Cache-Aware and Roofline-Ideal Automatic Differentiation

Automatic differentiation software libraries augment arithmetic operations with their derivatives, thereby relieving the programmer of deriving, implementing, debugging, and maintaining derivative code. With this encapsulation however, the responsibility of code optimization relies more heavily on the AD system itself (as opposed to the programmer and the compiler). Moreover, given that there are multiple contexts in reservoir simulation software for which derivatives are required (e.g. property package and discrete operator evaluations), the AD infrastructure must also be adaptable. An Operator Overloading AD design is proposed and tested to provide scalability and computational efficiency seemlessly across memory- and compute-bound applications. This is achieved by 1) use of portable and standard programming language constructs (C++17 and OpenMP 4.5 standards), 2) adopting a vectorized programming interface, 3) lazy evaluation via expression templates, and 4) multiple memory alignment and layout policies. Empirical analysis is conducted on various kernels spanning various arithmetic intensity and working set sizes. Cache- aware roofline analysis results show that the performance and scalability attained are reliably ideal. In terms of floapting point operations executed per second, the performance of the AD system matches optimized hand-code. Finally, the implementation is benchmarked using the Automatically Differentiable Expression Templates Library (ADETL).

A proof of the completeness of Lamb modes

The aim of this paper is to give a precise proof of the completeness of Lamb modes and associated modes. This proof is relatively simple and short but relies on two powerful mathematical theorems. The first one is a theorem on elliptic systems with a parameter due to Agranovich and Vishik. The second one is a theorem due to Locker which gives a criterion to show the completeness of the set of generalized eigenvectors of a Hilbert-Schmidt discrete operator.

Discrete spectra for some complex infinite band matrices

Under suitable assumptions the eigenvalues for an unbounded discrete operator \(A\) in \(l_2\), given by an infinite complex band-type matrix, are approximated by the eigenvalues of its orthogonal truncations. Let \[\Lambda (A)=\{\lambda \in {\rm Lim}_{n\to \infty} \lambda _n : \lambda _n \text{ is an eigenvalue of } A_n \text{ for } n \geq 1 \},\] where \({\rm Lim}_{n\to \infty} \lambda_n\) is the set of all limit points of the sequence \((\lambda_n)\) and \(A_n\) is a finite dimensional orthogonal truncation of \(A\). The aim of this article is to provide the conditions that are sufficient for the relations \(\sigma(A) \subset \Lambda(A)\) or \(\Lambda (A) \subset \sigma (A)\) to be satisfied for the band operator \(A\).

Approximation by pseudo-linear discrete operators

<p style='text-indent:20px;'>In this note, we construct a pseudo-linear kind discrete operator based on the continuous and nondecreasing generator function. Then, we obtain an approximation to uniformly continuous functions through this new operator. Furthermore, we calculate the error estimation of this approach with a modulus of continuity based on a generator function. The obtained results are supported by visualizing with an explicit example. Finally, we investigate the relation between discrete operators and generalized sampling series.</p>

Different dimensional fractional-order discrete chaotic systems based on the Caputo h-difference discrete operator: dynamics, control, and synchronization

The paper investigates control and synchronization of fractional-order maps described by the Caputo h-difference operator. At first, two new fractional maps are introduced, i.e., the Two-Dimensional Fractional-order Lorenz Discrete System (2D-FoLDS) and Three-Dimensional Fractional-order Wang Discrete System (3D-FoWDS). Then, some novel theorems based on the Lyapunov approach are proved, with the aim of controlling and synchronizing the map dynamics. In particular, a new hybrid scheme is proposed, which enables synchronization to be achieved between a master system based on a 2D-FoLDS and a slave system based on a 3D-FoWDS. Simulation results are reported to highlight the effectiveness of the conceived approach.

On the Numerical Implementation of the Multiplicative Regularization in Microwave Imaging

<p> We consider the widely-used weighted L2 norm total variation multiplicative regularizer (MR) for both the Gauss-Newton inversion (GNI) and contrast source inversion (CSI) algorithms in microwave imaging (MWI). It is shown that the proper numerical implementation of the discretized MR operator is important for the GNI algorithm whereas the CSI algorithm is more robust with respect to different implementations of this MR. For the GNI algorithm, the MR operator should be discretized such that high spatial frequency components are not present in its nullspace, and also the resulting discrete operator is positive definite.</p>

On the Numerical Implementation of the Multiplicative Regularization in Microwave Imaging

<p> We consider the widely-used weighted L2 norm total variation multiplicative regularizer (MR) for both the Gauss-Newton inversion (GNI) and contrast source inversion (CSI) algorithms in microwave imaging (MWI). It is shown that the proper numerical implementation of the discretized MR operator is important for the GNI algorithm whereas the CSI algorithm is more robust with respect to different implementations of this MR. For the GNI algorithm, the MR operator should be discretized such that high spatial frequency components are not present in its nullspace, and also the resulting discrete operator is positive definite.</p>

Extending square conservation to arbitrarily structured C-grids with shallow water equations

The square conservation law is implemented in atmospheric dynamic cores on latitude–longitude grids, but it is rarely implemented on quasi-uniform grids, given the difficulty involved in constructing anti-symmetrical spatial discrete operators on these grids. Increasingly more models are being developed on quasi-uniform grids, such as arbitrarily structured C-grids. Thuburn–Ringler–Skamarock–Klemp (TRiSK) is a shallow water dynamic core on an arbitrarily structured C-grid. The spatial discrete operator of TRiSK is able to naturally maintain the conservation properties of total mass and total absolute vorticity and conserving total energy with time truncation error; the first two integral invariants are exactly conserved during integration, but the total energy dissipates when using the dissipative temporal integration schemes, i.e., Runge–Kutta (RK). The method of strictly conserving the total energy simultaneously, which means conserving energy in the round-off error over the entire temporal integration period, uses both an anti-symmetrical spatial discrete operator and a square conservative temporal integration scheme. In this study, we demonstrate that square conservation is equivalent to energy conservation in both a continuous shallow water system and a discrete shallow water system of TRiSK. After that, we attempt to extend the square conservation law to the TRiSK framework. To overcome the challenge of constructing an anti-symmetrical spatial discrete operator, we unify the unit of evolution variables of shallow water equations using the Institute of Atmospheric Physics (IAP) transformation, and the temporal derivatives of new evolution variables can be expressed by a combination of temporal derivatives of original evolution variables, which means the square conservative spatial discrete operator can be obtain by using original spatial discrete operators in TRiSK. Using the square conservative Runge–Kutta scheme, the total energy is completely conserved, and there is no influence on the properties of conserving total mass and total absolute vorticity. In the standard shallow water numerical test, the square conservative scheme not only helps maintain total conservation of the three integral invariants but also creates less simulation error norms.

On a generalized function-to-sequence transform

By attaching a sequence {?n}n?N0 to the binomial transform, a new operator D? is obtained. We use the same sequence to define a new transform T? mapping derivatives to the powers of D?, and integrals to D-1?. The inverse transform B? of T? is introduced and its properties are studied. For ?n = (-1)n, B? reduces to the Borel transform. Applying T? to Bessel's differential operator d/dx x d/dx, we obtain Bessel's discrete operator D?nN?. Its eigenvectors correspond to eigenfunctions of Bessel's differential operator.