Efficient time-dependent vibrational coupled cluster computations with time-dependent basis sets at the two-mode coupling level: Full and hybrid TDMVCC[2].

The computation of the nuclear quantum dynamics of molecules is challenging, requiring both accuracy and efficiency to be applicable to systems of interest. Recently, theories have been developed for employing time-dependent basis functions (denoted modals) with vibrational coupled cluster theory (TDMVCC). The TDMVCC method was introduced along with a pilot implementation, which illustrated good accuracy in benchmark computations. In this paper, we report an efficient implementation of TDMVCC, covering the case where the wave function and Hamiltonian contain up to two-mode couplings. After a careful regrouping of terms, the wave function can be propagated with a cubic computational scaling with respect to the number of degrees of freedom. We discuss the use of a restricted set of active one-mode basis functions for each mode, as well as two interesting limits: (i) the use of a full active basis where the variational modal determination amounts essentially to the variational determination of a time-dependent reference state for the cluster expansion; and (ii) the use of a single function as an active basis for some degrees of freedom. The latter case defines a hybrid TDMVCC/TDH (time-dependent Hartree) approach that can obtain even lower computational scaling. The resulting computational scaling for hybrid and full TDMVCC[2] is illustrated for polyaromatic hydrocarbons with up to 264 modes. Finally, computations on the internal vibrational redistribution of benzoic acid (39 modes) are used to show the faster convergence of TDMVCC/TDH hybrid computations towards TDMVCC compared to simple neglect of some degrees of freedom.

The influence of exercise on clinical pain and pain mechanisms in patients with subacromial pain syndrome.

BACKGROUND: Few studies have investigated the underlying mechanisms for unilateral subacromial pain syndrome (SAPS). Therefore, this study examined (1) if 8-weeks of exercise could modulate clinical pain or temporal summation of pain (TSP), conditioned pain modulation (CPM), and exercise-induced hypoalgesia (EIH) and (2) if any of these parameters could predict the effect of 8-weeks of exercise in patients with unilateral SAPS. METHODS: Thirty-seven patients completed a progressive abduction exercise program every other day for 8-weeks. Worst shoulder pain in full abduction was rated on a numeric rating scale (NRS). Pain pressure thresholds (PPTs), TSP, CPM, EIH, Shoulder Pain and Disability Index (SPADI), Pain Catastrophizing Scale (PCS), PainDETECT questionnaire (PD-Q), Pain Self-Efficacy Questionnaire (PSE-Q) and Pittsburgh Sleep Quality Index (PSQI) were assessed before and after intervention. RESULTS: The intervention improved worst pain intensity (p < 0.001), increased the CPM (p < 0.001), improved the sleep scores (p < 0.005) and reduced the PainDETECT ratings (p < 0.001). No changes were observed in PPT, TSP, EIH, SPADI, PCS and PSE-Q (all p > 0.05). In a linear regression, the combination of all baseline parameters predicted 23.2% variance in absolute change in pain after 8 weeks. Applying backwards elimination to the linear regression yielded that baseline pain intensity combined with TSP predicted 33.8% variance. CONCLUSION: This explorative study suggested reduction in pain, improved sleep quality and increased CPM after 8-weeks of exercise. Furthermore, the results suggests that low pain intensity and high TSP scores (indicative for pain sensitisation) may predict a lack of pain improvement after exercise.

Sortilin Modulates Schwann Cell Signaling and Remak Bundle Regeneration Following Nerve Injury.

Peripheral nerve regeneration relies on the ability of Schwann cells to support the regrowth of damaged axons. Schwann cells re-differentiate when reestablishing contact with the sprouting axons, with large fibers becoming remyelinated and small nociceptive fibers ensheathed and collected into Remak bundles. We have previously described how the receptor sortilin facilitates neurotrophin signaling in peripheral neurons via regulated trafficking of Trk receptors. This study aims to characterize the effects of sortilin deletion on nerve regeneration following sciatic crush injury. We found that Sort1 - / - mice displayed functional motor recovery like that of WT mice, with no detectable differences in relation to nerve conduction velocities and morphological aspects of myelinated fibers. In contrast, we found abnormal ensheathment of regenerated C-fibers in injured Sort1 - / - mice, demonstrating a role of sortilin for Remak bundle formation following injury. Further studies on Schwann cell signaling pathways showed a significant reduction of MAPK/ERK, RSK, and CREB phosphorylation in Sort1 - / - Schwann cells after stimulation with neurotrophin-3 (NT-3), while Schwann cell migration and myelination remained unaffected. In conclusion, our results demonstrate that loss of sortilin blunts NT-3 signaling in Schwann cells which might contribute to the impaired Remak bundle regeneration after sciatic nerve injury.

Bioacoustic Detection of Wolves: Identifying Subspecies and Individuals by Howls.

Wolves (Canis lupus) are generally monitored by visual observations, camera traps, and DNA traces. In this study, we evaluated acoustic monitoring of wolf howls as a method for monitoring wolves, which may permit detection of wolves across longer distances than that permitted by camera traps. We analyzed acoustic data of wolves' howls collected from both wild and captive ones. The analysis focused on individual and subspecies recognition. Furthermore, we aimed to determine the usefulness of acoustic monitoring in the field given the limited data for Eurasian wolves. We analyzed 170 howls from 16 individual wolves from 3 subspecies: Arctic (Canis lupus arctos), Eurasian (C. l. lupus), and Northwestern wolves (C. l. occidentalis). Variables from the fundamental frequency (f0) (lowest frequency band of a sound signal) were extracted and used in discriminant analysis, classification matrix, and pairwise post-hoc Hotelling test. The results indicated that Arctic and Eurasian wolves had subspecies identifiable calls, while Northwestern wolves did not, though this sample size was small. Identification on an individual level was successful for all subspecies. Individuals were correctly classified with 80%-100% accuracy, using discriminant function analysis. Our findings suggest acoustic monitoring could be a valuable and cost-effective tool that complements camera traps, by improving long-distance detection of wolves.

Estimating Discard Mortality in Commercial Fisheries without Fish Dying: A 3R Challenge.

Globally, it is estimated that around 10% of the fish that are caught are discarded. This is considered to be a wasteful human marine activity since these fish are often dead or dying. To reduce the high discard rates of commercial fisheries, the European Union (E.U.) has enacted a landing obligation that includes the ability to exempt "species for which scientific evidence demonstrates high survival rates". Therefore, discard survival studies (henceforth DSSs) have become one of the most politically prioritized fisheries research areas in European fisheries. International expert groups have produced guidance reports to promote best practices and to harmonize the methodologies. Nevertheless, there has not been any focus on how to implement animal welfare (AW) regulations experimentally. Discard survival studies are "frontrunners" in fisheries science research areas that are embedded by animal research welfare requirements and are expected to be more restrictive in the future because of an increased public focus on fish welfare. This paper focuses on AW regulations in relation to conducting DSSs, but the outreach is much broader. We investigate experimental procedures by bringing in relevant examples, using output results, and relating this information to relevant AW guidelines and regulations by focusing on implementing 3R principles.

Antimicrobial PDMS Surfaces Prepared through Fast and Oxygen-Tolerant SI-SARA-ATRP, Using Na2SO3 as a Reducing Agent.

Poly(dimethylsiloxane) (PDMS) is an attractive, versatile, and convenient material for use in biomedical devices that are in direct contact with the user. A crucial component in such a device is its surface in terms of antimicrobial properties preventing infection. Moreover, due to its inherent hydrophobicity, PDMS is rather prone to microbial colonization. Thus, developing an antimicrobial PDMS surface in a simple, large-scale, and applicable manner is an essential step in fully exploiting PDMS in the biomedical device industry. Current chemical modification methods for PDMS surfaces are limited; therefore, we present herein a new method for introducing an atom transfer radical polymerization (ATRP) initiator onto the PDMS surface via the base-catalyzed grafting of [(chloromethyl)phenylethyl]trimethoxysilane to the PDMS. The initiator surface was grafted with poly[2-(dimethylamino)ethyl methacrylate] (PDMAEMA) brushes via a surface-initiated supplemental activator and reducing agent ATRP (SI-SARA-ATRP). The use of sodium sulfite as a novel reducing agent in SI-SARA-ATRP allowed for polymerization during complete exposure to air. Moreover, a fast and linear growth was observed for the polymer over time, leading to a 400 nm thick polymer layer in a 120 min reaction time. Furthermore, the grafted PDMAEMA was quaternized, using various alkylhalides, in order to study the effect on surface antimicrobial properties. It was shown that antimicrobial activity not only depended highly on the charge density but also on the amphiphilicity of the surface. The fast reaction rate, high oxygen tolerance, increased antimicrobial activity, and the overall robustness and simplicity of the presented method collectively move PDMS closer to its full-scale exploitation in biomedical devices.

Calculating vibrational excitation energies using tensor-decomposed vibrational coupled-cluster response theory.

The first implementation of tensor-decomposed vibrational coupled cluster (CP-VCC) response theory for calculating vibrational excitation energies is presented. The CP-VCC algorithm, which has previously been applied to solving the vibrational coupled cluster (VCC) ground-state equations without explicitly constructing any tensors of order three or higher, has been generalized to allow transformations with the Jacobian matrix necessary for computation of response excitation energies by iterative algorithms. A new eigenvalue solver for computing CP-VCC excitation energies is introduced, and the different numerical thresholds used for controlling the accuracy of the obtained eigenvalues are discussed. Numerical results are presented for calculations of the 20 lowest eigenvalues on a set of 10 four-atomic molecules, as well as for a number of polycyclic aromatic hydrocarbons (PAHs) of increasing size, up to PAH8 with 120 modes. It is shown that the errors introduced by the tensor decomposition can be controlled by the choice of numerical thresholds. Furthermore, all thresholds can be defined relative to the requested convergence threshold of the equation solver, which allows black-box calculations with minimal user input to be performed. Eigenstates of PAHs were efficiently computed without any explicitly constructed tensors, showing improvements in both memory and central processing unit time compared to the existing full-tensor versions.

A general implementation of time-dependent vibrational coupled-cluster theory.

The first general excitation level implementation of the time-dependent vibrational coupled cluster (TDVCC) method introduced in a recent publication [J. Chem. Phys. 151, 154116 (2019)] is presented. The general framework developed for time-independent vibrational coupled cluster (VCC) calculations has been extended to the time-dependent context. This results in an efficient implementation of TDVCC with general coupling levels in the cluster operator and Hamiltonian. Thus, the convergence of the TDVCC[k] hierarchy toward the complete-space limit can be studied for any sum-of-product Hamiltonian. Furthermore, a scheme for including selected higher-order excitations for a subset of modes is introduced and studied numerically. Three different definitions of the TDVCC autocorrelation function (ACF) are introduced and analyzed in both theory and numerical experiments. Example calculations are presented for an array of systems including imidazole, formyl fluoride, formaldehyde, and a reduced-dimensionality bithiophene model. The results show that the TDVCC[k] hierarchy converges systematically toward the full-TDVCC limit and that the implementation allows accurate quantum-dynamics simulations of large systems to be performed. Specifically, the intramolecular vibrational-energy redistribution of the 21-dimensional imidazole molecule is studied in terms of the decay of the ACF. Furthermore, the importance of product separability in the definition of the ACF is highlighted when studying non-interacting subsystems.

Time-dependent vibrational coupled cluster with variationally optimized time-dependent basis sets.

We develop time-dependent vibrational coupled cluster with time-dependent modals (TDMVCC), where an active set of one-mode basis functions (modals) is evolved in time alongside coupled-cluster wave-function parameters. A biorthogonal second quantization formulation of many-mode dynamics is introduced, allowing separate biorthogonal bases for the bra and ket states, thus ensuring complex analyticity. We employ the time-dependent bivariational principle to derive equations of motion for both the one-mode basis functions and the parameters describing the cluster (T) and linear de-excitation (L) operators. The choice of constraint (or gauge) operators for the modal time evolution is discussed. In the case of untruncated cluster expansion, the result is independent of this choice, but restricting the excitation space removes this invariance; equations for the variational determination of the constraint operators are derived for the latter case. We show that all single-excitation parts of T and L are redundant and can be left out in the case of variationally determined constraint-operator evolution. Based on a pilot implementation, test computations on Henon-Heiles model systems, the water molecule, and a reduced-dimensionality bi-thiophene model are presented, showing highly encouraging results for TDMVCC. It is demonstrated how TDMVCC in the limit of a complete cluster expansion becomes equivalent to multiconfiguration time-dependent Hartree for the same active-space size. Similarly, it is discussed how TDMVCC generally gives better and more stable results than its time-independent-modals counterpart, while equivalent results are obtained for complete expansions and full one-mode basis sets.

Oxygen-dependence of upper thermal limits in crustaceans from different thermal habitats.

The critical thermal maximum (CTMAX) is the temperature at which animals exhibit loss of motor response because of a temperature-induced collapse of vital physiological systems. A central mechanism hypothesised to underlie the CTMAX of water-breathing ectotherms is insufficient tissue oxygen supply for vital maintenance functions because of a temperature-induced collapse of the cardiorespiratory system. The CTMAX of species conforming to this hypothesis should decrease with declining water oxygen tension (PO2) because they have oxygen-dependent upper thermal limits. However, recent studies have identified a number of fishes and crustaceans with oxygen-independent upper thermal limits, their CTMAX unchanged in progressive aquatic hypoxia. The previous studies, which were performed separately on cold-water, temperate and tropical species, suggest the oxygen-dependence of upper thermal limits and the acute thermal sensitivity of the cardiorespiratory system increases with decreasing habitat temperature. Here we directly test this hypothesis by assessing the oxygen-dependence of CTMAX in the polar Antarctic krill (Euphausia superba), as well as the temperate Baltic prawn (Palaemon adspersus) and brown shrimp (Crangon crangon). We found that P. adspersus and C. crangon maintain CTMAX in progressive hypoxia down to 40 mmHg, and that only E. superba have oxygen-dependent upper thermal limits at normoxia. In E. superba, the observed decline in CTMAX with water PO2 is further supported by heart-rate measurements showing a plateauing, and subsequent decline and collapse of heart performance at CTMAX. Our results support the hypothesis that the oxygen-dependence of upper thermal limits in water-breathing ectotherms and the acute thermal sensitivity of their cardiorespiratory system increases with decreasing habitat temperature.

Extended vibrational coupled cluster: Stationary states and dynamics.

For the first time, equations are derived for computing stationary vibrational states with extended vibrational coupled cluster (EVCC) and for propagating nuclear wave packets using time-dependent EVCC (TDEVCC). Expressions for energies, properties, and auto-correlation functions are given. For TDEVCC, convergence toward the ground state for imaginary-time propagation is shown, as well as separability in the case of non-interacting subsystems. The analysis focuses substantially on the difference between bra and ket parameterizations for EVCC and TDEVCC compared to normal vibrational coupled cluster (VCC) and time-dependent VCC (TDVCC). A pilot implementation is presented within a new full-space framework that offers easy access to completely general, albeit not efficient, implementations of alternative VCC variants, such as EVCC. The new methods were tested on 35 three- and six-mode molecular systems. Both EVCC[k] and TDEVCC[k] showed good, hierarchical convergence toward the exact limit. This convergence was generally better than for normal VCC[k] and TDVCC[k] and better still than for (time-dependent) vibrational configuration interaction, though this should be balanced with the higher computational complexity of EVCC. The results highlight the importance of exponential parameterizations and separability in general, as seen, in particular, for the TDEVCC bra parameterization, which is in contrast to the partially linear one of TDVCC. With the results being rooted in the general structures of coupled cluster (CC) theory, they are expected to be relevant to other applications of both normal and extended CC theory as well.

MR-MCTDH[n]: Flexible Configuration Spaces and Nonadiabatic Dynamics within the MCTDH[n] Framework.

Solving the time-dependent Schrödinger equation (TDSE) for large molecular systems is a complicated task due to the inherent exponential scaling of the problem. One of the most successful and versatile methods for obtaining numerically converged solutions for small to medium-sized systems is multiconfiguration time-dependent Hartree (MCTDH). In a recent publication [J. Chem. Phys. 2020, 152, 084101] we introduced a hierarchy of approximations to the MCTDH method which mitigate the exponential scaling by truncating the configuration space based on a maximum excitation level w.r.t. a selected reference configuration. The MCTDH[n] methods are able to treat large systems, but the single-reference Ansatz is not optimal in cases where one (or a few) degrees of freedom are special. Examples could be double-well systems, intramolecular vibrational-energy redistribution (IVR) calculations, or nonadiabatic dynamics. In this work we introduce a multireference (MR) extension to the MCTDH[n] methods where selected higher-order excitations for the special degrees of freedom can be introduced in a simple but flexible way. The resulting MR-MCTDH[n] methods allow for, for example, treating nonadiabatic dynamics within the single-set formalism with the wave packets on each electronic surface described using the same level of approximation. Example calculations are performed on formyl fluoride (IVR), salicylaldimine (double well), and pyrazine (nonadiabatic dynamics). The results show that fast convergence is achieved by extending the configuration space in the special modes that govern the quantum dynamics.

Development of an individualized asynchronous sensor-based telerehabilitation program for patients undergoing total knee replacement: Participatory design.

Telerehabilitation programs can be employed to establish communication between patients and healthcare professionals and empower patients performing their training remotely. This study aimed to identify patients' requirements after a total knee replacement following a self-training rehabilitation program, leading to the design and development of a telerehabilitation program that can meet the stakeholders' actual needs. System design, development, and testing were conducted in five iterations based on a participatory design approach. Data collection was performed using interviews, observations, prototyping, and questionnaires. It was found that the main barriers facing the existing rehabilitation program were a lack of clear communication, lack of relevant information, and healthcare professional's feedback. The participants emphasized the main themes of communication, information, training, and motivation in the process of design and development. In using the telerehabilitation program, the patients reported a high level of user-friendliness, flexibility, and a sense of security. This study has identified obstacles in the current rehabilitation program and revealed the potential effectiveness of using asynchronous communication and sensor-based technologies by employing participatory design and development. A higher level of portability and flexibility were observed. However, future studies and development are required to investigate the overall usability and reliability of the telerehabilitation program.

Systematic and variational truncation of the configuration space in the multiconfiguration time-dependent Hartree method: The MCTDH[n] hierarchy.

The multiconfiguration time-dependent Hartree (MCTDH) method is a powerful method for solving the time-dependent Schrödinger equation in quantum molecular dynamics. It is, however, hampered by the so-called curse of dimensionality which results in exponential scaling with respect to the number of degrees of freedom in the system and, thus, limits its applicability to small- and medium-sized molecules. To avoid this scaling, we derive equations of motion for a series of truncated MCTDH methods using a many-mode second-quantization formulation where the configuration space is restricted based on mode-combination levels as also done in the vibrational configuration interaction and vibrational coupled cluster methods for solving the time-independent Schrödinger equation. The full MCTDH wave function is invariant with respect to the choice of constraint (or gauge) operators, but restricting the configuration space removes this invariance. We, thus, analyze the remaining redundancies and derive equations for variationally optimizing the non-redundant matrix elements of the constraint operators. As an alternative, we also present a constraint that keeps the density matrices block diagonal during the propagation and the two choices are compared. Example calculations are performed on formyl fluoride and a series of high-dimensional Henon-Heiles potentials. The results show that the MCTDH[n] methods can be applied to large systems and that an optimal choice of constraint operators is key to obtaining the correct physical behavior of the wave function.

Time-dependent vibrational coupled cluster theory: Theory and implementation at the two-mode coupling level.

Equations are derived for the time evolution of time-dependent vibrational coupled cluster (TDVCC) wave functions covering both the TDVCC ket state and the associated so-called Λ bra state. The equations are implemented in the special case of both the Hamiltonian and the cluster operator containing at most two-mode coupling terms. The nontrivial behavior of the evolution of norm, energy, and expectation values due to the nonunitary time-evolution of the nonvariational TDVCC theory is analyzed theoretically and confirmed in numerical experiments that also include time-dependent Hamiltonians. In the spirit of time-independent size-consistency analysis, the separability of both the coupled cluster and Λ states for noninteracting systems is studied. While the coupled cluster state clearly has the correct behavior, the behavior of the Λ state is more intricate, and the consequence for different properties is shown theoretically and numerically. Overall, the numerical experiments show that TDVCC in incomplete expansions gives higher accuracy than a standard linear variational wave function parameterization with the same number of independent parameters, while equivalent results are obtained for complete expansions. The efficiency of the methodology is illustrated in computations on polycyclic aromatic hydrocarbons with up to 156 modes.

Developing a telerehabilitation programme for postoperative recovery from knee surgery: specifications and requirements.

INTRODUCTION: Telerehabilitation programmes have been attracting increasing attention as a potential alternative to conventional rehabilitation. Video conferencing can facilitate communication between healthcare professionals and patients. However, in certain cases, video conferencing may face practical limitations. As an alternative to real-time conferencing, sensor-based technologies can transmit the acquired data to healthcare providers. This study aimed to design and develop a sensor-based telerehabilitation programme and to outline the corresponding requirements for such a system. DEVELOPMENT: The development of the sensor-based telerehabilitation programme was carried out based on user needs. The programme includes a portable platform for the patient as well as a web-based platform for the healthcare professional, thus allowing for an individualised rehabilitation programme. Communication, training, reporting, and information services were provided for the patients. Moreover, the portability and usability of the programme were enhanced by utilising the system in offline mode as well. APPLICATION: The programme is currently being tested in the North Denmark Region to assess the feasibility and acceptance of a telerehabilitation programme as an alternative solution to the self-training programme for patients who have been discharged from knee surgery. The preliminary results of our assessment showed a high level of acceptance among the users. DISCUSSION: In this study, a semi-online sensor-based telerehabilitation programme was tested. It is argued that a similar sensor-based telerehabilitation programme can be utilised as an alternative solution for self-training rehabilitation in the future; however; further studies and development are required to ensure the quality and reliability of sensor-based services.

Exponential parameterization of wave functions for quantum dynamics: Time-dependent Hartree in second quantization.

We derive equations for describing the time evolution of variational wave functions in linear and exponential parameterization with a second-quantization (SQ) formulation. The SQ formalism covers time-dependent Hartree (TDH), while exact states and approximate vibrational configuration interaction wave functions are described using state-transfer operators. We present detailed expressions for efficient evaluation of TDH in linear (L-TDH) and exponential (X-TDH) parametrization and an efficient implementation supporting linear scaling with respect to the number of degrees of freedom M when the Hamiltonian operator contains a constant number of terms per mode independently of the size of the system. The computational cost of the X-TDH method is reduced significantly compared to the L-TDH method for systems with many operator terms per mode such as is typical for accurate molecular potential-energy surfaces. Numerical results for L-TDH and X-TDH are presented which confirm the theoretical reduction of the M scaling compared to standard first-quantization formulations. Calculations on Henon-Heiles potentials with more than 105 dimensions and polycyclic aromatic hydrocarbons with up to 264 modes have been performed. Thus, the SQ formulation and the X-TDH method pave the way for studying the time-resolved quantum dynamics of large molecules.

Tensor-decomposed vibrational coupled-cluster theory: Enabling large-scale, highly accurate vibrational-structure calculations.

A new implementation of vibrational coupled-cluster (VCC) theory is presented, where all amplitude tensors are represented in the canonical polyadic (CP) format. The CP-VCC algorithm solves the non-linear VCC equations without ever constructing the amplitudes or error vectors in full dimension but still formally includes the full parameter space of the VCC[n] model in question resulting in the same vibrational energies as the conventional method. In a previous publication, we have described the non-linear-equation solver for CP-VCC calculations. In this work, we discuss the general algorithm for evaluating VCC error vectors in CP format including the rank-reduction methods used during the summation of the many terms in the VCC amplitude equations. Benchmark calculations for studying the computational scaling and memory usage of the CP-VCC algorithm are performed on a set of molecules including thiadiazole and an array of polycyclic aromatic hydrocarbons. The results show that the reduced scaling and memory requirements of the CP-VCC algorithm allows for performing high-order VCC calculations on systems with up to 66 vibrational modes (anthracene), which indeed are not possible using the conventional VCC method. This paves the way for obtaining highly accurate vibrational spectra and properties of larger molecules.

Efficient algorithms for solving the non-linear vibrational coupled-cluster equations using full and decomposed tensors.

Vibrational coupled-cluster (VCC) theory provides an accurate method for calculating vibrational spectra and properties of small to medium-sized molecules. Obtaining these properties requires the solution of the non-linear VCC equations which can in some cases be hard to converge depending on the molecule, the basis set, and the vibrational state in question. We present and compare a range of different algorithms for solving the VCC equations ranging from a full Newton-Raphson method to approximate quasi-Newton models using an array of different convergence-acceleration schemes. The convergence properties and computational cost of the algorithms are compared for the optimization of VCC states. This includes both simple ground-state problems and difficult excited states with strong non-linearities. Furthermore, the effects of using tensor-decomposed solution vectors and residuals are investigated and discussed. The results show that for standard ground-state calculations, the conjugate residual with optimal trial vectors algorithm has the shortest time-to-solution although the full Newton-Raphson method converges in fewer macro-iterations. Using decomposed tensors does not affect the observed convergence rates in our test calculations as long as the tensors are decomposed to sufficient accuracy.