Evolution of brain tumours backwards in time is studied using well-established brain tumour growth models being semilinear parabolic equations of reaction-diffusion type. To run the models backwards, the tumour cell density data at a fixed (final) time is used, rendering an inverse ill-posed problem. This problem is recast as the minimisation of a cost functional matching the data against the solution at a final time of a forward parabolic model having the initial cell density as a control function. Regularisation is incorporated via penalising terms involving Sobolev norms. Mathematical properties of the semilinear parabolic equations are shown in Sobolev-Bochner spaces including uniqueness of a solution to the inverse problem. Differentiability of the control-to-state map is established rendering a sensitivity problem. The derivative of the cost functional is calculated and the adjoint state is derived via the Lagrange formalism. A non-linear conjugate gradient method (NCG) is presented for the minimisation. Numerical realisation of the minimisation on the BraTS'20 dataset is included using a standard finite difference discretisation of the space and time derivatives, showing that tumour evolution backwards in time can be accomplished and that the initial tumour cell density can be reconstructed. Comparison is done with a non-linear Landweber method.

We study a novel degenerate and singular elliptic operator Δ˜(τ,χ) defined by Δ˜(τ,χ)u=τ(x,Du)(|Du|Δ1u+χ(x,Du)Δ∞u), where the singular weights τ(x,s)>0 and χ(x,s)≥0 are continuous functions on Ω×Rn∖{0}. The operator Δ˜(τ,χ) is an extension of Δ(p,q)u=|Du|qΔ1u+(p−1)|Du|p−2Δ∞u,p≥1,q≥0, introduced by the authors in Baravdishet al. (2020), which in turn is an extension of the p-Laplace operator Δp. We establish the well-posedness of the Neumann boundary value problem for the parabolic equation ut=Δ˜(τ,χ)u in the framework of viscosity solutions. For the solution u, the weight χ controls the evolution along the tangential and the normal directions, respectively, on the level surface of u. The weight τ controls the total speed of the evolution of u. We also prove the consistency and the convergence of the numerical scheme for the finite differences method of the parabolic equation above. Numerical simulations show that our novel nonlinear operator Δ˜(τ,χ) gives better results than both the Perona–Malik (Perona and Malik, 1990) and total variation (TV) methods (Chan and Shen, 2005) when applied to image enhancement.

We have in a previous study introduced a novel elliptic operator Δ(p,q)u=|∇u|^qΔ₁u+(p−1)|∇u|^p−2Δ_∞u, p≥1, q≥0, as a generalization of the p-Laplace operator. In this paper, we establish the well-posedness of the parabolic equation u_t=|∇u|^1−qΔ(_1+q,q), where q=q(|∇u|) is continuous and has range in [0,1],in the framework of viscosity solutions. We prove the consistency and convergence of the numerical scheme of finite differences of this parabolic equation. Numerical simulations shows the advantage of this operator applied to image enhancement.

In this work we introduce a novel operator $$\displaystyle \varDelta _(p,q)$$ as an extended family of operators that generalize the p-Laplace operator. The operator is derived with an emphasis on image processing applications, and particularly, with a focus on image denoising applications. We propose a non-linear transition function, coupling p and q, which yields a non-linear filtering scheme analogous to adaptive spatially dependent total variation and linear filtering. Well-posedness of the final parabolic PDE is established via pertubation theory and connection to classical results in functional analysis. Numerical results demonstrates the applicability of the novel operator $$\displaystyle \varDelta _(p,q)$$ .

In this study, we investigate the backward p(x)-parabolic equation as a new methodology to enhance images. We propose a novel iterative regularization procedure for the backward p(x)-parabolic equation based on the nonlinear Landweber method for inverse problems. The proposed scheme can also be extended to the family of iterative regularization methods involving the nonlinear Landweber method. We also investigate the connection between the variable exponent p(x) in the proposed energy functional and the diffusivity function in the corresponding Euler-Lagrange equation. It is well known that the forward problems converges to a constant solution destroying the image. The purpose of the approach of the backward problems is twofold. First, solving the backward problem by a sequence of forward problems we obtain a smooth image which is denoised. Second, by choosing the initial data properly we try to reduce the blurriness of the image. The numerical results for denoising appear to give improvement over standard methods as shown by preliminary results.

In this paper we present a new method for denoising audio signals. The method is based on the Singular Value Decomposition (SVD) of the frame matrix representing the signal inthe Overlap Add decomposition. Denoising is performed by modifying both the singular values, using a tapering model, and the singular vectors of the representation, using a nonlinear PDE method. The performance of the method is evaluated and compared with denoising obtained by filtering.

We study the sharpness of the Stolz approach for the a.e. convergence of functions in the Hardy spaces in the unit disc, first settled in the rotation invariant case by J. E. Littlewood in 1927 and later examined, under less stringent, quantitative hypothesis, by H. Aikawa in 1991. We introduce a new regularity condition, of a qualitative type, under which we prove a version of Littlewood's theorem for tangential approach whose shape may vary from point to point. Our regularity condition can be extended in those contexts where no group is involved, such as NTA domains in Rn. We show exactly in what sense our regularity condition is sharp.

We show that for every potential space LK1(ℝ n), there exists an approach region for which the associated maximal function is of weak-type, but the boundedness for the completed region is false, which is in contrast with the nontangential case. © 2004 American Mathematical Society.