Method of lines transpose: High order L-stable O(N) schemes for parabolic equations using successive convolution

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a single-dimensional heat equation solver that uses fast O(N) convolution. This fundamental solver has arbitrary order of accuracy in space, and is based on the use of the Green's function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multi-dimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite-Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen-Cahn, and the Fitzhugh-Nagumo system of equations in one and two dimensions

Higher Order A-Stable Schemes for the Wave Equation Using a Successive Convolution Approach

In several recent works, we developed a new second order, A-stable approach to wave propagation problems based on the method of lines transpose (MOL$^T$) formulation combined with alternating direction implicit (ADI) schemes. Because our method is based on an integral solution of the ADI splitting of the MOL$^T$ formulation, we are able to easily embed non-Cartesian boundaries and include point sources with exact spatial resolution. Further, we developed an efficient $O(N)$ convolution algorithm for rapid evaluation of the solution, which makes our method competitive with explicit finite difference (e.g., finite difference time domain) solvers, in terms of both accuracy and time to solution, even for Courant numbers slightly larger than 1. We have demonstrated the utility of this method by applying it to a range of problems with complex geometry, including cavities with cusps. In this work, we present several important modifications to our recently developed wave solver. We obtain a family of wave solvers which are unconditionally stable, accurate of order $2P$, and require $O(P^d N)$ operations per time step, where $N$ is the number of spatial points and $d$ the number of spatial dimensions. We obtain these schemes by including higher derivatives of the solution, rather than increasing the number of time levels. The novel aspect of our approach is that the higher derivatives are constructed using successive applications of the convolution operator. We develop these schemes in one spatial dimension, and then extend the results to higher dimensions, by reformulating the ADI scheme to include recursive convolution. Thus, we retain a fast, unconditionally stable scheme, which does not suffer from the large dispersion errors characteristic to the ADI method. We demonstrate the utility of the method by applying it to a host of wave propagation problems. This method holds great promise for developing higher order, parallelizable algorithms for solving hyperbolic PDEs and can also be extended to parabolic PDEs

Method of Lines Transpose: High Order L-Stable {O}(N) Schemes for Parabolic Equations Using Successive Convolution

We present a new solver for nonlinear parabolic problems that is L-stable and achieves high order accuracy in space and time. The solver is built by first constructing a one-dimensional heat equation solver that uses fast $\mathcal O(N)$ convolution. This fundamental solver has arbitrary order of accuracy in space and is based on the use of the Green\u27s function to invert a modified Helmholtz equation. Higher orders of accuracy in time are then constructed through a novel technique known as successive convolution (or resolvent expansions). These resolvent expansions facilitate our proofs of stability and convergence, and permit us to construct schemes that have provable stiff decay. The multidimensional solver is built by repeated application of dimensionally split independent fundamental solvers. Finally, we solve nonlinear parabolic problems by using the integrating factor method, where we apply the basic scheme to invert linear terms (that look like a heat equation), and make use of Hermite--Birkhoff interpolants to integrate the remaining nonlinear terms. Our solver is applied to several linear and nonlinear equations including heat, Allen--Cahn, and the FitzHugh--Nagumo system of equations in one and two dimensions

Control of Neural Stem Cell Survival by Electroactive Polymer Substrates

Stem cell function is regulated by intrinsic as well as microenvironmental factors, including chemical and mechanical signals. Conducting polymer-based cell culture substrates provide a powerful tool to control both chemical and physical stimuli sensed by stem cells. Here we show that polypyrrole (PPy), a commonly used conducting polymer, can be tailored to modulate survival and maintenance of rat fetal neural stem cells (NSCs). NSCs cultured on PPy substrates containing different counter ions, dodecylbenzenesulfonate (DBS), tosylate (TsO), perchlorate (ClO4) and chloride (Cl), showed a distinct correlation between PPy counter ion and cell viability. Specifically, NSC viability was high on PPy(DBS) but low on PPy containing TsO, ClO4 and Cl. On PPy(DBS), NSC proliferation and differentiation was comparable to standard NSC culture on tissue culture polystyrene. Electrical reduction of PPy(DBS) created a switch for neural stem cell viability, with widespread cell death upon polymer reduction. Coating the PPy(DBS) films with a gel layer composed of a basement membrane matrix efficiently prevented loss of cell viability upon polymer reduction. Here we have defined conditions for the biocompatibility of PPy substrates with NSC culture, critical for the development of devices based on conducting polymers interfacing with NSCs

A bootstrap method for sum-of-poles approximations

A bootstrap method is presented for finding efficient sum-of-poles approximations of causal functions. The method is based on a recursive application of the nonlinear least squares optimization scheme developed in (Alpert et al. in SIAM J. Numer. Anal. 37:1138–1164, 2000), followed by the balanced truncation method for model reduction in computational control theory as a final optimization step. The method is expected to be useful for a fairly large class of causal functions encountered in engineering and applied physics. The performance of the method and its application to computational physics are illustrated via several numerical examples

Effects of antiplatelet therapy on stroke risk by brain imaging features of intracerebral haemorrhage and cerebral small vessel diseases: subgroup analyses of the RESTART randomised, open-label trial

Background Findings from the RESTART trial suggest that starting antiplatelet therapy might reduce the risk of recurrent symptomatic intracerebral haemorrhage compared with avoiding antiplatelet therapy. Brain imaging features of intracerebral haemorrhage and cerebral small vessel diseases (such as cerebral microbleeds) are associated with greater risks of recurrent intracerebral haemorrhage. We did subgroup analyses of the RESTART trial to explore whether these brain imaging features modify the effects of antiplatelet therapy

Effects of antiplatelet therapy after stroke due to intracerebral haemorrhage (RESTART): a randomised, open-label trial

Background: Antiplatelet therapy reduces the risk of major vascular events for people with occlusive vascular disease, although it might increase the risk of intracranial haemorrhage. Patients surviving the commonest subtype of intracranial haemorrhage, intracerebral haemorrhage, are at risk of both haemorrhagic and occlusive vascular events, but whether antiplatelet therapy can be used safely is unclear. We aimed to estimate the relative and absolute effects of antiplatelet therapy on recurrent intracerebral haemorrhage and whether this risk might exceed any reduction of occlusive vascular events. Methods: The REstart or STop Antithrombotics Randomised Trial (RESTART) was a prospective, randomised, open-label, blinded endpoint, parallel-group trial at 122 hospitals in the UK. We recruited adults (≥18 years) who were taking antithrombotic (antiplatelet or anticoagulant) therapy for the prevention of occlusive vascular disease when they developed intracerebral haemorrhage, discontinued antithrombotic therapy, and survived for 24 h. Computerised randomisation incorporating minimisation allocated participants (1:1) to start or avoid antiplatelet therapy. We followed participants for the primary outcome (recurrent symptomatic intracerebral haemorrhage) for up to 5 years. We analysed data from all randomised participants using Cox proportional hazards regression, adjusted for minimisation covariates. This trial is registered with ISRCTN (number ISRCTN71907627). Findings: Between May 22, 2013, and May 31, 2018, 537 participants were recruited a median of 76 days (IQR 29–146) after intracerebral haemorrhage onset: 268 were assigned to start and 269 (one withdrew) to avoid antiplatelet therapy. Participants were followed for a median of 2·0 years (IQR [1·0– 3·0]; completeness 99·3%). 12 (4%) of 268 participants allocated to antiplatelet therapy had recurrence of intracerebral haemorrhage compared with 23 (9%) of 268 participants allocated to avoid antiplatelet therapy (adjusted hazard ratio 0·51 [95% CI 0·25–1·03]; p=0·060). 18 (7%) participants allocated to antiplatelet therapy experienced major haemorrhagic events compared with 25 (9%) participants allocated to avoid antiplatelet therapy (0·71 [0·39–1·30]; p=0·27), and 39 [15%] participants allocated to antiplatelet therapy had major occlusive vascular events compared with 38 [14%] allocated to avoid antiplatelet therapy (1·02 [0·65–1·60]; p=0·92). Interpretation: These results exclude all but a very modest increase in the risk of recurrent intracerebral haemorrhage with antiplatelet therapy for patients on antithrombotic therapy for the prevention of occlusive vascular disease when they developed intracerebral haemorrhage. The risk of recurrent intracerebral haemorrhage is probably too small to exceed the established benefits of antiplatelet therapy for secondary prevention