles or between a particle and the boundary is modeled by a repulsive potential force. The drag exerted to the fluid by a particle is assumed proportional to its velocity. When the particle is rigid, its state is described by a zero velocity gradient tensor within the nonzero phase field that defines its profile and a constraining stress exists therein. While the particle is elastic, a linear constitutive equation for the elastic stress is provided within the particle domain. A hybrid, thermodynamically consistent hydrodynamic model valid in the entire computational domain is then derived for the fluid-particle ensemble using the generalized Onsager principle accounting for both rigid and elastic particles. Structure-preserving numerical algorithms are subsequently developed for the thermodynamically consistent model. Numerical tests in 2D and 3D space are carried out to verify the rate of convergence and numerical examples are given to demonstrate the usefulness of the computational framework for simulating fluid-structure interactions for passive as well as self-propelling active particles in a viscous fluid matrix.
