Coupling parallel asynchronous multisplitting methods with Krylov methods to solve pseudo-linear evolution 3D problems

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

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Rational Krylov methods for fractional diffusion problems on graphs

BIT Numerical Mathematics ◽

10.1007/s10543-021-00881-0 ◽

2021 ◽

Author(s):

Michele Benzi ◽

Igor Simunec

Keyword(s):

Analytic Function ◽

Diffusion Equation ◽

Graph Laplacian ◽

Fractional Diffusion ◽

Singular Matrix ◽

Krylov Methods ◽

Fractional Powers ◽

Directed Networks ◽

Rank One ◽

Diffusion Problems

AbstractIn this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product $$f(L^T) \varvec{b}$$ f ( L T ) b , where f is a non-analytic function involving fractional powers and $$\varvec{b}$$ b is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $$f(L^T) \varvec{b}$$ f ( L T ) b to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

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Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

Open Mathematics ◽

10.1515/math-2021-0027 ◽

2021 ◽

Vol 19 (1) ◽

pp. 111-120

Author(s):

Qinghua Zhang ◽

Zhizhong Tan

Keyword(s):

Cauchy Problem ◽

Asymptotic Behavior ◽

Evolution Equation ◽

Nonlinear Operator ◽

Evolution Equations ◽

Bounded Function ◽

Inhomogeneous Term ◽

Linear Evolution ◽

Abstract Evolution Equations ◽

Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

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Thalamic morphology predicts the onset of freezing of gait in Parkinson’s disease

npj Parkinson s Disease ◽

10.1038/s41531-021-00163-0 ◽

2021 ◽

Vol 7 (1) ◽

Author(s):

Nicholas D’Cruz ◽

Griet Vervoort ◽

Sima Chalavi ◽

Bauke W. Dijkstra ◽

Moran Gilat ◽

...

Keyword(s):

Parkinson’S Disease ◽

Parkinson's Disease ◽

Resting State ◽

Freezing Of Gait ◽

Study Entry ◽

Resting State Functional Connectivity ◽

Behavioral Correlates ◽

Local Volume ◽

Linear Evolution ◽

Risk Of Falls

AbstractThe onset of freezing of gait (FOG) in Parkinson’s disease (PD) is a critical milestone, marked by a higher risk of falls and reduced quality of life. FOG is associated with alterations in subcortical neural circuits, yet no study has assessed whether subcortical morphology can predict the onset of clinical FOG. In this prospective multimodal neuroimaging cohort study, we performed vertex-based analysis of grey matter morphology in fifty-seven individuals with PD at study entry and two years later. We also explored the behavioral correlates and resting-state functional connectivity related to these local volume differences. At study entry, we found that freezers (N = 12) and persons who developed FOG during the course of the study (converters) (N = 9) showed local inflations in bilateral thalamus in contrast to persons who did not (non-converters) (N = 36). Longitudinally, converters (N = 7) also showed local inflation in the left thalamus, as compared to non-converters (N = 36). A model including sex, daily levodopa equivalent dose, and local thalamic inflation predicted conversion with good accuracy (AUC: 0.87, sensitivity: 88.9%, specificity: 77.8%). Exploratory analyses showed that local thalamic inflations were associated with larger medial thalamic sub-nuclei volumes and better cognitive performance. Resting-state analyses further revealed that converters had stronger thalamo-cortical coupling with limbic and cognitive regions pre-conversion, with a marked reduction in coupling over the two years. Finally, validation using the PPMI cohort suggested FOG-specific non-linear evolution of thalamic local volume. These findings provide markers of, and deeper insights into conversion to FOG, which may foster earlier intervention and better mobility for persons with PD.

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