Related papers: Reconstruction of a Space-Time Dependent Source in…
Recovering high-dimensional signals from corrupted measurements is a central challenge in inverse problems. Recent advances in generative diffusion models have shown remarkable empirical success in providing strong data-driven priors, but…
We study two new classes of inverse problems for a time-switched system in which a fractional wave equation (with Caputo derivative of order $\alpha \in (1,2)$) governs the dynamics on the interval $[0,a)$, and a fractional diffusion…
This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of…
In this paper we study the simultaneous reconstruction of two coefficients in a reaction-subdiffusion equation, namely a nonlinearity and a space dependent factor. The fact that these are coupled in a multiplicative matter makes the…
As the number of processor cores on supercomputers becomes larger and larger, algorithms with high degree of parallelism attract more attention. In this work, we propose a novel space-time coupled algorithm for solving an inverse problem…
In this paper, we investigate a nonlinear inverse problem aimed at recovering a coefficient $a(t, x)$, dependent on both time and a subset of spatial variables, in a diffusion equation \( u_t - \Delta_x u - u_{yy} +a(t, x) u = f(t,x,y) \),…
In this work, we investigate a numerical procedure for recovering a space-dependent diffusion coefficient in a (sub)diffusion model from the given terminal data, and provide a rigorous numerical analysis of the procedure. By exploiting…
This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which…
Existing approaches to diffusion-based inverse problem solvers frame the signal recovery task as a probabilistic sampling episode, where the solution is drawn from the desired posterior distribution. This framework suffers from several…
In this article, we consider the diffusion equation with multi-term time-fractional derivatives. We first derive that the solution is positive when the source term is nonpositive by a subordination principle for the solution. As an…
This paper deals with the distributed order time-fractional diffusion equations with non-homogeneous Dirichlet (Nuemann) boundary condition. We first prove the wellposedness of the weak solution to the initial boundary value problem for the…
In this work, we study an inverse problem of recovering information about the weight in distributed-order time-fractional diffusion from the observation at one single point on the domain boundary. In the absence of an explicit knowledge of…
We derive optimal $L^2$-error estimates for semilinear time-fractional subdiffusion problems involving Caputo derivatives in time of order $\alpha\in (0,1)$, for cases with smooth and nonsmooth initial data. A general framework is…
We consider the first and half order time fractional equation with the zero initial condition. We investigate an inverse source problem of determining the time-independent source factor by the data at an arbitrarily fixed time and we…
This paper is concerned with inverse source problems for the time-dependent Lam\'e system in an unbounded domain corresponding to the exterior of a bounded cavity or the full space $\R^3$. If the time and spatial variables of the source…
Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a…
We study the recovery of a spatially dependent source in a one-dimensional space-time fractional wave equation using boundary measurement data collected at a single endpoint. The main challenge arises from the fact that the eigenfunctions…
In this paper, we obtain the sharp uniqueness for an inverse $x$-source problem for a one-dimensional time-fractional diffusion equation with a zeroth-order term by the minimum possible lateral Cauchy data. The key ingredient is the unique…
This paper is concerned with the inverse acoustic scattering problems of reconstructing time-dependent multiple point sources and sources on a curve $L$ of the form $\lambda(t)\tau(x)\delta_L(x)$. A direct sampling method with a novel…
Diffusion models are powerful tools for sampling from high-dimensional distributions by progressively transforming pure noise into structured data through a denoising process. When equipped with a guidance mechanism, these models can also…