This work analyzes the convergence of probability densities solving uncertainty quantification problems for computational models where the mapping between input and output spaces is itself approximated. Specifically, we assume the exact mapping is replaced by a sequence of approximate maps that converges in L^p for some 1 ≤ p < ∞. To each approximate map, we then consider probability densities associated with push-forward and pullback measures solving forward and inverse problems, respectively. While the push-forward density is uniquely defined for each map, this is generally not the case for pullback densities since the maps are not typically bijective. A recently developed data-consistent inversion approach is therefore utilized to construct a specific sequence of pullback densities associated with the approximate maps. Convergence results for the push-forward and pullback densities are then proven under some additional assumptions. This significantly advances the results from a previous study that analyzed the convergence of such probability densities under the restrictive assumption that the approximate maps converged in an essentially uniform sense. Moreover, this greatly expands the realm of data-consistent inversion to problems requiring surrogate techniques that only guarantee L^p convergence for some 1 ≤ p < ∞. Numerical examples are also included along with numerical diagnostics of solutions and numerical verification of assumptions.

中文翻译：

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不确定性量化中正反问题的近似图和概率密度的 Lp 收敛性

这项工作分析了概率密度的收敛性，解决了计算模型的不确定性量化问题，其中输入和输出空间之间的映射本身是近似的。具体来说，我们假设精确映射被一系列近似映射代替，这些近似映射在L ^p中收敛于一些 1 ≤ p< ∞。对于每个近似图，我们然后考虑与分别解决正向和逆向问题的前推和回调措施相关的概率密度。虽然前推密度是为每个地图唯一定义的，但对于回调密度通常不是这种情况，因为地图通常不是双射的。因此，使用最近开发的数据一致反演方法来构建与近似图相关的特定回调密度序列。然后在一些额外的假设下证明了前推和回调密度的收敛结果。这显着推进了先前研究的结果，该研究分析了在近似映射以基本一致的方式收敛的限制性假设下这种概率密度的收敛性。而且，L ^p收敛于一些 1 ≤ p < ∞。还包括数值示例以及解决方案的数值诊断和假设的数值验证。