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An automated framework is presented for the numerical solution of optimal control problems with PDEs as constraints, in both the stationary and instationary settings. The associated code can solve both linear and non-linear problems, and examples for incompressible flow equations are considered. The software, which is based on a Python interface to the Firedrake system, allows for a compact definition of the problem considered by providing a few lines of code in a high-level language. The software is provided with efficient iterative linear solvers for optimal control problems with PDEs as constraints. The use of advanced preconditioning techniques results in a significant speed-up of the solution process for large-scale problems. We present numerical examples of the applicability of the software on classical control problems with PDEs as constraints.

In this paper, we consider effective discretization strategies and iterative solvers for nonlinear PDE-constrained optimization models for pattern evolution within biological processes. Upon a Sequential Quadratic Programming linearization of the optimization problem, we devise appropriate time-stepping schemes and discrete approximations of the cost functionals such that the discretization and optimization operations are commutative, a highly desirable property of a discretization of such problems. We formulate the large-scale, coupled linear systems in such a way that efficient preconditioned iterative methods can be applied within a Krylov subspace solver. Numerical experiments demonstrate the viability and efficiency of our approach., Comment: 33 pages, 3 figures, 7 tables

Beyond estimating parameters of interest from data, one of the key goals of statistical inference is to properly quantify uncertainty in these estimates. In Bayesian inference, this uncertainty is provided by the posterior distribution, the computation of which typically involves an intractable high-dimensional integral. Among available approximation methods, sampling-based approaches come with strong theoretical guarantees but scale poorly to large problems, while variational approaches scale well but offer few theoretical guarantees. In particular, variational methods are known to produce overconfident estimates of posterior uncertainty and are typically non-identifiable, with many latent variable configurations generating equivalent predictions. Here, we address these challenges by showing how diffusion-based models (DBMs), which have recently produced state-of-the-art performance in generative modeling tasks, can be repurposed for performing calibrated, identifiable Bayesian inference. By exploiting a previously established connection between the stochastic and probability flow ordinary differential equations (pfODEs) underlying DBMs, we derive a class of models, inflationary flows, that uniquely and deterministically map high-dimensional data to a lower-dimensional Gaussian distribution via ODE integration. This map is both invertible and neighborhood-preserving, with controllable numerical error, with the result that uncertainties in the data are correctly propagated to the latent space. We demonstrate how such maps can be learned via standard DBM training using a novel noise schedule and are effective at both preserving and reducing intrinsic data dimensionality. The result is a class of highly expressive generative models, uniquely defined on a low-dimensional latent space, that afford principled Bayesian inference., Comment: 10 pages, 6 figures

We present a new approach for nonlocal image denoising, based around the application of an unnormalized extended Gaussian ANOVA kernel within a bilevel optimization algorithm. A critical bottleneck when solving such problems for finely-resolved images is the solution of huge-scale, dense linear systems arising from the minimization of an energy term. We tackle this using a Krylov subspace approach, with a Nonequispaced Fast Fourier Transform utilized to approximate matrix-vector products in a matrix-free manner. We accelerate the algorithm using a novel change of basis approach to account for the (known) smallest eigenvalue-eigenvector pair of the matrices involved, coupled with a simple but frequently very effective diagonal preconditioning approach. We present a number of theoretical results concerning the eigenvalues and predicted convergence behavior, and a range of numerical experiments which validate our solvers and use them to tackle parameter learning problems. These demonstrate that very large problems may be effectively and rapidly denoised with very low storage requirements on a computer., Comment: 30 pages, 8 figures, 3 tables

We derive a new parallel-in-time approach for solving large-scale optimization problems constrained by time-dependent partial differential equations arising from fluid dynamics. The solver involves the use of a block circulant approximation of the original matrices, enabling parallelization-in-time via the use of fast Fourier transforms, and we devise bespoke matrix approximations which may be applied within this framework. These make use of permutations, saddle-point approximations, commutator arguments, as well as inner solvers such as the Uzawa method, Chebyshev semi-iteration, and multigrid. Theoretical results underpin our strategy of applying a block circulant strategy, and numerical experiments demonstrate the effectiveness and robustness of our approach on Stokes and Oseen problems. Noteably, satisfying results for the strong and weak scaling of our methods are provided within a fully parallel architecture., Comment: 24 pages

In this paper, we describe and analyze the spectral properties of a symmetric positive definite inexact block preconditioner for a class of symmetric, double saddle-point linear systems. We develop a spectral analysis of the preconditioned matrix, showing that its eigenvalues can be described in terms of the roots of a cubic polynomial with real coefficients. We illustrate the efficiency of the proposed preconditioners, and verify the theoretical bounds, in solving large-scale PDE-constrained optimization problems.

In this paper we consider the numerical solution to the soft-margin support vector machine optimization problem. This problem is typically solved using the SMO algorithm, given the high computational complexity of traditional optimization algorithms when dealing with large-scale kernel matrices. In this work, we propose employing an NFFT-accelerated matrix-vector product using an ANOVA decomposition for the feature space that is used within an interior point method for the overall optimization problem. As this method requires the solution of a linear system of saddle point form we suggest a preconditioning approach that is based on low-rank approximations of the kernel matrix together with a Krylov subspace solver. We compare the accuracy of the ANOVA-based kernel with the default LIBSVM implementation. We investigate the performance of the different preconditioners as well as the accuracy of the ANOVA kernel on several large-scale datasets., Comment: Official Code https://github.com/wagnertheresa/NFFTSVMipm

We consider a numerical framework tailored to identifying optimal parameters in the context of modelling disease propagation. Our focus is on understanding the behaviour of optimisation algorithms for such problems, where the dynamics are described by a system of ordinary differential equations associated with the epidemiological SIRD model. We examine properties of the solution operator and determine existence of optimal parameters for the problem considered. Further, first-order optimality conditions are derived, the solution of which provides a certificate of goodness of fit, which is not always guaranteed with parameter tuning techniques. We then propose strategies for the numerical solution of such problems, based on projected gradient descent, Fast Iterative Shrinkage-Thresholding Algorithm (FISTA), and limited memory BFGS trust region approaches. We carry out a thorough computational study for a range of problems of interest, determining the relative performance of these numerical methods. Our results provide insights into the efficacy of these strategies, contributing to ongoing research into optimising parameters for accurate and reliable disease spread modelling. Moreover, our approach paves the way for calibration of more intricate compartmental models., Comment: 26 pages, 7 figures, 3 tables

We introduce a design modification to conventional geometry of the cryogenic three-terminal switch, the nanocryotron (nTron). The conventional geometry of nTrons is modified by including parallel current-carrying channels, an approach aimed at enhancing the device's performance in magnetic field environments. The common challenge in nTron technology is to maintain efficient operation under varying magnetic field conditions. Here we show that the adaptation of parallel channel configurations leads to an enhanced gate signal sensitivity, an increase in operational gain, and a reduction in the impact of superconducting vortices on nTron operation within magnetic fields up to 1 Tesla. Contrary to traditional designs that are constrained by their effective channel width, the parallel nanowire channels permits larger nTron cross sections, further bolstering the device's magnetic field resilience while improving electro-thermal recovery times due to reduced local inductance. This advancement in nTron design not only augments its functionality in magnetic fields but also broadens its applicability in technological environments, offering a simple design alternative to existing nTron devices., Comment: 6 pages, 5 figures. Accepted by Applied Physics Letters. Supplemental information includes additional supporting figures and tables

Magnon interference is a signature of coherent magnon interactions for coherent information processing. In this work, we demonstrate programmable real-time magnon interference, with examples of nearly perfect constructive and destructive interference, between two remotely coupled yttrium iron garnet spheres mediated by a coplanar superconducting resonator. Exciting one of the coupled resonators by injecting single- and double-microwave pulse leads to the coherent energy exchange between the remote magnonic resonators and allows us to realize a programmable magnon interference that can define an arbitrary state of coupled magnon oscillation. The demonstration of time-domain coherent control of remotely coupled magnon dynamics offers new avenues for advancing coherent information processing with circuit-integrated hybrid magnonic networks.

We use co-sputtering to directly synthesize thin films of the A15 phase intermetallic compound Ta3Sb, which has been predicted to have a giant spin Hall conductivity. We identify a large window of Ta:Sb flux ratio that stabilizes single-phase A15 Ta3Sb. Composition analyses of these films show a Ta:Sb atomic ratio of 4:1, which is consistent with the known Ta-Sb phase diagram. The spin Hall conductivity of thin film Ta3Sb is -3400+/-400 (hbar/2e) S/cm and the spin-orbit torque efficiency is -0.6+/-0.1 at 20 K, as determined from harmonic Hall measurements of Ta3Sb/permalloy bilayer structures. These giant values make Ta3Sb a promising material for efficient charge-to-spin conversion in spintronic applications. Large field-like spin-orbit effective fields that are independent of the ferromagnetic layer thickness have also been measured in the Ta3Sb/permalloy bilayers. We attribute the field-like spin-orbit effective field to the Rashba effect at the interface.

Tunability of coherent coupling between fundamental excitations is an important prerequisite for expanding their functionality in hybrid quantum systems. In hybrid magnonics, the dipolar interaction between magnon and photon usually persists and cannot be switched off. Here, we demonstrate this capability by coupling a superconducting resonator to a layered hybrid perovskite antiferromagnet, which exhibits a magnon band gap due to its intrinsic Dzyaloshinskii-Moriya interaction. The pronounced temperature sensitivity of the magnon band gap location allows us to set the photon mode within the gap and to disable magnon-photon hybridization. When the resonator mode falls into the magnon band gap, the resonator damping rate increases due to the nonzero coupling to the detuned magnon mode. This phenomena can be used to quantify the magnon band gap using an analytical model. Our work brings new opportunities in controlling coherent information processing with quantum properties in complex magnetic materials., Comment: 8 pages, 4 figures

In this article, we derive fast and robust parallel-in-time preconditioned iterative methods for the all-at-once linear systems arising upon discretization of time-dependent PDEs. The discretization we employ is based on a Runge--Kutta method in time, for which the development of parallel solvers is an emerging research area in the literature of numerical methods for time-dependent PDEs. By making use of classical theory of block matrices, one is able to derive a preconditioner for the systems considered. The block structure of the preconditioner allows for parallelism in the time variable, as long as one is able to provide an optimal solver for the system of the stages of the method. We thus propose a preconditioner for the latter system based on a singular value decomposition (SVD) of the (real) Runge--Kutta matrix $A_{\mathrm{RK}} = U \Sigma V^\top$. Supposing $A_{\mathrm{RK}}$ is invertible, we prove that the spectrum of the system for the stages preconditioned by our SVD-based preconditioner is contained within the right-half of the unit circle, under suitable assumptions on the matrix $U^\top V$ (the assumptions are well posed due to the polar decomposition of $A_{\mathrm{RK}}$). We show the numerical efficiency of our SVD-based preconditioner by solving the system of the stages arising from the discretization of the heat equation and the Stokes equations, with sequential time-stepping. Finally, we provide numerical results of the all-at-once approach for both problems, showing the speed-up achieved on a parallel architecture.

Nonreciprocal magnon propagation has recently become a highly potential approach of developing chip-embedded microwave isolators for advanced information processing. However, it is challenging to achieve large nonreciprocity in miniaturized magnetic thin-film devices because of the difficulty of distinguishing propagating surface spin waves along the opposite directions when the film thickness is small. In this work, we experimentally realize unidirectional microwave transduction with sub-micron-wavelength propagating magnons in a yttrium iron garnet (YIG) thin film delay line. We achieve a non-decaying isolation of 30 dB with a broad field-tunable band-pass frequency range up to 14 GHz. The large isolation is due to the selection of chiral magnetostatic surface spin waves with the Oersted field generated from the coplanar waveguide antenna. Increasing the geometry ratio between the antenna width and YIG thickness drastically reduces the nonreciprocity and introduces additional magnon transmission bands. Our results pave the way for on-chip microwave isolation and tunable delay line with short-wavelength magnonic excitations., Comment: 6 pages, 4 figures

Titanium nitride is a material of interest for many superconducting devices such as nanowire microwave resonators and photon detectors. Thus, controlling the growth of TiN thin films with desirable properties is of high importance. In previous work on niobium nitride, ion beam-assisted sputtering (IBAS) reduced nitrogen sensitivity during deposition in tandem with an increase in nominal critical temperature. We have deposited thin films of titanium nitride by both, the conventional method of DC reactive magnetron sputtering and the IBAS method and compare their superconducting critical temperatures Tc as functions of thickness, sheet resistance, and nitrogen flow rate. We perform electrical and structural characterizations by electric transport and X-ray diffraction measurements. Compared to the conventional method of reactive sputtering, the IBAS technique has demonstrated a 10% increase in nominal critical temperature and 33% reduced sensitivity to nitrogen flow, without noticeable variation in the lattice structure. Additionally, we explore the behavior of superconducting Tc in ultra-thin films. Trends in films grown at high nitrogen concentrations follow predictions of mean-field theory in disordered films and show suppression of superconducting Tc due to geometric effects, while nitride films grown at low nitrogen concentrations strongly deviate from the theoretical models.

A numerical framework is developed to solve various types of PDEs on complicated domains, including steady and time-dependent, non-linear and non-local PDEs, with different boundary conditions that can also include non-linear and non-local terms. This numerical framework, called MultiShape, is a class in Matlab, and the software is open source. We demonstrate that MultiShape is compatible with other numerical methods, such as differential--algebraic equation solvers and optimization algorithms. The numerical implementation is designed to be user-friendly, with most of the set-up and computations done automatically by MultiShape and with intuitive operator definition, notation, and user-interface. Validation tests are presented, before we introduce three examples motivated by applications in Dynamic Density Functional Theory and PDE-constrained optimization, illustrating the versatility of the method.

Superconductivity forms out of the condensation of Cooper pairs of electrons. The mechanism by which Cooper pairs are created in non-conventional superconductors is often elusive because experimental signatures that connect a specific pairing mechanism to the properties of superconducting state are rare. The recently discovered superconducting oxide-insulator/KTaO$_3$ interface may offer clues about its origins. Here we observe distinct dependences of the superconducting transition temperature Tc on carrier density n$_{2D}$ for electron gases formed at KTaO$_3$ (111), (001) and (110) interfaces. For the KTaO$_3$ (111) interface, a remarkable linear dependence of Tc on n$_{2D}$ is observed over a range of nearly one order of magnitude. Further, our study of the dependence of superconductivity on gate electric fields reveals the role of the interface in mediating superconductivity, which also allows for a reversible electric switching of superconductivity at T = 2 K. We found that the extreme sensitivity of superconductivity to crystallographic orientation can be explained by Cooper pairing via inter-orbital interactions induced by the inversion-breaking transverse optical (TO1) phonons and quantum confinement. This mechanism is also consistent with the dependence of Tc on n$_{2D}$ at the KTaO$_3$ (111) interface. Our study may shed light on the pairing mechanism in other superconducting quantum-paraelectrics.

We demonstrate microwave-mediated distant magnon-magnon coupling on a superconducting circuit platform, incorporating chip-mounted single-crystal Y$_3$Fe$_5$O$_{12}$ (YIG) spheres. Coherent level repulsion and dissipative level attraction between the magnon modes of the two YIG spheres are demonstrated. The former is mediated by cavity photons of a superconducting resonator, and the latter is mediated by propagating photons of a coplanar waveguide. Our results open new avenues towards exploring integrated hybrid magnonic networks for coherent information processing on a quantum-compatible superconducting platform., Comment: 6 pages, 4 figures, Accepted in Phys. Rev. Lett

Variational autoencoders are among the most popular methods for distilling low-dimensional structure from high-dimensional data, making them increasingly valuable as tools for data exploration and scientific discovery. However, unlike typical machine learning problems in which a single model is trained once on a single large dataset, scientific workflows privilege learned features that are reproducible, portable across labs, and capable of incrementally adding new data. Ideally, methods used by different research groups should produce comparable results, even without sharing fully trained models or entire data sets. Here, we address this challenge by introducing the Rosetta VAE (R-VAE), a method of distilling previously learned representations and retraining new models to reproduce and build on prior results. The R-VAE uses post hoc clustering over the latent space of a fully-trained model to identify a small number of Rosetta Points (input, latent pairs) to serve as anchors for training future models. An adjustable hyperparameter, $\rho$, balances fidelity to the previously learned latent space against accommodation of new data. We demonstrate that the R-VAE reconstructs data as well as the VAE and $\beta$-VAE, outperforms both methods in recovery of a target latent space in a sequential training setting, and dramatically increases consistency of the learned representation across training runs., Comment: 16 pages, 7 figures, Bayesian Deep Learning Workshop at Neurips 2021

We derive an extension of the sequential homotopy method that allows for the application of inexact Krylov methods for the linear (double) saddle-point systems arising in the local semismooth Newton method for the homotopy subproblems. For the class of problems that exhibit (after suitable partitioning of the variables) a zero in the off-diagonal blocks of the Hessian of the Lagrangian, we propose and analyze an efficient, parallelizable, symmetric positive definite preconditioner based on a double Schur complement approach. For discretized optimal control problems with PDE constraints, this structure is often present with the canonical partitioning of the variables in states and controls. We conclude with numerical results for a badly conditioned and highly nonlinear benchmark optimization problem with elliptic partial differential equations and control bounds. The resulting method is faster than using direct linear algebra for the 2D benchmark and allows for the parallel solution of large 3D problems., Comment: 25 pages

Spin torque is a promising tool for driving magnetization dynamics for novel computing technologies. These torques can be easily produced by spin-orbit effects, but for most conventional spin source materials, a high degree of crystal symmetry limits the geometry of the spin torques produced. Magnetic ordering is one way to reduce the symmetry of a material and allow exotic torques, and antiferromagnets are particularly promising because they are robust against external fields. We present spin torque ferromagnetic resonance measurements and second harmonic Hall measurements characterizing the spin torques in antiferromagnetic iron rhodium alloy. We report extremely large, strongly temperature-dependent exotic spin torques with a geometry apparently defined by the magnetic ordering direction. We find the spin torque efficiency of iron rhodium to be (330$\pm$150) % at 170 K and (91$\pm$32) % at room temperature. We support our conclusions with theoretical calculations showing how the antiferromagnetic ordering in iron rhodium gives rise to such exotic torques., Comment: 21 pages, 6 figures

While most classic studies of function in experimental neuroscience have focused on the coding properties of individual neurons, recent developments in recording technologies have resulted in an increasing emphasis on the dynamics of neural populations. This has given rise to a wide variety of models for analyzing population activity in relation to experimental variables, but direct testing of many neural population hypotheses requires intervening in the system based on current neural state, necessitating models capable of inferring neural state online. Existing approaches, primarily based on dynamical systems, require strong parametric assumptions that are easily violated in the noise-dominated regime and do not scale well to the thousands of data channels in modern experiments. To address this problem, we propose a method that combines fast, stable dimensionality reduction with a soft tiling of the resulting neural manifold, allowing dynamics to be approximated as a probability flow between tiles. This method can be fit efficiently using online expectation maximization, scales to tens of thousands of tiles, and outperforms existing methods when dynamics are noise-dominated or feature multi-modal transition probabilities. The resulting model can be trained at kiloHertz data rates, produces accurate approximations of neural dynamics within minutes, and generates predictions on submillisecond time scales. It retains predictive performance throughout many time steps into the future and is fast enough to serve as a component of closed-loop causal experiments., Comment: Version of the work appearing in NeurIPS 2021

We derive novel, fast, and parameter-robust preconditioned iterative methods for steady and time-dependent Navier--Stokes control problems. Our approach may be applied to time-dependent problems which are discretized using backward Euler or Crank--Nicolson, and is also a valuable candidate for Stokes control problems discretized using Crank--Nicolson. The key ingredients of the solver are a saddle-point type approximation for the linear systems, an inner iteration for the $(1,1)$-block accelerated by a preconditioner for convection--diffusion control, and an approximation to the Schur complement based on a potent commutator argument applied to an appropriate block matrix. A range of numerical experiments validate the effectiveness of our new approach.

In this paper we present general-purpose preconditioners for regularized augmented systems arising from optimization problems, and their corresponding normal equations. We discuss positive definite preconditioners, suitable for CG and MINRES. We consider "sparsifications" which avoid situations in which eigenvalues of the preconditioned matrix may become complex. Special attention is given to systems arising from the application of regularized interior point methods to linear or nonlinear convex programming problems.

We consider symmetric positive definite preconditioners for multiple saddle-point systems of block tridiagonal form, which can be applied within the MINRES algorithm. We describe such a preconditioner for which the preconditioned matrix has only two distinct eigenvalues, 1 and -1, when the preconditioner is applied exactly. We discuss the relative merits of such an approach compared to a more widely studied block diagonal preconditioner, specify the computational work associated with applying the new preconditioner inexactly, and survey a number of theoretical results for the block diagonal case. Numerical results validate our theoretical findings., Comment: 19 pages

We demonstrate the electrical detection of magnon-magnon hybrid dynamics in yttrium iron garnet/permalloy (YIG/Py) thin film bilayer devices. Direct microwave current injection through the conductive Py layer excites the hybrid dynamics consisting of the uniform mode of Py and the first standing spin wave ($n=1$) mode of YIG, which are coupled via interfacial exchange. Both the two hybrid modes, with Py or YIG dominated excitations, can be detected via the spin rectification signals from the conductive Py layer, providing phase resolution of the coupled dynamics. The phase characterization is also applied to a nonlocally excited Py device, revealing the additional phase shift due to the perpendicular Oersted field. Our results provide a device platform for exploring hybrid magnonic dynamics and probing their phases, which are crucial for implementing coherent information processing with magnon excitations

Data assimilation algorithms combine information from observations and prior model information to obtain the most likely state of a dynamical system. The linearised weak-constraint four-dimensional variational assimilation problem can be reformulated as a saddle point problem, which admits more scope for preconditioners than the primal form. In this paper we design new terms which can be used within existing preconditioners, such as block diagonal and constraint-type preconditioners. Our novel preconditioning approaches: (i) incorporate model information, and (ii) are designed to target correlated observation error covariance matrices. To our knowledge (i) has not previously been considered for data assimilation problems. We develop new theory demonstrating the effectiveness of the new preconditioners within Krylov subspace methods. Linear and non-linear numerical experiments reveal that our new approach leads to faster convergence than existing state-of-the-art preconditioners for a broader range of problems than indicated by the theory alone. We present a range of numerical experiments performed in serial., Comment: 22 pages, 2 figures

A superconducting diode is an electronic device that conducts supercurrent and exhibits zero resistance primarily for one direction of applied current. Such a dissipationless diode is a desirable unit for constructing electronic circuits with ultralow power consumption. However, realizing a superconducting diode is fundamentally and technologically challenging, as it usually requires a material structure without a centre of inversion, which is scarce among superconducting materials. Here, we demonstrate a superconducting diode achieved in a conventional superconducting film patterned with a conformal array of nanoscale holes, which breaks the spatial inversion symmetry. We showcase the superconducting diode effect through switchable and reversible rectification signals, which can be three orders of magnitude larger than that from a flux-quantum diode. The introduction of conformal potential landscapes for creating a superconducting diode is thereby proven as a convenient, tunable, yet vastly advantageous tool for superconducting electronics. This could be readily applicable to any superconducting materials, including cuprates and iron-based superconductors that have higher transition temperatures and are desirable in device applications., Comment: 25 pages, 4 figures